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**|**HHS Author Manuscripts**|**PMC2897162

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- Abstract
- 1. Introduction
- 2. The Basic Set Up
- 3. Euler Equation and Conservation of the Momentum for Lie Groups of Matrices
- 4. Geodesic Evolution of the Diffeomorphism and Conservation of Momentum
- 5. The Normal Momentum Motion Constraint
- 6. Geodesic Evolution in the Orbit
- 7. Computational Results
- References

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J Math Imaging Vis. Author manuscript; available in PMC 2010 July 6.

Published in final edited form as:

J Math Imaging Vis. 2006 January 31; 24(2): 209–228.

doi: 10.1007/s10851-005-3624-0PMCID: PMC2897162

NIHMSID: NIHMS189824

MICHAEL I. MILLER, Center of Imaging Science & Department of Biomedical Engineering, The John Hopkins University, 301 Clark Hall, Baltimore, MD 21218, USA;

MICHAEL I. MILLER: ude.uhj.sic@mim; ALAIN TROUVÉ: rf.nahcac-sne.almc@evuort; LAURENT YOUNES: ude.uhj@senuoy.tnerual

See other articles in PMC that cite the published article.

Studying large deformations with a Riemannian approach has been an efficient point of view to generate metrics between deformable objects, and to provide accurate, non ambiguous and smooth matchings between images. In this paper, we study the geodesics of such large deformation diffeomorphisms, and more precisely, introduce a fundamental property that they satisfy, namely the conservation of momentum. This property allows us to generate and store complex deformations with the help of one initial “momentum” which serves as the initial state of a differential equation in the group of diffeomorphisms. Moreover, it is shown that this momentum can be also used for describing a deformation of given visual structures, like points, contours or images, and that, it has the same dimension as the described object, as a consequence of the normal momentum constraint we introduce.

Over the past several years we have been studying natural shapes using homogeneous orbits of imagery constructed via the action of transformation groups on exemplars or templates. The mathematical structure of group action as a model in image analysis has been pioneered by Grenander [13], the idea being to introduce the group actions in the very nature of the objects themselves, through the notion of *deformable templates*. Roughly speaking, a deformable template simply is an “object or exemplar” *I*_{temp} on which a group *G* acts and generates, through the orbit = *G* · *I*_{temp}, a whole family of new objects. The interest of this approach is to concentrate the modeling effort on the group *G*, and not on the family of objects .

Since the earliest introduction by Silicon Graphics Incorporated of special purpose graphics hardware for object rendering, group action as a model in image analysis has been the subject of a wide range of research in computer vision. Naturally, the analytical and computational properties of the low-dimensional matrix Lie groups form the core dogma of modern Computer graphics. In sharp contrast, however, for the study of imagery generated from natural or biological shapes, the finite dimensional matrix groups are replaced by their infinite dimensional analogue, the more general diffeomorphisms [7, 11, 16, 22, 23, 36].

The anatomical orbit or deformable template is made into a metric space with a metric distance between elements by constructing curves through the space of diffeomorphisms connecting them; the length of the curve becomes the basis for the construction, the metric distance corresponding to the geodesic shortest length curves. This gives rise to a natural variational problem describing the geodesic flows between elements in the orbit, with the solution of the associated Euler-Lagrange equations giving the optimal flow of diffeomorphisms and thus the metric between the shapes. The obtained setting shares several similarities with the mechanics of perfect fluids, for which the Euler-Lagrange equation has been derived by Arnold (Eq. (1) of [2]) for the group of divergence-free volume-preserving diffeomorphisms. As well these results become another example of the general Euler-Poincaré principle of [19] applied to an infinite dimensional setting.

Interestingly, as emphasized by Arnold [1] in his study, one of the most beautiful aspects of studying diffeomorphisms with a Lie group point of view is that many fundamental aspects which can be proved in the finite dimensional case can be formally extended to retrieve well-known equations of mechanics. One of the purposes of this paper is to develop infinite dimensional analogues, for the study of high dimensional shapes via diffeomorphisms, of several of the well known properties of Lie groups in rigid body mechanics. In particular we shall focus on the interpretation of the Euler equation as an expression of the evolution of the generalized momentum of diffeomorphoc flow of least energy in both Eulerian and Lagrangian coordinates.

