Metamorphoses Through Lie Group Action

© 2005 SFoCMDOI: 10.1007/s10208-004-0128-z

Found. Comput. Math. 173–198 (2005)

The Journal of the Society for the Foundations of Computational Mathematics

FOUNDATIONS OFCOMPUTATIONALMATHEMATICS

Metamorphoses Through Lie Group Action

Alain Trouve1 and Laurent Younes2

1CMLAENS de Cachan61, avenue du President Wilson94235 Cachan CEDEX, [email protected]

2Department of Applied Mathematics and Statistics and Center for Imaging ScienceThe Johns Hopkins University3400 N-Charles StreetBaltimore, MD 21218-2686, [email protected]

Abstract. We formally analyze a computational problem which has importantapplications in image understanding and shape analysis. The problem can be sum-marized as follows. Starting from a group action on a Riemannian manifold M , weintroduce a modification of the metric by partly expressing displacements on Mas an effect of the action of some group element. The study of this new structurerelates to evolutions on M under the combined effect of the action and of resid-ual displacements, called metamorphoses. This can and has been applied to imageprocessing problems, providing in particular diffeomorphic matching algorithms forpattern recognition.

Contents1. Notation 1742. A New Metric on M 178

2.1. General Form 1782.2. Examples 179

Date received: March 9, 2004. Final version received: August 20, 2004. Date accepted: September 23,2004. Communicated by Peter Olver. Online publication February 11, 2005.AMS classification: 68T45, 53B21, 58E10, 58E40.Key words and phrases: Groups of diffeomorphisms, Infinite-dimensional shape spaces, Geodesics,Shape recognition, Image registration.

174 A. Trouve and L. Younes

3. Geodesic Equations 1823.1. General Form 1823.2. Examples 184

4. Evolution Equations 1884.1. Evolution in z 1884.2. Evolution of the Velocity 1904.3. Examples 191

5. A Last Example: Deforming Geometric Curves 192Appendix A. Existence of Solutions of dgt/dt = de Rgtvt 195Appendix B. Proof of Proposition 2 197References 197

1. Notation

Many situations in image analysis model a set of visual objects, like images,shapes, patterns of points, which can be affected by “deformations.” Followingthe seminal approach of Grenander’s deformable template theory [7], [8], we havedesigned during the last decade, theoretical and numerical methods to analyze theaction of diffeomorphisms on geometric structures, like landmarks, shapes, andimages [20], [25], [26], [21], [16], [2], [22], [15]. The basic assumption of thetheory of deformable templates, is that observable objects are assumed to belongto the orbit of a fixed template, under the action of a group which would herebe a group of diffeomorphisms. Here, following [16], [2], [22], the point of viewis slightly different, because the base-point of the deformation (the template) isallowed to vary during a process called a metamorphosis (this term is borrowedfrom computer graphics where it indicates a “morphing” process, generally on faceimages, but we use it here in a fairly larger framework). Metamorphoses thereforeare deformable templates with varying templates. In this paper, we study theirmetric and geometric properties from an abstract, general point of view, which hasthe advantage of embedding several apparently distinct situations into a unifyingframework.

Here are the basic assumptions. The deformations belong to a Lie group G withLie algebra denoted by g, acting on a Riemannian manifold M which containsvisual objects. We assume that g is a Hilbert space with norm | |g; the metric onM at a given point m ∈ M is denoted by 〈·, ·〉m and the corresponding norm | · |m .

Notation 1. Assuming that the action of G ×M → M is C1, for any g ∈ G, themapping A(g) : m → gm is a diffeomorphism on M . Then, A : G → Diff(M)is an homomorphism of groups, so that we can identify G with a subgroup ofDiff(M). We will use the notation A(g) = g so that g(m) = gm.

Moreover, since for m fixed, the mapping Rm : g → gm is differentiable withrespect to g at the identity element e of G, then for any v ∈ g, de Rm(v) ∈ Tm M andm → de Rm(v) ∈ χ(M), the set of continuous vector fields on M . Hence, g can be

Metamorphoses Through Lie Group Action 175

identified with a subspace of χ(M) equipped with a Hilbertian metric inheritedfrom g. Using this identification, we will denote for any v ∈ g, v(m) .= de Rm(v).It will sometimes be convenient to also use the notation δm instead of de Rm so that

δmv = v(m).

If (gt , t ∈ [0, 1]) is a differentiable curve on G, and Rg is the right-multiplicat-ion in G (Rg(g′) = g′g), we define its velocity vt (which is a curve on g) by therelation

dgt

dt= de Rgtvt , (1)

or, using the previous identification,

dgt

dt= vt ◦ gt . (2)

A metamorphosis on M is a pair of curves (gt , µt ), respectively, on G and M ,with g0 = e. Its image is the curve mt on M defined by mt = gt (µt ). We callgt the deformation part of the metamorphosis, and µt the template evolution part.When µt is constant, we say that the metamorphosis is a pure deformation. Inthe rest of the paper, we will study how metamorphoses can be used to define anew metric on M , and obtain the corresponding geodesic equations. These willbe essentially abstract developments, but they will be illustrated by the followingexamples, which correspond to useful situations in image analysis.

Example 1 (Landmark Matching with Affine Robustness). We here considerlandmarks which are collections of labeled points in Rk . Elements of M are N -tuples of points, so that M can be identified to (Rk)N . Let the Riemannian metricon M be the usual Euclidean metric, so that, if m = (m1, . . . ,m N ) ∈ M andη = (η1, . . . , ηN ) ∈ Tm M ∼ (Rk)N , we have

|η|2m =N∑

i=1

|ηi |2Rk .

The group G is the affine group on Rk : elements of G are pairs (B, T ) whereB is an invertible k × k matrix and T ∈ Rk , with the (semidirect) product(B, T )(B ′, T ′) = (B B ′, BT ′+T ). The action of G on M is defined for g = (B, T )and m = (m1, . . . ,m N ) by

g(m) = (Bm1 + T, . . . , Bm N + T ).

The Lie algebra on G is equal toMk(R)×Rk . There are many possible definitionsfor the norm on g, and we shall choose the simplest one, for which, letting v =(β, τ ) ∈ g,

|v|2g = trace(tββ)+ ‖τ‖2Rk .


The velocity (βt , τt ) of a curve (Bt , Tt ) on G is characterized by the system

d Bt

dt= βt Bt ,

dTt

dt= βt Bt + τt .

