Sobolev metrics on shape space of surfaces
Let and be connected manifolds without boundary with , and let compact. Then shape space in this work is either the manifold of submanifolds of that are diffeomorphic to , or the orbifold of unparametrized immersions of in . We investigate the Sobolev Riemannian metrics on shape space: These are induced by metrics of the following form on the space of immersions:
where is some fixed metric on , is the induced metric on , are tangent vectors at to the space of embeddings or immersions, and is a positive, selfadjoint, bijective scalar pseudo differential operator of order depending smoothly on . We consider later specifically the operator , where is the Bochner-Laplacian on induced by the metric . For these metrics we compute the geodesic equations both on the space of immersions and on shape space, and also the conserved momenta arising from the obvious symmetries. We also show that the geodesic equation is well-posed on spaces of immersions, and also on diffeomorphism groups. We give examples of numerical solutions.
Key words and phrases:
2000 Mathematics Subject Classification:Primary 58B20, 58D15, 58E12
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Many procedures in science, engineering, and medicine produce data in the form of shapes. If one expects such a cloud to follow roughly a submanifold of a certain type, then it is of utmost importance to describe the space of all possible submanifolds of this type (we call it a shape space hereafter) and equip it with a significant metric which is able to distinguish special features of the shapes. Most of the metrics used today in data analysis and computer vision are of an ad-hoc and naive nature; one embeds shape space in some Hilbert space or Banach space and uses the distance therein. Shortest paths are then line segments, but they leave shape space quickly.
Riemannian metrics on shape space itself are a better solution. They lead to geodesics, to curvature and diffusion. Eventually one also needs statistics on shape space like means of clustered subsets of data (called Karcher means on Riemannian manifolds) and standard deviations. Here curvature will play an essential role; statistics on Riemannian manifolds seems hopelessly underdeveloped just now.
1.1. The shape spaces used in this work
Thus, initially, by a shape we mean a smoothly embedded surface in which is diffeomorphic to . The space of these shapes will be denoted and viewed as the quotient (see  for more details)
of the open subset of smooth embeddings of in , modulo the group of smooth diffeomorphisms of . It is natural to consider all possible immersions as well as embeddings, and thus introduce the larger space as the quotient of the space of smooth immersions by the group of diffeomorphisms of (which is, however, no longer a manifold, but an orbifold with finite isotropy groups, see ).
More generally, a shape will be an element of the Cauchy completion (i.e., the metric completion for the geodesic distance) of with respect to a suitably chosen Riemannian metric. This will allow for corners. In practice, discretization for numerical algorithms will hide the need to go to the Cauchy completion.
1.2. Where this work comes from
In , Michor and Mumford have investigated a variety of Riemannian metrics on the shape space
of unparametrized immersion of the circle into the plane. In [19, section 3.10] they found that the simplest such metric has vanishing geodesic distance; this is the metric induced by on :
In  they found that the vanishing geodesic distance phenomenon for the -metric occurs also in the more general shape space where is replaced by a compact manifold and Euclidean is replaced by Riemannian manifold ; it also occurs on the full diffeomorphism group , but not on the subgroup of volume preserving diffeomorphisms, where the geodesic equation for the -metric is the Euler equation of an incompressible fluid. In [20, sections 3, 4 and 5] three classes of metrics were investigated: Almost local metrics on planar curves, Sobolev metrics on planar curves, and metrics induced from Sobolev metrics on the diffeomorphism group of the plane. The results about almost local metrics from [20, section 3] were generalized by the authors to the case of surfaces in .
Now we take up the investigations from [20, section 4]. The immersion-Sobolev metric considered there is
The interesting special case and has been studied in [22, 25] and in  where an isometry to an infinite dimensional Grassmannian with the Fubini-Study metric was described. In this case, the metric reduces to:
The cases and have also been treated in , where estimates on the geodesic distance are proven and the metric completion of the space of curves is characterized.
In this work we generalize the immersion-Sobolev metrics from [20, section 4] to higher dimensions and to non-flat ambient space, namely to the shape space of surfaces of type in ; here is a compact orientable connected manifold of smaller dimension than , for example a sphere .
1.3. Riemannian metrics
The tangent space at an immersion consists of all vector fields along :
A Riemannian metric on is a family of positive definite inner products where and . Each metric is weak in the sense that , viewed as linear map from into its dual consisting of distributional sections of is injective. (But it can never be surjective.) We require that our metrics will be invariant under the action of , hence the quotient map dividing by this action will be a Riemannian submersion. This means that the tangent map of the quotient map is a metric quotient mapping between all tangent spaces. Thus we will get Riemannian metrics on . For any those vectors in which are -perpendicuar to the -orbit through are called horizontal (with respect to ). They form the -orthogonal space to the orbit. A priori we do not know that it is a complementary space. For the metrics considered in this work it will turn out to be a complement.