Such a point of view will link our geodesic formulation to a *conservation of momentum law* in Lagrangian coordinates providing a powerful method for studying and modeling diffeomorphic evolution of shape. It will imply that the momentum of the diffeomorphic flow at any place along the geodesic can be generated from the momentum at the origin, thus providing the vehicle for *geodesic generation via shooting*.

This same conservation of momentum of the diffeomorphic flow, allows us to derive equations for the geodesic evolution of the elements in the orbit
${I}_{t}=I\circ {\phi}_{t}^{-1}$, *t* [0, 1], *I* . This unifies various geodesic evolutions associated with orbits of sparse finite dimensional landmarked shapes as well as the evolutions of dense images. Of special interest is the fact that for the special case of image matching, geodesic evolution of elements in the orbit links us to the notion of normal motion familiar to the rapidly growing community working in *level set methods*. Interestingly, as we show, the momentum of the diffeomorphic flow is normal to the level sets associated with geodesic motion. By solving the partial differential equations which are associated with the conservation of momentum, we will be able to control by specifying the initial conditions (within a specific class of momentum which depends on the considered imaging problem) a wide range of arbitrarily large deformations; this provides new possibilities for learning shape models of deformable templates, or for designing new numerical matching procedures.

This second point of view in terms of the conservation of momentum law also sheds new light on a large number of high dimensional evolution based Active Model Methods in Computer Vision, including active snakes and contours [6, 12, 18, 20, 29, 31, 37, 40, 41, 43], active surfaces and deformable models [8, 9, 21, 24, 25, 28, 30, 33, 39, 40, 42]. In such approaches vector fields are defined which give the boundary manifold of the shape some velocity of motion, usually following the gradient of an energy to form an attractive force to pull the boundary. The power of such methods is that they parameterize motion only associated with a submanifold of the imagery, not the entire extrinsic background space. For example, to deform a planar simply connected shape via an active contour method, the dimension of the motion is determined by the dimension of the boundary of the region, which is substantially less than the dimension of the plane. Historically such approaches have not been studied globally as diffeomorphic action. In fact it is well known that such methods cannot prevent self intersection nor ensure topological consistency, for which the addition of other constraints become necessary [14, 15]. From the conservation law in Lagrangian coordinates describing geodesic motion in the metric space of diffeomorphic action, we introduce the *normal momentum motion* which constrains the momentum to the bounding manifold, and extends the velocity of motion of the shape to the entire background space, thereby giving the global property that the resulting integrated vector field generates a diffeomorphism on the entire extrinsic space. This in turn carries the smooth submanifold diffeomorphically. As the analysis shows, this global property seems to be required to generate geodesic motions.

The basic component of our models is the group of one-to-one, smooth, transformations (diffeomorphisms) of a bounded subset Ω * ^{d}*. In this paper, we consider diffeomorphisms emerging as flows of non-autonomous differential equations. A time dependent vector field on Ω is a function:

$$\begin{array}{r}v:[0,1]\times \mathrm{\Omega}\to {\mathbb{R}}^{d}\\ (t,x)\mapsto v(t,x)\end{array}$$

(*v*(*t*, *x*) will also be denoted *v _{t}*(

$$\frac{dy}{dt}={v}_{t}(y).$$

The flow of this ODE is a function * ^{v}* which depends on time and space, such that

$$\frac{\partial {\phi}^{v}}{\partial t}(t,x)={v}_{t}\left({\phi}_{t}^{v}(x)\right)$$

and (0, *x*) = *x* for all *x* Ω. We will also use the notation
${\phi}_{t}^{v}(x)$ for (*t*, *x*) and

$${\phi}_{s}^{v-1}={\phi}_{t}^{v}\circ {\phi}_{s}^{v-1}.$$

(1)

It is well-known that, under some smoothness conditions on *v*, such
${\phi}_{t}^{v}$ is at all times a diffeomorphism of Ω.