According to Notation 1, G is identified with a subgroup of Diff(M), i.e.,with a subgroup of diffeomorphisms on the space of N -tuples of points in Rk .More precisely, G is identified with a subgroup of Aff(M), the group of affinetransformations on M .

Although we shall follow this example with G being the complete group of affinetransformations on Rk , reducing it to a subgroup is a straightforward operation,which simply requires restricting the velocity vectors to those belonging to theLie algebra of G. The group of similitudes (translations, rotations, and scaling) isof special interest, because it corresponds to a large literature on shape theory, inwhich N -shapes are defined as collections of N labeled points modulo the actionof G. This theory, initiated in [10], and presented in textbooks such as [18], [3],studies what remains from the collections of points once similitudes are removed,which is a different point of view from our approach here, since we are using affinedeformations as part of the modeling process, and not as a nuisance factor. Thesame remark applies to [11], in which the more ambitious issue of consideringclosed plane curves modulo similitudes and change of parameter is addressed.

Example 2 (Deformable Landmarks on a Manifold [9], [2], [23]). We considerhere general deformations acting on collections of points (y1, . . . , yN ), each yi

being assumed to belong to an open and bounded subset of a smooth Riemannianmanifold M0 (the most common application being M0 = Rk , but the general casealso has some interest [6]). We therefore set M = N and, as in Example 1,the metric on M is the product metric, setting, for m = (y1, . . . , yN ) ∈ M andη = (η1, . . . , ηN ) ∈ Tm M ∼∏N

i=1 Tyi M0,

|η|2m =N∑

i=1

|ηi |2yi .

The group G we consider here is a group of diffeomorphisms of , with actiong(y1, . . . , yN ) = (g(y1), . . . , g(yN )). Considering such an infinite-dimensionalgroup brings additional difficulties, which will not be addressed in the forthcom-ing abstract derivations. The underlying construction we are assuming here hasbeen proposed in [19] and [4]. The problem is that it is not possible to obtainall together the Lie group properties and the (Riemannian) metric property whichare immediate in finite dimensions. Lie groups of diffeomorphisms on manifoldshas been the subject of intensive studies in the framework of global analysis (e.g.,[17], [5], [12]). Strictly enforcing the Lie group properties comes at the cost of


only considering smooth (C∞) diffeomorphisms, and of the absence of any nicemetric properties (like completeness and existence and geodesics) for the kind ofRiemannian metric one is likely to consider. This metric aspect is more importantfor applications, since one ends up solving variational problems, for which it isimportant to know that solutions exist. Trying to build a group of diffeomorphismswhich has most of the properties of a complete Riemannian manifold now im-plies relaxing some of the Lie group conditions. This is the construction we nowsummarize.

This starts with selecting g, as a Hilbert space (not necessarily a Lie algebra),assuming that it is continuously embedded into X 1( ), the Banach space of con-tinuously differentiable vector fields on . We also assume that elements in g havenull boundary conditions, in the sense that X∞c ( ) (the space of C∞ vector fieldswith compact support on ) is dense in g. A generic way to build such a space isbased on the Friedrich extension of an admissible operator [27]; a vector field on is square integrable if

∫

‖v(y)‖2y dν0(y) <∞,

where ν0 is the volume form on M0. We denote by L2( ) the set of square integrablevector fields on . We say that a symmetric operator L : X∞c ( ) → L2( ) isadmissible if, for all y ∈ , Lv(y) ∈ Ty M and, for some constant K ,

(Lv, v) :=∫

⟨Lv(y), v(y)

⟩y

dν0(y) ≥ K max{‖v(y)‖2y + ‖∇yv‖2

y, x ∈ }.

Under this assumption, X∞c ( ) and L can be completed into a Hilbert spaceg ⊂ X 1( ) and an operator L : g→ g∗ such that

‖v‖2g = (Lv, v).

If vt ∈ L1([0, 1], g), the differential equation dyt/t = vt (yt ), y0 = x hassolutions over [0, 1]. For t ∈ [0, 1], this solution is denoted gvt (x), and the setG = {gv1 , v ∈ L1([0, 1], g)} is a group (with product gh = g ◦ h), which will behereafter our group of diffeomorphisms of . Note that, in this setting (1) preciselygives

dgt

dt= vt ◦ gt .

Here, note that for any g ∈ G ⊂ Diff( ), g belongs to Diff( M).

Example 3 (Deformable Images [16], [22]). We consider, here again, a groupG of diffeomorphisms of , restricting, for simplicity, to the case M0 = Rk , andM is a set of square integrable functions m : → R. We let G act on M bygm = m ◦ g−1, and use the L2 norm on the Hilbert space L2( ,Rk). FollowingNotation 1, m → gm defines an element g ∈ L(L2), i.e., the linear group on


L2. Note that in Example 1, g also belongs to a linear group (but on a finite-dimensional vector space), and we shall see that both examples share importantstructural properties.

2. A New Metric on M

2.1. General Form

We return to the abstract setting to show how metamorphoses can be used to placea new Riemannian metric on M , which will take the action of G into account. Asimple motivation for this can be taken from the following situation in Example 1:if m = (y1, . . . , yN ) belongs to M , and (T 1, . . . , T N ) are randomly chosen unitvectors inRk , m will be at an equal Euclidean distance, N , from the configurations(y1 + T 1, . . . , yN + T 1) and (y1 + T 1, . . . , yN + T N ). However, in situationswhen m represents a shape, it is natural to consider that the first configuration,which is a translation of m, should be much closer to m than the second one. Inother words, we want to assign a different cost to the variation in cases when itcan be, at least partially, explained by the action of G.

Metamorphoses, by the evolution of their image, provide a convenient repre-sentation of combinations of a group action and of a variation on M . Indeed, if((gt , µt ), t ∈ [0, 1]) is given (with gt ∈ G and µt ∈ M) and mt = gt (µt ) is itsimage, a straightforward computation yields, vt being the velocity of gt (definedin equation (1)):

dmt

dt= dµt gt

(dµt

dt

)+ dgt

dt(µt )

= dµt gt

(dµt

dt

)+ vt (mt ). (3)

In particular, when t = 0,

dmt

dt

∣∣∣∣t=0

= dµt

dt

∣∣∣∣t=0

+ v0(m0). (4)

This expression provides the decomposition of a generic element η ∈ Tm M interms of an infinitesimal metamorphosis, represented by an element of g× Tm M .Indeed, for m ∈ M , introduce the map

�m : g× Tm M → Tm M,(v, δ) �→ v(m)+ δ,

so that (4) can be written

dmt

dt

∣∣∣∣t=0

= �m

(dµt

dt

∣∣∣∣t=0

, v0

).