The simplest inner product on the tangent bundle to is
where is the Euclidean inner product on . Since the volume form reacts equivariantly to the action of the group , this metric is invariant, and the map to the quotient is a Riemannian submersion for this metric. The -horizontal vectors in are just those vector fields along which are pointwise -normal to ; we will call them normal fields.
All of the metrics we will look at will be of the form (see section 6):
where is a positive bijective operator depending smoothly on , which is selfadjoint unbounded in the Hilbert space with inner product . We will assume that is in addition equivariant with respect to reparametrizations, i.e.
The -horizontal vectors will be those such that is normal.
The tangent map of the quotient map is then an isometry when restricted to the horizontal spaces, just as in the finite dimensional situation. Riemannian submersions have a very nice effect on geodesics: the geodesics on the quotient space are exactly the images of the horizontal geodesics on the top space ; by a horizontal geodesic we mean a geodesic whose tangent lies in the horizontal bundle. The induced metric is invariant under the action of and therefore induces a unique metric on . See for example [2, section 1]. Later in section 8 we shall consider the special case .
1.4. Inner versus outer metrics
The metrics studied in this work are induced from on shape space. One might call them inner metrics since the differential operator governing the metric is defined intrinsic to . Intuitively, these metrics can be seen as describing some elastic or viscous behaviour of the shape itself.
In contrast to these metrics, there are also metrics induced from on shape space. (The widely used LDDMM algorithm uses such a metric.) The differential operator governing these metrics is defined on all of , even outside of the shape. Intuitively, these metrics can be seen as describing some elastic or viscous behaviour of the ambient space that gets deformed as the shape changes. One might call these metrics outer metrics.
1.5. Contributions of this work.
This work is the first to treat Sobolev inner metrics on spaces of immersed surfaces and on higher dimensional shape spaces.
It contains the first description of how the geodesic equation can be formulated in terms of gradients of the metric with respect to itself when the ambient space is not flat. To achieve this, a covariant derivative on some bundles over immersions is defined. This covariant derivative is induced from the Levi-Civita covariant derivative on ambient space.
The geodesic equation is formulated in terms of this covariant derivative. Well-posedness of the geodesic equation is shown under some regularity assumptions that are verified for Sobolev metrics. Well-posedness also follows for the geodesic equation on diffeomorphism groups, where this result has not yet been obtained in that full generality.
To derive the geodesic equation, a variational formula for the Laplacian operator is developed. The variation is taken with respect to the metric on the manifold where the Laplacian is defined. This metric in turn depends on the immersion inducing it.
It is shown that Sobolev inner metrics separate points in shape space when the order of the differential operator governing the metric is high enough. (The metric needs to be as least as strong as the -metric.) Thus Sobolev inner metrics overcome the degeneracy of the -metric.
The path-length distance of Sobolev inner metrics is compared to the Fréchet distance. It would be desirable to bound Féchet distance by some Sobolev distance. This however remains an open problem.
Finally it is demonstrated in some examples that the geodesic equation for the -metric on shape space of surfaces in can be solved numerically.
2. Content of this work
This work progresses from a very general setting to a specific one in three steps. In the beginning, a framework for general inner metrics is developed. Then the general concepts carry over to more and more specific inner metrics.
First, shape space is endowed with a general inner metric, i.e with a metric that is induced from a metric on the space of immersions, but that is unspecified otherwise. It is shown how various versions of the geodesic equation can be expressed using gradients of the metric with respect to itself and how conserved quantities arise from symmetries. (This is section 4.)
Then it is assumed that the inner metric is defined via an elliptic pseudo-differential operator. Such a metric will be called a Sobolev-type metric. The geodesic equation is formulated in terms of the operator, and existence of horizontal paths of immersions within each equivalence class of paths is proven. (This is section 6.) Then estimates on the path-length distance are derived. Most importantly it is shown that when the operator involves high enough powers of the Laplacian, then the metric does not have the degeneracy of the -metric. (This is section 7.)
Motivated by the previous results it is assumed that the elliptic pseudo-differential operator is given by the Laplacian and powers of it. Again, the geodesic equation is derived. The formulas that are obtained are ready to be implemented numerically. (This is section 8.)
The remaining sections cover the following material:
Section 3 treats some differential geometry of surfaces that is needed in this work. It is also a good reference for the notation that is used. The biggest emphasis is on a rigorous treatment of the covariant derivative. Some material like the adjoint covariant derivative is not found in standard text books.