The groups that we consider are precisely composed with such flows
${\phi}_{1}^{v}$ for *v* belonging to a specified functional class. More precisely, we assume that a Hilbert space is given, the element of which being smooth enough vector fields on Ω, and denote the norm and inner product on this space by || ||_{}, ,_{}. We now define (following [35]) the group *G* as the set of functions
${\phi}_{1}^{v}$ for time-dependent vector fields *v* satisfying

$${\int}_{0}^{1}{\left|\right|{v}_{t}\left|\right|}_{\text{g}}dt<\infty ,$$

i.e. for *v* belonging to *L*^{1}([0, 1], ). We will always assume that can be embedded in the space of (
${C}_{0}^{1}(\mathrm{\Omega},{\mathbb{R}}^{d})$, || ||_{1, ∞}), containing vector fields on Ω, which vanish on Ω, where

$${\left|\right|w\left|\right|}_{1,\infty}=\phantom{\rule{0.16667em}{0ex}}{\left|\right|w\left|\right|}_{\infty}+{\left|\right|dw\left|\right|}_{\infty}.$$

From this definition, it appears that the main ingredient in the construction of *G* is the Hilbert space .

Fixing *v* , one can define the linear form *w* *v*, *w** _{g}*, which will be denoted

$$(Lv,w)\doteq {\langle v,w\rangle}_{\text{g}}$$

(we use the standard notation (*M*, *w*) for the linear form *M* applied to *w*). By definition, *Lv* belongs to the dual, ^{*} of , and *L* can be seen as an operator *L*: → ^{*} (this is the canonical duality operator of on its dual). As we shall see, this operator turns out to be a key feature in our analysis. For the moment, we point out the fact that *Lv* is a linear form on which is a space of smooth vector fields. Therefore, *Lv* itself can be a singular object (a generalized function, or a distribution). Here are a few examples of distributions *M* which qualify as elements of ^{*}, under our running assumption that is embedded in the space of *C*^{1} functions:

*L*^{1}vector fields of Ω: if*ψ*: Ωis integrable, define^{d}$$(M,v)={\int}_{\mathrm{\Omega}}{\langle \psi (x),v(x)\rangle}_{{\mathbb{R}}^{d}}dx$$- Let now
*μ*be any measure on Ω, and*ψ*be*μ*integrable. Define$$(M,v)={\int}_{\mathrm{\Omega}}{\langle \psi (x),v(x)\rangle}_{{\mathbb{R}}^{d}}d\mu (x)$$ - Dirac measures: as a particular case of the previous, define, for
*x*Ω and*a*,^{d}$$(M,v)={\langle a,v(x)\rangle}_{{\mathbb{R}}^{d}}$$This will be denoted $M={\delta}_{x}^{\ast}(a)\in {\text{g}}^{\ast}$. - Differential operators: if (
*f*_{i}_{,}, 1 ≤_{j}*i*,*j*≤*d*) are integrable functions, define$$(M,v)=\sum _{i,j=1}^{d}{\int}_{\mathrm{\Omega}}{f}_{ij}\frac{\partial {v}_{i}}{\partial {x}_{j}}dx$$

It is important to notice that, although *L* is defined in a rather abstract way in the previous lines, numerical procedures to compute geodesics can be derived most of the time from the knowledge of its inverse (of Green kernel) *K* = *L*^{−1}. This *K* is a smoothing kernel, the choice of which, within a specific range of available kernels, being the starting point of any practical procedure. We do not detail numerical algorithms in this paper, but the reader can refer to [4, 5, 17, 23] for examples of choices of *K*.

Consider a time-dependent diffeomorphism *v* *L*^{1}([0, 1], ), and let (
${\phi}_{0t}^{v}$, *t* [0, 1]) be the associated flow, defined in the previous section. Along time, each point *x* Ω, considered as a particle, evolves on the trajectory
$t\mapsto {\phi}_{0t}^{v}(x)$, its velocity at time *t* being by definition
${v}_{t}({\phi}_{0t}^{v}(x))$. In other terms, for *y* Ω, *v _{t}*(

So, at each time, we have an Eulerian velocity field, *y* *v _{t}*(

$${E}_{\text{total}}(v)=\frac{1}{2}{\int}_{0}^{1}{\left|\right|{v}_{t}\left|\right|}_{\text{g}}^{2}dt.$$