Fix σ 2 > 0. Since �m is onto (�m(0, δ) = δ), we can define a new metric on Mby

‖η‖2m = inf

{|v|2g +

1

σ 2|δ|2m : η = �m(v, δ)

}

(note that we are using double lines instead of simple ones to distinguish betweenthe new metric and the original one).

Define Vm = �−1m (0). This is a linear subspace of g × Tm M and ‖η‖2

m is thenorm of the linear projection of (0, η) on Vm , for the Hilbert structure on g×Tm Mdefined by

‖(v, δ)‖2e,m = |v|2g +

1

σ 2|δ|2m .

Thus ‖ · ‖m = ‖πVm (0, η)‖e,m is associated to an inner product. Since it is aprojection on a close subspace, the infimum is attained and, by definition, cannotvanish unless η = 0. This therefore provides a new Riemannian metric on M .

With this metric, the energy of a curve is

E(mt ) =∫ 1

0

∥∥∥∥dmt

dt

∥∥∥∥2

mt

dt = inf

(∫ 1

0|vt |2gdt + 1

σ 2

∫ 1

0

∣∣∣∣dmt

dt−vt (mt )

∣∣∣∣2

mt

dt

)(5)

the infimum being over all curves t �→ vt on g.The distance between two elements m and m ′ in M can therefore be computed

by minimizing

U (vt ,mt ) =∫ 1

0|vt |2gdt + 1

σ 2

∫ 1

0

∣∣∣∣dmt

dt− vt (mt )

∣∣∣∣2

mt

dt

over all curves {(vt ,mt ), t ∈ [0, 1]} on g×M , with boundary conditions m0 = mand m1 = m ′.

Introducing gt , the solution1 of (1), and the metamorphosis (gt , µt ), with µt =g−1

t (mt ), equation (3) provides a second expression for the energy:

E(mt ) = inf

(∫ 1

0|vt |2gdt + 1

σ 2

∫ 1

0

∣∣∣∣dµt gt

(dµt

dt

)∣∣∣∣2

mt

dt

). (6)

2.2. Examples

We now specialize this computation to the three examples we are considering. Thisessentially means computing δm and v �→ v in these situations.

1 The existence of this solution when v ∈ L2([0, 1],g) is validated in Appendix A.


2.2.1. Example 1. If m = (m1, . . . ,m N ), differentiating

(B, T ) �→ (Bm1 + T, . . . , Bm N + T )

at (B, T ) = (Id, 0) gives, for v = (β, τ ) ∈ g,

v(m) = (βm1 + τ, . . . , βm N + τ), (7)

so that, letting mt = (m1t , . . . ,m N

t ),

U (vt ,mt ) =∫ 1

0(trace(tβtβt )+ |τt |2Rk ) dt + 1

σ 2

N∑i=1

∫ 1

0

∣∣∣∣dmit

dt− βmi

t − τ∣∣∣∣2

Rk

dt.

Similarly, for g = (B, T ) ∈ G, differentiating µ �→ g(µ) with respect toµ = (µ1, . . . , µN ) yields

dµg(η) = (Bη1, . . . , BηN ),

where η = (η1, . . . , ηN ), so that, the alternate form of the energy, (6), is

U (gt , µt ) =∫ 1

0(trace(tβtβt )+ |τt |2Rk ) dt + 1

σ 2

N∑i=1

∫ 1

0

∣∣∣∣B−1t

dµit

dt

∣∣∣∣2

Rk

dt.

2.2.2. Example 2. Since g(m) = (g(m1), . . . , g(m N )), we have v(m) =(v(m1), . . . , v(m N )) for v ∈ g, so that

U (vt ,mt ) =∫ 1

0|vt |2gdt + 1

σ 2

N∑i=1

∫ 1

0

∣∣∣∣dmit

dt− v(mi

t )

∣∣∣∣2

mi

dt,

and (6) may also be computed from the expression dµg(η) = (dµ1 g(η1), . . . ,

dµN g(ηN )) for η = (η1, · · · , ηN ). The result of minimizing the geodesic en-ergy between two sets of one hundred two-dimensional landmarks is provided inFigure 1.

2.2.3. Example 3. This example is more problematic: m being a function on ,we have for g ∈ G, g(m) = m ◦ g−1; formal differentiation in the neighborhoodof the identity would yield v(m) = −〈∇m, v〉Rk for v ∈ g. For m ∈ L2( ,R),this is not necessarily defined in the strong sense, and can be given a generalizedmeaning, η = v(m) being identified to the linear form, defined, over all smoothfunctions ϕ on with compact support, by

ϕ �→∫

m(x) divx (ϕv) dx .

This coincides with − ∫ 〈∇x m, v(x)〉Rkϕ(x) dx when m is C1. Note that, for the

energy U (vt ,mt ) to be finite, we still need the sum dmt/t − vt (m) to be squareintegrable as a function of two variables t and x .


Fig. 1. An example of the geodesic between landmarks. The first row provides the initial requirements(100 landmarks placed on a horizontal ellipse to be displaced to a vertical ellipse) followed, in thesecond image, by the geodesic trajectories. The next two rows provide the intermediate positions ofthe landmarks (red dots) and the effect of the underlying diffeomorphism on a grid.

The second form of the energy is simpler, since dµg(η) = η ◦ g−1, so that (6) is

∫ 1

0|vt |2g dt + 1

σ 2

∫ 1

0

∣∣∣∣dµt

dt◦ g−1

t

∣∣∣∣2

dt,

which has a meaning whenever dµt/dt is square integrable. However, it is nowthe boundary condition µ1 ◦ g−1

1 = m ′ which may be hard to fulfill if m and m ′

are simply assumed to belong to L2. This implies that mt and vt can no longer betreated as independent variables in the variational analysis of this problem, whichbecomes, because of this, much more intricate.

To simplify computations, we will restrict ourselves, in the remainder of thispaper, to the case when the compared images are smooth (at least C1). Under theseconditions, it may be assumed that mt is C1 at all times, because it can be shown tobe so for the optimal path. More general situations are taken into account in [22].An example of a geodesic obtained by minimizing this energy between two givenimages is given in Figure 2.