Section 5 contains formulas for the variation of the metric, volume form, covariant derivative and Laplacian with respect to the immersion inducing them. These formulas are used extensively later.
Section 9 covers the special case of flat ambient space. The geodesic equation is simplified and conserved momenta for the Euclidean motion group are calculated. Sobolev-type metrics are compared to the Fréchet metric which is available in flat ambient space.
Section 10 treats diffeomorphism groups of compact manifolds as a special case of the theory that has been developed so far.
In section 11 it is shown in some examples that the geodesic equation on shape space can be solved numerically.
3. Differential geometry of surfaces and notation
In this section the differential geometric tools that are needed to deal with immersed surfaces are presented and developed. The most important point is a rigorous treatment of the covariant derivative and related concepts.
3.1. Basic assumptions and conventions
It is always assumed that and are connected manifolds of finite dimensions and , respectively. Furthermore it is assumed that is compact, and that is endowed with a Riemannian metric .
In this work, immersions of into will be treated, i.e. smooth functions with injective tangent mapping at every point. The set of all such immersions will be denoted by . It is clear that only the case is of interest since otherwise would be empty.
Immersions or paths of immersions are usually denoted by . Vector fields on or tangent vectors with foot point , i.e., vector fields along , will be called , for example. Subscripts like denote differentiation with respect to the indicated variable, but subscripts are also used to indicate the foot point of a tensor field.
3.2. Tensor bundles and tensor fields
The tensor bundles
will be used. Here denotes the bundle of -tensors on , i.e.
and is the pullback of the bundle via , see [17, section 17.5]. A tensor field is a section of a tensor bundle. Generally, when is a bundle, the space of its sections will be denoted by .
To clarify the notation that will be used later, some examples of tensor bundles and tensor fields are given now. and are the bundles of symmetric and alternating -tensors, respectively. is the space of differential forms, is the space of vector fields, and
is the space of vector fields along .
3.3. Metric on tensor spaces
Let denote a fixed Riemannian metric on . The metric induced on by is the pullback metric
where are vector fields on . The dependence of on the immersion should be kept in mind. Let
can be extended to the cotangent bundle by setting
for . The product metric
extends to all tensor spaces , and yields a metric on .
The trace contracts pairs of vectors and co-vectors in a tensor product:
A special case of this is the operator inserting a vector into a co-vector or into a covariant factor of a tensor product. The inverse of the metric can be used to define a trace
contracting pairs of co-vecors. Note that depends on the metric whereas does not. The following lemma will be useful in many calculations:
(In the expression under the trace, and are seen as maps .)
Express everything in a local coordinate system of .
Note that only the symmetry of has been used. ∎
3.5. Volume density
Let be the density bundle over , see [17, section 10.2]. The volume density on induced by is
The volume of the immersion is given by
The integral is well-defined since is compact. If is oriented the volume density may be identified with a differential form.
3.6. Metric on tensor fields
A metric on a space of tensor fields is defined by integrating the appropriate metric on the tensor space with respect to the volume density:
for , and
for , . The integrals are well-defined because is compact.
3.7. Covariant derivative
Covariant derivatives on vector bundles as explained in [17, sections 19.12, 22.9] will be used. Let be the Levi-Civita covariant derivatives on and , respectively. For any manifold and vector field on , one has
Usually the symbol will be used for all covariant derivatives. It should be kept in mind that depends on the metric and therefore also on the immersion . The following properties hold [17, section 22.9]:
respects base points, i.e. , where is the projection of the tangent space onto the base manifold.
is -linear in . So for a tangent vector , makes sense and equals .
is -linear in .
for , the derivation property of .
For any manifold and smooth mapping and one has . If and are -related, then .
The two covariant derivatives and can be combined to yield a covariant derivative acting on by additionally requiring the following properties [17, section 22.12]:
respects the spaces .
, a derivation with respect to the tensor product.
commutes with any kind of contraction (see [17, section 8.18]). A special case of this is
Property (1) is important because it implies that respects spaces of sections of bundles. For example, for and , one gets
3.8. Swapping covariant derivatives
Some formulas allowing to swap covariant derivatives will be used repeatedly. Let be an immersion, a vector field along and vector fields on . Since is torsion-free, one has [17, section 22.10]:
Furthermore one has [17, section 24.5]:
where is the Riemann curvature tensor of .
These formulas also hold when is a path of immersions, is a vector field along and the vector fields are vector fields on . A case of special importance is when one of the vector fields is and the other , where is a vector field on . Since the Lie bracket of these vector fields vanishes, (1) and (2) yield
3.9. Second and higher covariant derivatives
When the covariant derivative is seen as a mapping
then the second covariant derivative is simply . Since the covariant derivative commutes with contractions, can be expressed as
Higher covariant derivates are defined accordingly as , .