Note that, in classical fluid mechanics, the kinetic energy is the sum of particle kinetic energies, which, for a homogeneous fluid with mass density given by *ρ* yields

$${\left|\right|{v}_{t}\left|\right|}^{2}=\frac{\rho}{2}{\int}_{\mathrm{\Omega}}\mid {v}_{t}(y){\mid}^{2}dy.$$

This is the *L*^{2} norm of *v*, which cannot be used in our context, since we require that is embedded in
${C}_{0}^{1}$ (we need some kind of Sobolev norms). However, in analogy with standard mechanical systems, we may define the global momentum of the system at time *t* to be the linear form *M _{t}*

$${M}_{t}=L{v}_{t}.$$

So, if *v _{t}* is the Eulerian velocity field at time

The Eulerian frame, as introduced above, describes mechanical quantities as they are observed in the current configuration at each time. The Lagrangian frame, on the contrary, describes quantities as seen from the initial configuration. For example, the diffeomorphism
${\phi}_{0t}^{v}(x)$ provides the position at time *t* of the particle which was at *x* at time 0, which is a Lagrangian notion. For the velocity, we create a Lagrangian velocity field by pulling back the previously defined velocity *v _{t}*, setting

$$\begin{array}{l}{v}_{t}^{l}(x)=\frac{d}{ds}{\left({\phi}_{t}^{-1}({\phi}_{t+s}(x))\right)}_{\mid s=0},\\ \text{i}.\text{e}.{v}_{t}^{l}={(d{\phi}_{t})}^{-1}({v}_{t}\circ {\phi}_{t}).\end{array}$$

The operation

$$v\mapsto (d\phi )v\circ {\phi}^{-1}$$

defines a fundamental Lie group operation, and is called the adjoint action of *G* on its Lie algebra (which here is ), denoted Ad* _{}v*. We have obtained the relation

$${v}_{t}=A{d}_{{\phi}_{t}}{v}_{t}^{l}.$$

To interpret the adjoint action pictorially, the new vector field under the adjoint action *v* → (*d*)*v* ^{−1} has to be interpreted as the transformation of *v* under the deformation generated by . Figure 1 shows how the field *v ^{l}* at location

The momentum *M _{t}* =

$$(L{v}_{t},{v}_{t})=(L{v}_{t},A{d}_{{\phi}_{t}}{v}_{t}^{l})=(A{d}_{{\phi}_{t}}^{\ast}L{v}_{t},{v}_{t}^{l}).$$

This leads to the definition ${M}_{t}^{l}=A{d}_{{\phi}_{t}}^{\ast}L{v}_{t}$ for the momentum in Lagrangian coordinates. The Lagrangian frame takes here the role of a Galilean, or reference frame, and we will retrieve below the fundamental principle of mechanics, which states that the Lagrangian momentum is constant over time along any energy minimizing path. Before this, we make a brief interuption in our discussion to describe the relation between the classical mechanics of a rigid body, and geodesic equation in matrix Lie groups. This simple description will help to understand the formalism in our infinite dimensional group of diffeomorphisms.

In this part, we derive the Euler equation for extremal paths of the kinetic energy in the case of Lie groups of matrices. This derivation is well-known in the context of classical solid mechanics [1], but this simpler case, which can be derived completly without too much technicalities, may be helpful to understand the case of diffeomorphisms.

Let *G* * _{d}*() be a Lie group of

$$\frac{d{\phi}_{t}}{dt}={v}_{t}\circ {\phi}_{t}$$

in which the (left) product of matrices is replaced by the (right) composition of functions.

Returning to the matrix case, we define the kinetic energy at time *t* to be (*Ja _{t}*,

$$(J{a}_{t},{a}_{t})=(J{\omega}_{t},{\omega}_{t}).$$

(note that here the notation (,) refers to the sum of products of coordinates, i.e. the usual inner product on Euclidean spaces, after identification between and ^{*}). The total energy is

$$E(g)=\frac{1}{2}{\int}_{0}^{1}(J{a}_{t},{a}_{t})dt.$$

We retrieve again the analogy with the diffeomorphisms by letting

$${\langle a,b\rangle}_{\text{g}}=(Ja,b)$$

so that *J* takes the role of *L* in the previous section.