Fig. 2. An example of a geodesic between images (original images taken from the Olivetti facedatabase). The three intermediate images are generated by the optimization algorithm.

3. Geodesic Equations

3.1. General Form

In this section, we compute the Euler–Lagrange equations for the energy

U (vt ,mt ) =∫ 1

0|vt |2g dt + 1

σ 2

∫ 1

0

∣∣∣∣dmt

dt− vt (mt )

∣∣∣∣2

mt

dt.

These equations are important because, on the one hand, they characterizegeodesics on M for the new metric and, on the other hand, they also provide gradi-ent increments which can be used for the numerical computation of the geodesics.

To obtain the first equation, start with a variation vt �→ vt + εwt in g. Letf (ε) = U (vt + εwt ,mt ), so that

f ′(0) = 2∫ 1

0〈vt , wt 〉g dt − 2

∫ 1

0〈zt ,wt (mt )〉mt

dt,

where we have introduced the notation

σ 2zt = dmt

dt− vt (mt ) = dµt gt

(dµt

dt

). (8)

Hence, we get, for any w ∈ g,

〈vt , w〉g = 〈zt ,w(mt )〉mt. (9)

When A is a continuous operator between two Hilbert spacesH andH′, the adjointof A, denoted A† : H′ → H, is uniquely defined by

〈h, Av〉H′ = 〈A†h, v〉H.Using this notation (with H = g and H′ = Tm M), and recalling the notationδm : g → Tm M for the continous linear operator defined by δm(v) = v(m), theidentity f ′(0) = 0 yields our first Euler–Lagrange equation:

vt − δ†mt(zt ) = 0 (10)


which gives (from the expression of zt )

dmt

dt= vt (mt )+ σ 2zt = (δmδ

†m + σ 2Id)zt . (11)

Let Hσm : Tm M → Tm M be equal to (δmδ

†m + σ 2Id). This is a symmetric and

positive operator. If Tm M has finite dimensions, or if δmδ†m is compact, Hσ

m isinvertible and equation (11) implies, as shown below,

E(mt ) =∫ 1

0

⟨dmt

dt, (Hσ

m )−1 dmt

dt

⟩mt

dt. (12)

Indeed, since (1/σ 2)|dmt/dt − vt (mt )|2mt= σ 2〈zt , zt 〉mt

, we deduce from (9) and(11) that

|v|2g +1

σ 2

∣∣∣∣dmt

dt− vt (mt )

∣∣∣∣2

mt

= 〈zt , vt (m)+ σ 2zt 〉mt

=⟨zt ,

dmt

dt

⟩mt

=⟨

dmt

dt, (Hσ

m )−1 dmt

dt

⟩mt

.

Equality (12) provides an intrinsic expression for the new metric. In other words,

‖η‖2m = 〈η, (Hσ

m )−1η〉m .

We pass to the variation with respect to the curve m on M . For this we considera variation mt,ε with mt,0 = mt , m0,ε = m0, and m1,ε = m1 and let f (ε) =(σ 2/2)U (v,mt,ε). We have, letting ηt = dmt,ε/dε,

f ′(0) =∫ 1

0

⟨zt , ∇∂m/∂ε

(∂m

∂t− vt

)⟩mt

dt

=∫ 1

0

⟨zt , ∇∂m/∂t

∂m

∂ε− ∇∂m/∂εvt

⟩mt

dt

= −∫ 1

0〈∇∂m/∂t zt , ηt 〉mt

dt −∫ 1

0〈zt ,∇ηt vt 〉mt

dt.

This implies that, for all t and all η ∈ Tmt M ,

〈∇∂m/∂t zt , η〉mt= −〈zt ,∇ηvt 〉mt

. (13)

If we define ∇†ξ by 〈∇†

ξ χ, η〉m = 〈ξ,∇ηχ〉m (ξ, η, χ being vector fields on M)this may be written

∇∂m/∂t zt + ∇†zt

vt = 0. (14)


The following proposition summarizes the previous results:

Proposition 1. The geodesic equations for the new Riemannian structure on Mare

dmt

dt= vt (mt ) = dµt gt

(dµt

dt

)+ σ 2zt ,

∇∂m/∂t zt + ∇†zt

vt = 0,vt = δ†

mt(zt ).

(15)

Note also that the previous computation provides the gradient of the energy Uwith respect to v and m, which is given by

{Gradv U = 2(vt − δ†

mt(zt )),

Gradm U = −2(∇∂m/∂t zt + ∇†zt

vt ).

3.2. Examples

3.2.1. Example 1. Letting vt = (βt , τt ), mt = (m1t , . . . ,m N

t ), we have zt =(z1

t , . . . , zNt ) with

σ 2zit =

dmit

dt− βmi

t − τ.

To clarify (10) we compute, for ξ = (ξ 1, . . . , ξ N ) and m = (m1, . . . ,m N ), theelement δ†

mξ = (β, τ ) ∈ g. This is characterized by the identity: for (β ′, τ ′) ∈ g:

trace(tββ ′)+ 〈τ, τ ′〉Rk =N∑

i=1

〈ξ i , β ′mi + τ ′〉Rk

which immediately provides

β =N∑

i=1

ξ i tmi , τ =

N∑i=1

ξ i ,

so that equation (10) is

βt =N∑

i=1

zittmi

t ,

τt =N∑

i=1

zit .

Moreover, we have

δmδ†mξ = (βm1 + τ, . . . , βm N + τ)


and

βmi + τ =N∑

j=1

(1+ 〈mi ,m j 〉Rk )ξ j ,

so that the matrix Hσm can be identified to the Nk × Nk block matrix, for which

the (i, j) k × k block is (1 + 〈mi ,m j 〉Rk )Id if i �= j and (1 + σ 2 + |mi |2Rk ).Id if

i = j .We now compute equation (14). Since M is Euclidean, we have ∇∂m/∂t zt =

d Z/dt and ∇ηχ = dχη. For v = (β, τ ), and χ = v, as given by equation (7), thisyields

∇ηv = (βη1, . . . , βηn),

so that

〈z,∇ηv〉m =N∑

i=1

〈zi , βηi 〉Rk =

N∑i=1

〈tβzi , ηi 〉Rk .