3.10. Adjoint of the covariant derivative
The covariant derivative
admits an adjoint
with respect to the metric , i.e.:
In the same way, can be defined when is acting on . In either case it is given by
where the trace is contracting the first two tensor slots of . This formula will be proven now:
The definition of the Laplacian used in this work is the Bochner-Laplacian. It can act on all tensor fields and is defined as
3.12. Normal bundle
The normal bundle of an immersion is a sub-bundle of whose fibers consist of all vectors that are orthogonal to the image of :
If then the fibers of the normal bundle are but the zero vector. Any vector field along can be decomposed uniquely into parts tangential and normal to as
where is a vector field on and is a section of the normal bundle .
3.13. Second fundamental form and Weingarten mapping
Let and be vector fields on . Then the covariant derivative splits into tangential and a normal parts as
is the second fundamental form of . It is a symmetric bilinear form with values in the normal bundle of . When is seen as a section of one has since
The trace of is the vector valued mean curvature .
4. Shape space
Briefly said, in this work the word shape means an unparametrized surface. (The term surface is used regardless of whether it has dimension two or not.) This section is about the infinite dimensional space of all shapes. First some spaces of parametrized and unparametrized surfaces are described, and it is shown how to define Riemannian metrics on them. The geodesic equation and conserved quantities arising from symmetries are derived.
The agenda that is set out in this section will be pursued in section 6 when the arbitrary metric is replaced by a Sobolev-type metric involving a pseudo-differential operator and later in section 8 when the pseudo-differential operator is replaced by an operator involving powers of the Laplacian.
4.1. Riemannian metrics on immersions
The space of smooth immersions of the manifold into the manifold will be denoted by or briefly . It is a smooth Fréchet manifold containing the space of embeddings of into as an open subset [14, theorem 44.1]. Consider the following natural bundles of -multilinear mappings:
These bundles are isomorphic to the bundles
where denotes the -completed bornological tensor product of locally convex vector spaces [14, section 5.7, section 4.29]. Note that is not isomorphic to since the latter bundle corresponds to multilinear mappings with finite rank.
It is worth to write down more explicitly what some of these bundles of multilinear mappings are. The tangent space to is given by
Thus is the space of vector fields along the immersion . Now the cotangent space to will be described. The symbol means that the tensor product is taken over the algebra .
The bundle is of interest for the definition of a Riemannian metric on . (The subscripts and indicate symmetric and alternating multilinear maps, respectively.) Letting denotes the symmetric tensor product and the -completed bornological symmetric tensor product, one has
A Riemannian metric on is a section of the bundle such that at every , is a symmetric positive definite bilinear mapping
Each metric is weak in the sense that , seen as a mapping
is injective. (But it can never be surjective.)
4.2. Covariant derivative on immersions
The covariant derivative defined in section 3.7 induces a covariant derivative over immersions as follows. Let be a smooth manifold. Then one identifies
As described in section 3.7 one has the covariant derivative
Thus one can define
It is helpful to point out some special cases of how this construction can be used. The case will be important to formulate the geodesic equation. The expression that will be of interest in the formulation of the geodesic equation is , which is well-defined when is a path of immersions and is its velocity.
Another case of interest is . Let . Then the covariant derivative is well-defined and tensorial in . Requiring to respect the grading of the spaces of multilinear maps, to act as a derivation on products and to commute with compositions of multilinear maps, one obtains as in section 3.7 a covariant derivative acting on all mappings into the natural bundles of multilinear mappings over . In particular, and are well-defined for
by the usual formulas
4.3. Metric gradients
The metric gradients are uniquely defined by the equation
where are vector fields on and the covariant derivative of the metric tensor is defined as in the previous section. (This is a generalization of the definition used in  that allows for a curved ambient space .)
Nevertheless it will be assumed for now that the metric gradients exist.
4.4. Geodesic equation on immersions
Given as defined in the previous section and as defined in section 4.2, the geodesic equation reads as
This is the same result as in [20, section 2.4], but in a more general setting.
Let be a one-parameter family of curves of immersions with fixed endpoints. The variational parameter will be denoted by and the time-parameter by . In the following calculation, let denote composed with , i.e.
Remember that the covariant derivative on that has been introduced in section 4.2 is torsion-free so that one has
Thus the first variation of the energy of the curves is
If is energy-minimizing, then one has at that
4.5. Geodesic equation on immersions in terms of the momentum
In the previous section the geodesic equation for the velocity has been derived. In many applications it is more convenient to formulate the geodesic equation as an equation for the momentum . is an element of the smooth cotangent bundle, also called smooth dual, which is given by