We now compute the Euler equation for least energy paths between two fixed endpoints *g*_{0} and *g*_{1}. We recall that the Lie bracket on is [*a*, *b*] = *ab* − *ba*.

The Euler-Lagrange equation for the kinetic energy is given by

$$\frac{\partial Ja}{\partial t}-a{d}_{a}^{\ast}(Ja)=0.$$

(2)

where $a{d}_{a}^{\ast}:{\text{g}}^{\ast}\to {\text{g}}^{\ast}$ is defined by duality through the equalities $(a{d}_{a}^{\ast}f,b)=(f,a{d}_{a}b)=(f,[a,b])$.

Let (*t* *g*_{0}(*t*)) be an extremal curve for the kinetic energy and ((*t*, *h*) *g*(*t*, *h*)) be a smooth deformation around *h* = 0 (*g*(*t*, 0) = *g*_{0}(*t*)): Let *a*(*t*, *h*) and *A*(*t*, *h*) be such that

$$\frac{\partial g}{\partial t}=ga\phantom{\rule{0.38889em}{0ex}}\text{and}\phantom{\rule{0.38889em}{0ex}}\frac{\partial g}{\partial h}=gA.$$

(3)

Writing ${\scriptstyle \frac{{\partial}^{2}g}{\partial t\partial h}}={\scriptstyle \frac{{\partial}^{2}g}{\partial h\partial t}}$, we get $\mathit{gAa}+g{\scriptstyle \frac{\partial a}{\partial h}}=\mathit{gaA}+g{\scriptstyle \frac{\partial A}{\partial t}}$ i.e.

$$\frac{\partial a}{\partial h}=\frac{\partial A}{\partial t}+[a,A]=\frac{\partial A}{\partial t}+{\text{ad}}_{a}A.$$

(4)

The curve *A*(*t*, *h*) can vary freely in , with boundary conditions *A*(0, *h*) = *A*(1, *h*) = 0. From

$$\frac{d}{dh}{\left(\int {\langle a,a\rangle}_{\text{g}}dt\right)}_{\mid h=0},$$

we get

$$\int {\langle a,\frac{\partial A}{\partial t}+a{d}_{a}A\rangle}_{\text{g}}dt=\int \left(Ja,\frac{\partial A}{\partial t}+{\text{ad}}_{a}A\right)dt=0$$

(5)

Using the duality relation, we get $(Ja,{\text{ad}}_{a}A)=({\text{ad}}_{a}^{\ast}(Ja),A)$ so that by integration by part, we finally obtain the Euler equation

$$\frac{\partial Ja}{\partial t}-{\text{ad}}_{a}^{\ast}(Ja)=0.$$

(6)

We know from Lagrangian mechanics that the motion of a body with inertial operator *J* without external forces are extremal paths of the kinetic energy. Hence, Eq. (6) is the evolution equation of a body. We recognize in this equation the momentum to the body
${M}_{t}^{b}\doteq J{a}_{t}$ and the Euler equation is then:

$$\frac{\partial {M}^{b}}{\partial t}-{\text{ad}}_{a}^{\ast}({M}^{b})=0.$$

(7)

The momentum in the body here is to relate to the momentum in Eulerian coordinates for diffeomorphisms. However, if we study the motion of the body in a fixed static reference frame, *the momentum to the space* denoted here *M ^{s}* should remain constant in the absence of external forces. The momentum to the space is defined from

$${M}_{t}^{b}\doteq A{d}_{{g}_{t}^{-1}}^{\ast}({M}_{t}^{b})$$

(8)

where
${\text{Ad}}_{g}^{\ast}$ is the co-adjoint representation which is defined by duality through the equalities:
${\text{(Ad}}_{g}^{\ast}f,b)=(f,{\text{Ad}}_{g}b)=(f,gb{g}^{-1})$. We derive from the evolution equation for *M ^{b}*, given by the Euler equation (7), the conservation of the momentum to the space

Along extremal curves for the kinetic energy, M^{s} is constant:

$$\frac{d{M}^{s}}{dt}=0.$$

(9)