Thus, equation (14) gives, in this case,

dzit

dt+ tβt z

it = 0.

3.2.2. Example 2. G now is a group of diffeomorphisms on ⊂ M0. We have

σ 2zit =

dmit

dt− vt (m

it ).

To compute δ†mt(zt ), which is characterized by

〈z, δm(v)〉m = 〈v, δ†m(z)〉g,

we need to introduce the reproducing kernel, K , of g. It associates to (x, y) ∈ 2

a linear operator, denoted K (x, y), from Ty M0 to Tx M0 which satisfies:

• for all x ∈ and for η ∈ Tx M0, the vector field y �→ K (y, x)η, which ishereafter denoted Kxη, belongs to g;• for all x ∈ and η ∈ Tx ,

〈Kxη, v〉g = 〈η, v(x)〉x .Note that the kernel has the property K (x, y)† = K (y, x), i.e.,

〈K (x, y)η, η′〉x = 〈η, K (y, x)η′〉yfor η ∈ Ty M0 and η′ ∈ Tx M0.

Given this kernel, the product 〈ξ, δm(v)〉m may be written

〈ξ, δm(v)〉m =N∑

i=1

〈ξ i , v(mi )〉mi =N∑

i=1

〈Kmi ξ i , v〉g,


so that δ†m(ξ) =

∑Ni=1 Kmi ξ i and (10) is

vt =N∑

i=1

Kmitzi

t .

We also have, δm ◦ δ†m(ξ) = (ξ 1, . . . , ξ N ) with

ξ j =N∑

i=1

Kmi (m j )ξ i =N∑

i=1

K (m j ,mi )ξ i .

Thus, Hσm :

∏Ni=1 Tmi M0 →

∏Ni=1 Tmi M0 is defined by

(Hσm ξ)

i = σ 2ξ i +N∑

j=1

K (mi ,m j )ξ j .

This is an Nk × Nk block matrix, for which the (i, j) k × k block is K (mi ,m j )

if i �= j and σ 2Id+ K (mi ,m j ) if i = j . Note that the previous example appearsas a particular case, with K (x, y) = (1+ 〈x, y〉)Id.

We now clarify equation (14). We have, for η = (η1, . . . , ηN ) ∈ Tm M ,

∇ηv = (∇η1v, . . . ,∇ηN v),

so that

∇†z v = (∇†

z1v, . . . ,∇†zN v),

and equation (14) becomes, for i = 1, . . . , N ,

∇dmi/dt zit + ∇†

zitvt = 0

where the covariant derivative now is the one on M0. When M0 = Rk , we have∇ηv = dv(η) and this equation becomes

dzit

dt+ t dmi

tvt (z

it ) = 0.

Figure 3 provides an example of a geodesic for landmarks in R2 (obtainedby solving the evolution equation), with a comparison with the correspondinggeodesic in the Euclidean space.

3.2.3. Example 3. Here δm(v) = −〈∇m, v〉 (recall that we restrict ourselves tothe case when m is C1) and, using the reproducing kernel of g,

〈z, δm(v)〉L2 = −∫

z(x)〈∇x m, v(x)〉Rk dx

= −∫

〈Kx (z(x)∇x m), v〉g dx

= −〈K (z∇m), v〉g


Fig. 3. Comparison of geodesics under the new metric and the Euclidean one for landmarks in R2

under diffeomorphic action. The template is the central ellipse and both evolutions are designed tocoincide at time t = 1. Images are given at times t = 0.5, 1, 1.7, 2, 2.4. The geodesics for the newmetric progressively deform the ellipse in a way which is consistent with the notion of deformation(first column). Euclidean geodesics (second column) squeeze the initial ellipse, and then invert itto start expanding again (the shaded part has changed sides). Crossing-over ultimately occurs withmetamorphoses too, because of their Euclidean component, but after a much longer time.

with the notation K f =∫

Kx f (x) dx . Equation (10) therefore becomes

vt = −K (zt∇mt ).

We have δm ◦ δ†m(η) = 〈∇m, K (η∇m)〉Rk , i.e.,

(δm ◦ δ†m(η))(y) =

∫

〈∇ym, K (y, x)∇x m)〉Rkη(x) dx,

so that δm ◦ δ†m is a kernel operator, therefore compact, which implies that Hσ

m isinvertible and expression (12) is well defined.

Consider now equation (14). We have v(m) = δm(v) = −〈∇m, v〉, which islinear in m, so that ∇ηv = −x∇ηv and

〈z,∇ηv〉m = −∫

〈∇xη, v(x)〉z(x) dx =∫

η(x) divx (zv) dx,

which implies

dzt

dt+ div(ztvt ) = 0.


4. Evolution Equations

4.1. Evolution in z

Equation (14) already provides an evolution equation for z. We now proceed to aninterpretation of it in terms of conservation of a certain quantity. For any vectorfield η on M , we have, by equation (13), at all times t ,

〈∇∂m/∂t zt , η〉mt= −〈zt ,∇ηvt 〉mt

.

Let µt,ε be a perturbation of µt along η, i.e., µt,0 = µt and ∂µt,ε/∂ε|ε=0 = η.Now, define, α and β being positive numbers,

mα,β,ε = gα(µβ,ε),

so that mt,t,0 = mt . For d/dt = ∂/∂α+∂/∂β, we have around (α, β, ε) = (t, t, 0),

d

dt

⟨zt ,∂m

∂ε

⟩mt

=⟨∇dm/dt zt ,

∂m

∂ε

⟩mt

+⟨zt ,∇dm/dt

∂m

∂ε

⟩mt

.