Indeed, we have

$$\left(\frac{d{M}_{t}^{s}}{dt},b\right)=\frac{d}{dt}({M}_{t}^{s},b)=\frac{d}{dt}(J{a}_{t},{\text{Ad}}_{{g}_{t}^{-1}}b)$$

Since, ${\scriptstyle \frac{d}{dt}}({\text{Ad}}_{{g}_{t}^{-1}})=-{\text{ad}}_{{a}_{t}}{\text{Ad}}_{{g}_{t}^{-1}}$, we get finally using Euler Eq. (6),

$$\begin{array}{l}\left(\frac{d{M}_{t}^{s}}{dt},b\right)=\left(\frac{\partial Ja}{\partial t}-{\text{ad}}_{a}^{\ast}(Ja),{\text{Ad}}_{{g}_{t}^{-1}}b\right)\\ =0.\end{array}$$

(10)

Thus, from the conservation of the momentum to the space, ${M}_{t}^{s}\equiv {M}_{0}^{s}$, we deduce that

$$J{a}_{t}={\text{Ad}}_{{g}_{t}}^{\ast}(L{a}_{0}),$$

(11)

or equivalenty, for any *b* :

$$(J{a}_{t},b)=(J{a}_{0},{\text{Ad}}_{{g}_{t}}b)$$

(12)

These results are in fact true for any Lie-group with a left-invariant metric. As we now investigate, they can be formally extrapolated also for infinite dimensional groups of diffeomorphisms.

The derivation of the Euler equation for extremal paths of the kinetic energy in the case of finite-dimensional Lie groups can be carried out in full generality within the Lie theory framework, to lead to the law of conservation of momentum. A general computation can be found in [1]. In our infinite dimensional case, a rigorous derivation of this law is much harder, and must most of the time be obtained directly from variational and functional analysis arguments rather than with purely algebraic Lie group derivations. However, it is interesting, and quite informative, to use these derivations to obtain a formal proof of the conservation of momentum, without wondering too much about the well-posedness of the expressions. This will be done in the next paragraphs.

The first Euler equation provides the variations of the momentum in Eulerian coordinates. Before stating it, we need some definitions:

The adjoint action Ad of *G* on and the associated adjoint action ad of on itself are given with their dual operators Ad^{*}, ad^{*} by

$$\begin{array}{l}{\text{Ad}}_{\phi}w=(d\phi )w\circ {\phi}^{-1},\\ ({\text{Ad}}_{\phi}^{\ast}f,w)=(f,{\text{Ad}}_{\phi}w)\end{array}$$

(13)

$$\begin{array}{l}{\text{ad}}_{v}w=[v,w]=(dv)w-(dw)v,\\ ({\text{ad}}_{v}^{\ast}f,w)=(f,{\text{ad}}_{v}w).\end{array}$$

(14)

with *G*, *w* , *f* ^{*}.

Already at this point, one can point out the difficulty of the infinite dimensional problem: at the difference with the matrix case, if *v*, *w* belong to , it cannot be guaranteed that it is still so for [*w*, *v*] = (*dw*)*v* − (*dv*)*w*: in situations of interest, is, in fact, not a Lie algebra: Ad* _{}w* and ad

The geodesics are extremal curves for the kinetic energy. They satisfy an Euler equation giving the variation of the momentum in terms of the co-adjoint action operator on the momentum.

The Euler equation for the kinetic energy is given by

$$\frac{\mathit{dLv}}{dt}+a{d}_{v}^{\ast}(Lv)=0.$$

(15)

When *Lv* *H* (i.e. it is a function), one has

$${\text{ad}}_{v}^{\ast}Lv=\text{div}(Lv\otimes v)+d{v}^{\ast}Lv.$$

(16)

where div(*u* *v*) = *duv* + div(*v*)*u*.

These equations, which are derived below, are special cases of the Euler-Poincaré principle, described, for example in [19]. Equation (15) is formally identical to Eq. (2) in the matrix case, excepted for a sign difference arising from the switch from a left-invariance in the matrix case to a right-invariance in the diffeomorphism case.

This is exactly as in the matrix case. Here again, let (*t* * _{t}*) be extremal and ((

$$\frac{{\partial}^{2}\phi}{\partial \epsilon \partial t}=\frac{{\partial}^{2}\phi}{\partial t\partial \epsilon}$$

which yields

$${h}_{t}\circ {\phi}_{t}+{d}_{{\phi}_{t}}{v}_{t}{\eta}_{t}\circ {\phi}_{t}=\frac{\partial {\eta}_{t}}{\partial t}\circ {\phi}_{t}+{d}_{\phi t}{\eta}_{t}{v}_{t}\circ {\phi}_{t}$$

or (applying ${\phi}_{t}^{-1}$ on the right to both terms) gives

$${h}_{t}=\frac{\partial {\eta}_{t}}{\partial t}+d{\eta}_{t}{v}_{t}-d{v}_{t}{\eta}_{t}=\frac{\partial {\eta}_{t}}{\partial t}+[{\eta}_{t},{v}_{t}].$$

The^{1} first variation of the energy is given by

$$\begin{array}{l}\frac{d}{d\epsilon}{\int}_{0}^{1}{\left|\right|{v}_{t,\epsilon}\left|\right|}_{\text{g}}^{2}dt\\ =2{\int}_{0}^{1}{\langle {v}_{t},{h}_{t}\rangle}_{\text{g}}dt\\ ={\int}_{0}^{1}{\langle {v}_{t},\frac{d{\eta}_{t}}{dt}+[{\eta}_{t},{v}_{t}]\rangle}_{\text{g}}dt\\ ={\int}_{0}^{1}\left(L{v}_{t},\frac{d{\eta}_{t}}{dt}\right)dt-{\int}_{0}^{1}(L{v}_{t},a{d}_{{v}_{t}}{\eta}_{t})dt.\end{array}$$

Since * _{t}* is extremal, this expression vanishes for all

We now prove Eq. (16) under the assumption that *Lv* is a function. By definition

$$\begin{array}{l}(a{d}_{v}^{\ast}Lv,w)=(Lv,\mathit{dvw}-\mathit{dwv})\\ =(d{v}^{\ast}Lv,w)-(Lv,\mathit{dwv})\end{array}$$

and the conclusion comes from Stokes’ theorem which states that (since *v* and *w* vanish on Ω)

$$(\text{div}(Lv\otimes v),w)=-(Lv,dw.v).$$

It appears that the Euler equation (15) with
${\text{ad}}_{v}^{\ast}Lv=\text{div}\phantom{\rule{0.16667em}{0ex}}\mathit{vLv}+d{v}^{\ast}Lv$ has been derived in [26] and subsequently [22] directly as the Euler-Lagrange equation for the kinetic energy by analytical means. This has been originally proved by Arnold in [3] for the motion of impressible fluid which corresponds to the case *L* = Id with the constraint div *v* = 0.

The Euler equation (15) is the evolution of the momentum in Eulerian coordinates. We recognize in this equation the momentum *M _{t}*

Along extremal curves for the kinetic energy, ${M}_{t}^{l}$ is constant:

$$\frac{d{M}_{t}^{l}}{dt}=0.$$

(17)

In particular, we have for all *w* ,

$$(L{v}_{t},w)=(L{v}_{0},{(d{\phi}_{t})}^{-1}w\circ {\phi}_{t}).$$

(18)

Indeed, fix *w* and let
$f(\epsilon )=({M}_{t+\epsilon}^{l},w)$. We have, on the first hand
${f}^{\prime}(0)=({\scriptstyle \frac{d{M}_{t}^{l}}{dt}},w)$, and on the second hand (derivatives being evaluated at time *ε* = 0)

$$\begin{array}{l}{f}^{\prime}(0)=\frac{d}{d\epsilon}(L{v}_{t+\epsilon},{\text{Ad}}_{{\phi}_{t+\epsilon}}w)\\ =\left(\frac{dL{v}_{t}}{dt},{\text{Ad}}_{{\phi}_{t}}w\right)+\left(L{v}_{t},\frac{d}{d\epsilon}A{d}_{{\phi}_{t+\epsilon}}w\right)\end{array}$$