Since dm/dt = ∂m/∂α+∂m/∂β and∇∂m/∂α∂m/∂ε = ∇∂m/∂ε∂m/∂α, we deducethat

d

dt

⟨zt ,

∂m

∂ε

⟩mt

=⟨∇dm/dt zt ,

∂m

∂ε

⟩mt

+⟨zt , ∇∂m/∂ε

∂m

∂α

⟩mt

+⟨zt , ∇∂m/∂β

∂m

∂ε

⟩mt

,

so that, using equality (13) and the fact that vt (mt ) = ∂m/∂α|α=t,β=t,ε=0, we getfinally

d

dt

⟨zt ,

∂m

∂ε

⟩mt

=⟨zt , ∇∂m/∂β

∂m

∂ε

⟩mt

. (16)

Since ∂m/∂ε|α=t,β,ε=0 = dµβgt (η(µβ)) and ∂m/∂β|α=β=t,ε=0 = σ 2zt , we obtain

d

dt〈zt ,Adgt (η)〉mt

= 〈zt ,∇σ 2ztAdgt (η)〉mt

, (17)

where Ad : G × g → g is the adjoint action defined by Adg.w = de�gw with�g(h) = ghg−1, h ∈ G, and Adg is the induced action on χ(M) by the identifi-cation provided in Notation 1 (Adg(v) = dg−1 gv ◦ g−1). This equation receives aninteresting interpretation if we consider the pull-back of the metric by gt . Indeed,for any g ∈ G, let 〈 , 〉g denote the pull-back of the metric on M by g definedfor any u, u′ ∈ TµM by

〈u, u′〉gµ = 〈dµg(u), dµg(u′)〉g(µ)and let ∇g denote the associated Levi-Civita connection defined for any u ∈ TµMand η ∈ g by

(dµg)(∇gu η) = ∇dµg(u) Adg(η). (18)


Since σ 2zt = dµt gt (dµ/dt), we get from (17) that

d

dt

⟨dµ

dt, η

⟩gt

µt

=⟨

dµ

dt, ∇gt

dµ/dtη

⟩gt

µt

. (19)

We now introduce a new notion related to metamorphoses. We say that a vectorfield η on M , such that∇gt

dµt/dtη = 0, is morphoparallel along the metamorphosis.We therefore have obtained the fact that

d

dt

⟨dµ

dt, η

⟩gt

µt

= 0

whenever η is morphoparallel.It can easily be checked from the chart representation that morphoparallelism

only involves the values of η along the curve µt . Moreover, as a first-order lineardifferential equation on T M , this has a unique solution starting from a tangentvector η0 at Tµ0 M . The solution at time t of this ordinary differential equationwith value ηs ∈ Tµs M at time s can be called morphoparallel transport alongthe metamorphosis and will be denoted θstηs . Thus, if η is morphoparallel along(gt , µt ), we have

〈µt , θ0t (η0)〉gtµt= 〈µ0, η0〉µ0

, (20)

where µs = dµ/dt|t=s . This implies that 〈zt , dµt gt (θ0t (η0))〉mt= 〈µ0, η0〉µ0

.Hence, since µ0 = z0 and µ0 = m0 we get finally

zt = (dmt g−1t )† ◦ θ†

t0(z0). (21)

4.1.1. The Affine Case. In general, the morphoparallel transport equation cannotbe solved analytically, so that little is gained, from a practical point of view,compared to the initial equation (14). This will be illustrated in Example 2 below.But, as shown in Examples 1 and 3, morphoparallel translation may sometimescoincide with parallel transport along µt , in which case equation (21) becomesquite useful. This arises when the metric ∇g coincides with ∇ for all g ∈ G asdefined more formally below.

Definition 1. Let g : M → M be a C2 invertible mapping. We say that g leavesinvariant the Levi-Civita connexion if, for any X, Y ∈ χ1(M), we have

Adg(∇X Y ) = ∇Adg(X) Adg(Y ).

When M is a flat space, it is a known fact that g leaves the connexion invariantif and only if g is affine as stated in the following result (the proof is given forcompleteness in Appendix B):

Proposition 2. Assume that M is a Hilbert space considered as a Riemannianmanifold with metric induced by the inner product. Let g : M → M be a C2


invertible mapping. Then g leaves invariant the Levi-Civita connexion if and onlyif, i.e., for any u ∈ M , we have g(u) = g(0)+ d0g(u).

Thus, in this case, the morphoparallel transport is the parallel transport which issimply a translation. This implies that θ0t = Id and (21) becomes

zt = (dmt g−1t )†(z0). (22)

As we have seen, this is the situation of Examples 1 and 3 which can be equivalentlyderived as arising from a semidirect product G�M with G ⊂ L(M) [14]. So, theaffine case, or semidirect product case, corresponds to some important situations asmetamorphosis between functional data like grey-level images. However, Example2 cannot be handled in this framework, and more generally, when M is finitedimensional and the group acting on M is infinite dimensional, we leave the affineframework and, the morphoparallel transport is different from the parallel transport.

By extension, when M is not flat, the set of diffeomorphisms of M,which leavethe metric invariant, is still called the affine group of M , and denoted Aff(M). Thus,when G ⊂ Aff(M), equation (21) becomes computationally trivialized.

When M is a compact manifold, it is known [24] that the connected componentsof the identity in the affine group and in the group of isometries of M coincide. Notethat, by definition of the metamorphoses, we are only interested in diffeomorphismsg which belong to this connected component. The group of isometries of a compactfinite-dimensional manifold is itself a compact finite-dimensional manifold. Thus,if M is compact, morphoparallel transport coincides with parallel transport onlyin specific situations.

4.2. Evolution of the Velocity

From (10) and (21) we get immediately

vt = δ†m ◦ (dmt g

−1t )† ◦ θ†

t0(z0). (23)

From (9) we get, forw ∈ g: (d/dt)〈zt ,Adgt (w)〉mt= (d/dt)〈vt , Adgt (w)〉g. Thus

we deduce from (17) that

d

dt〈vt ,Adgt (w)〉g = σ 2〈zt ,∇zt Adgt w〉mt

. (24)

This equation without a second term only depends on the Lie group structure.It has the general form of a geodesic equation on a Lie group with a right-invariantmetric, as derived in [1]. The nonvanishing left-hand term modifies this conser-vative evolution to account for template evolution in the optimal metamorphosis.This term cannot vanish unless the metamorphosis is constant, since (9) and z0 = 0implies v0 = 0 as an initial condition. Thus, a pure metamorphosis can only beobtained as the limit process for which σ → 0, while z0 remains nonzero.


If we let Sz(w) = ∇zw, for w ∈ g, we may finally write

d

dt〈(Adgt )

†(vt ), w〉g = 〈(Adgt )† ◦ S†

zt(zt ), w〉g

ord

dt(Adgt )

†vt − (Adgt )†S†

ztzt = 0. (25)

Since, as we have seen 〈zt ,∇σ 2ztAdgt w〉mt

= 〈µt ,∇gtµtη〉gt

µt, we get from (20)

thatd

dt〈vt , Adgt (η)〉g = 〈z0, θt0(∇gt

µtw)〉

m0

which yields

〈vt , Adgtw〉V = 〈v0, w〉g +⟨z0,

∫ t

0θs0(∇gs

µsw) ds

⟩m0

. (26)

Finally, we can reintroduce the expression of µs in the function of µ0 = z0,yielding µs = θ†

t0(z0) to explain the dependency of vt as a function of v0 and z0.

4.3. Examples

4.3.1. Example 1. This first example enters exactly into the setting of an affineaction on M . Here M = (Rk)N with the usual dot product. If g = (B, T ) ∈ G, gcan be represented as an affine transformation on M whose differential is given by

dm g(η1, . . . , ηN ) = (Bη1, . . . , BηN ),

so that (25) gives

zit =

tB−1

t zi0,

and (23) gives

βt = t Bt−1

N∑i=1

zi0

tmi

t ,

τt = t Bt−1

N∑i=1

zi0.

4.3.2. Example 2. Here, the action of G is not affine. Let us verify this, assumingthat M0 = R

k to simplify. In this case, letting m = (m1, . . . ,m N ) and µi =g−1(mi ),

Adg(η)(m) = (dµ1 g(η1(µ1)), . . . , dµN g(ηN (µN ))).


As a consequence, working on the i th component, we have, for ξ i ∈ Tµi M,

(∇gξ iη

i ) = dmi g−1(dmi (dµi g ◦ ηi ◦ g−1)(dµi g(ξ i )))

= dmi g−1(d2µi g(ηi (µi ), ξ i )+ dµi g ◦ dµiηi (ξ i ))

= dµiηi (ξ i )+ dmi g−1(d2µi g(ηi (µi ), ξ i )).

Since ∇ξ iηi = dµiηi (ξ i ) the metric is not conserved. Morphoparallel transportalong (gt , µt ) is the solution of the equation

dηit

dt+ (dµi

tgt )−1d2

µitgt

(ηi ,

dµi

dt

)= 0.

4.3.3. Example 3. This is again (at least formally) an affine action: M is a linearspace and m → gm = m ◦ g−1 is linear in m. To explain equation (21), we needto compute, according to equation (22), zt = (dmt g

−1t )†z0, which is characterized

by∫

zt (y)ξ(y) dy =∫

z0(x)ξ ◦ gt (x) dx =∫

z0 ◦ g−1t (y)ξ(y)|dy g−1

t | dy,

so that equation (21) is zt (y) = |dy g−1t |z0 ◦ g−1

t (y).Figure 4 provides examples of solutions of this system with a fixed initial image

(template). The first two simply are reconstructions of precomputed geodesics,with two different target images. The third one provides two ways of averagingthe previous two: first by averaging the target images, which clearly providesunsatisfactory results, then by averaging the initial conditions z0 before solvingthe evolution equations, with a much more consistent result.

5. A Last Example: Deforming Geometric Curves

We conclude this paper with the presentation of a fourth example of metamorpho-sis, in which M is, as a set, a Hilbert space, but with a nontrivial metric. Indeed, weconsider regular plane curves in parametric form, i.e., functions m : [0, 1]→ ,with an open bounded subset of R2. We assume enough derivatives for thefollowing formal computations, and we restrict ourselves to closed curves.

On this “manifold” M , we will use the metric (with ξ, η : [0, 1]→ R2 having

the same regularity and periodicity condition)

〈ξ, η〉m =∫ 1

0

tξ(x)η(x)

∣∣∣∣dm

dx

∣∣∣∣dx .

ξ and η can in fact be considered as vector fields supported by the image of m inR

2 and the corresponding metric is with respect to arc-length integration on m.In the following, we will denote qm(x) = |dm/dx | = ds/dx , s being the arc-


Fig. 4. Solving the evolution equations with different initial conditions allows us to represent a largerange of image varieties. The first line, with a single image, is the template. The second line provides theinitial z0, the true target (from which z0 is computed), and the reconstruction of the target by geodesicevolution. The third line provides the same, with a different target. Finally, the last line compares thenaive averaging of the targets (second image) to the output of averaging the initial z0 before solvingthe geodesic equations.

length on m. This is a natural metric if one is interested in the geometric aspectsof the deformation. Recall that we restrict ourselves to regular curves so that qm

is nonvanishing on [0, 1].We now consider the action of a group G of diffeomorphisms of on M , letting

(gm)(x) = g ◦m(x). When G is defined as in Examples 2 and 3, on the basis of aHilbert space V of vector fields on , the infinitesimal action, defined for v ∈ V ,


is v(m)(x) = v ◦ m(x) and the energy of a metamorphosis (gt , µt ) is

∫ 1

0|vt |V dt + 1

σ 2

∫ 1

0

∣∣∣∣dmt

dt− vt ◦ mt

∣∣∣∣2

mt

dt.

The geodesic equations (10) and (13) require the computation of the dual eval-uation function, δ†

m , and of the Levi-Civita connnection on M . Starting with δ†m ,

we have (K being the reproducing kernel on V ),

〈ξ, δm(v)〉m =∫ 1

0

tξ(x)v(m(x))qm(x) dx

=∫ 1

0〈Km(x)ξ(x), v〉V qm(x) dx,

so that

δ†m(x) =

∫ 1

0qm(x)Km(x)ξ(x) dx,

and equation (10) becomes

vt (y) =∫ 1

0K (y,mt (x))zt (x)qmt (x) dx (27)

with σ 2zt = dmt/dt − vt ◦ mt .The covariant derivative can be computed from the standard formula

2〈ζ,∇ξ η〉=ξ〈η, ζ 〉+η〈ξ, η〉−ζ 〈ξ, η〉−〈[ξ, ζ ], η〉−〈[η, ζ ], ξ〉+〈[ξ, η], ζ 〉,which yields, after a simple computation,

∇ξ η = ξη + 1

2q(η(ξq)+ ξ(ηq)− ρ tηξ ),

where ρ f is defined by

∫ 1

0

tρfm(x)ζm(x) dx =

∫ 1

0fm(x)(ζq)m(x) dx .

Using the fact that qm(x) = |dm/dx |, we have (ζq)m(x) = (tdζm/dx)τm(x) withτm(x) = (dm/dx)/|dm/dx | (the unit tangent) so that, using integration by parts,

ρ fm = −

d

dx( fmτm)

and

∇ξ η = ξη + 1

2q

(η

(t dξ

dxτ

)+ ξ

(t dη

dxτ

)+ d

dx

((tξη)τ

)).


Introducing the arc-length derivative d/ds = (1/q)d/dx , this becomes

∇ξ η = ξη + 1

2

(η

(t dξ

dsτ

)+ ξ

(t dη

dsτ

)+ d

ds

((tξη)τ

)).

In particular, adopting, for short, the notation u = ∂u/∂t and u′ = ∂u/∂s, andusing the fact that (m)′ = σ 2z′ + dmvτ,

∇m z = z + 12 (z(

t (dvτ)τ )+ σ 2z(t z′τ)+ m(t z′τ)+ ((t zm)τ )′)

= z + 12 (z(

t (dvτ)τ )+ 2σ 2z(t z′τ)+ v(t z′τ)+ ((t zm)τ )′),

(v and dv being evaluated at m).Since 〈ζ,∇ξ η〉 = 〈∇†

ζ η, ξ〉, the same analysis provides

∇†ζ η = η†ζ + 1

2q(ζ(ηq)− η(ζq)+ ρ tηζ ),

with 〈η†ζ , ξ〉 = 〈ζ, ξη〉, so that

∇†ζ η = η†ζ + 1

2

(ζ

(t dη

dsτ

)− η

(t dζ

dsτ

)− d

ds((tηζ )τ)

).

Applying this to ζ = z and ηm = v(m) = v ◦ m yields z†v = t dvz and

∇†z v = t dvz + 1

2 (z(t (dvτ)τ )− v(z′τ)− ((t zv)τ)′).

This yields the equation, for zt ,

z + t dvz + (tτ dvτ)z + σ 2z(t z′τ) = 0. (28)

We therefore can summarize the geodesic equation for the studied action by(m)′ = σ 2z′ + dmvτ,

z + t dvz + (tτ dvτ)z + σ 2z(t z′τ) = 0,v(y) = ∫ 1

0 K (y,m)z ds.

Appendix A. Existence of Solutions of dgt/dt = de Rgtvt

In this appendix, we consider, on G, the right-invariant Riemannian structureassociated to the norm on g, and we assume that G is a complete Riemannianmanifold. We will also assume that there exists a constant C such that |[v,w]|g ≤C |v|g|w|g for v,w ∈ g.

Our goal is to define solutions of equation (1), when vt satisfies

‖v‖1,g :=∫ 1

0|vt |g dt <∞.


First note that if vt is continuous in time, a solution exists at least in small time,and we now show that it can be extended to [0, 1]. Along such a solution, we have

dG(gs, gt ) ≤∫ t

s|vs |g ds,

which implies that if [0, T [ is a maximal interval for the solution, a limit can befound for gt when t tends to T (because G is complete), and the solution can befurther extended beyond T , unless of course T = 1.

We now show that, for any h ∈ G and u ∈ g:

|Adwt u|g ≤ exp(CdG(e, wt ))|u|g.Indeed let ht be a geodesic between e and h (it exists because G is complete), andlet ut = (de Rht )

−1(dht/dt) so that∣∣∣∣ d

dtAdht u

∣∣∣∣g

= |[ut , Adht u]| ≤ C |ut |g|Adht u|g.

andd

dt|Adht u|2g ≤ 2C |ut |g|Adht u|2g.

By Gronwall’s lemma, this implies

|Adht u|g ≤ |u|g exp

(C∫ t

0|us |gds

)= |u|g exp (CdG(e, ht )).

Now, consider the map v �→ gvt which associates to (v : s �→ vs), assumedto be continuous and belonging to L1([0, 1], g), the solution of (1) at time t . Ifv,w ∈ L1([0, 1], g) we have, letting qt = gvt (g

wt )−1,

dG(gvt , gwt ) = dG(e, qt ) ≤

∫ t

0

∣∣∣∣(de Rqt )−1 dqt

dt

∣∣∣∣g

dt.

A straightforward computation shows that

ρs := (de Rqs )−1 dqs

ds= vs − Adq−1

sws

= vs − ws −∫ s

0

d

duAdq−1

uws du

= vs − ws −∫ s

0[ρu, Adq−1

uws] du.

Since dG(e, q−1u ) = dG(gvu , gwu ) ≤ dG(e, gvu)+ dG(e, gwu ) ≤ ‖v‖1,g + ‖w‖1,g,

we have

|ρs |g ≤ |ws − vs |g + C exp(C(‖v‖1,g + ‖w‖1,g))

∫ s

0|ρu |g|wu |gdu,


and by Gronwall’s lemma again, there exists a continuous function K (‖v‖1,g,‖w‖1,g)

such that

|ρs |g ≤ K (‖v‖1,g, ‖w‖1,g)‖w − v‖1,g,

so that

dG(gvt , gwt ) ≤ K (‖v‖1,g, ‖w‖1,g)‖w − v‖1,g.

This implies that gvt is uniformly continuous over bounded subsets of L1([0, 1], g)∩ C0([0, 1], g), and can therefore be extended by continuity to L1([0, 1], g).

Appendix B. Proof of Proposition 2

Proof. (⇒) Indeed, for any u ∈ M , let u : M → M be the constant vector fielddefined by u(m) = u for any m ∈ M . Let v = Adg(u). If xt = tu, then dx/dt = uand if yt = g(xt ), we have

dy

dt= dxt g

(dx

dt

)= dxt g(u) = Adg(u)(yt ) = v(yt ).

Then

∇dy/dtdy

dt= ∇vv(yt )

(a)= dxt g(∇uu(xt )) = dxt g

(∇dx/dt

dx

dt

)= 0,

where (a) comes from the invariance property of g. Thus, dy/dt is constant andg(u) = y0 +

∫ 10 (dy/dt) dt = g(0)+ d0g(u).

(⇐) Assume that g : M → M is defined by g(u) = g(0) + B(u) where B isa continuous invertible linear operator on M . For any X and Y ∈ χ1(M), since inthis flat case ∇X Y = dY (X), we get

Adg(∇X Y )(m) = B ◦ dg−1(m)Y (X (g−1(m))).

Now, dm(Adg(Y )) = B ◦ dg−1(m)Y ◦ B−1, so that

∇Adg(X) Adg(Y ) = B ◦ dg−1(m)Y (X (g−1(m))) = Adg(∇X Y )(m),

and the result is proved.

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Metamorphoses Through Lie Group Action

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