Note here that Ad_{}_{}_{}_{′}= Ad* _{}* Ad

$$\begin{array}{l}\frac{d}{d\epsilon}({\text{Ad}}_{{\phi}_{\epsilon}}{w}^{\prime}){\mid}_{\epsilon =0}=\frac{d}{d\epsilon}\left((d{\phi}_{\epsilon}){w}^{\prime}\circ {\phi}_{\epsilon}^{-1}\right){\mid}_{\epsilon =0}\\ =dv({w}^{\prime})-d{w}^{\prime}(v)=a{d}_{v}{w}^{\prime}.\end{array}$$

(19)

Applying this to *w*′ = Ad_{t−1} and *v* = *v _{t}*, we get

$$\begin{array}{l}{f}^{\prime}(0)=\left(\frac{dL{v}_{t}}{dt},{\text{Ad}}_{{\phi}_{t}}w\right)+(L{v}_{t},a{d}_{{v}_{t}}A{d}_{{\phi}_{t}}w)\\ =\left(\frac{dL{v}_{t}}{dt}+a{d}_{{v}_{t}}^{\ast},A{d}_{{\phi}_{t}}w\right)=0\end{array}$$

by Eq. (15). This completes the proof,

$$\left(\frac{d{M}_{t}^{l}}{dt},w\right)=0.$$

Although the conservation of momentum has only been derived from formal arguments, we can check that, when it is satisfied, the generated deformation paths do provide extremal curves of the kinetic energy. The perturbation of the end point of the path (
${\phi}_{0t}^{v}$, *t* [0, 1]) at time 1 under a perturbation
${v}_{t}^{\epsilon}$ of *v _{t}* is given by [22]:

$$\frac{d}{d\epsilon}{\phi}_{01}^{{v}^{\epsilon}}(x)={\int}_{0}^{1}{d}_{{\phi}_{0s}^{v}}{\phi}_{s1}^{v}({h}_{s}\circ {\phi}_{0s}^{v})ds.$$

(20)

with
${h}_{s}(x)={\scriptstyle \frac{d{v}_{s}^{\epsilon}(x)}{d\epsilon}}$, the derivative being taken at *ε* = 0 (we have used the notation of Eq. (1)). Assume that this expression vanishes (so that the end point
${\phi}_{01}^{{v}^{\epsilon}}$ remains unchanged). The first variation of the kinetic energy is given by
${\int}_{0}^{1}{\langle {v}_{t},{h}_{t}\rangle}_{L}dt={\int}_{0}^{1}(L{v}_{t},{h}_{t})dt={\int}_{0}^{1}(L{v}_{0},{d}_{{\phi}_{0t}}{\phi}_{t0}{v}_{t}\circ {\phi}_{0t})dt$. Now, using (20) and the fact that *d*_{0t} _{t}_{1} = *d*_{00} _{01}*d*_{0t} _{t}_{0}, we get easily that
${\int}_{0}^{1}{d}_{{\phi}_{0t}}{\phi}_{t0}{v}_{t}\circ {\phi}_{0t}dt=0$ so that, by linearity,

$${\int}_{0}^{1}{\langle {v}_{t},{h}_{t}\rangle}_{L}dt=\left(L{v}_{0},{\int}_{0}^{1}{d}_{{\phi}_{0t}}{\phi}_{t0}{h}_{t}\circ {\phi}_{0t}dt\right)=0.$$

(21)

For *M* ^{*}, the evolution
$t\mapsto {\text{Ad}}_{{\phi}_{t}^{-1}}^{\ast}M$ is called coadjoint transport. The fact that the momentum evolves by coadjoint transport along a geodesic implies the conservation of several properties whenever they are initially true, for *Lv*_{0}. These properties will turn out to be of main importance for image processing applications.

In this section, we assume that *M*_{0} = *Lv*_{0} is given, and that the coadjoint transport
${M}_{t}=L{v}_{t}=A{d}_{{\phi}_{t0}}^{\ast}({M}_{0})$ is well defined at all considered times.

Let Supp(*M*) denote the support of a momentum *M* ^{*}. It is defined by the complementary of the union of all open sets Ω′ Ω which are such that (*M*, *w*) = 0 whenever *w* vanishes outside Ω′. We have the property:

If ${M}_{t}={\text{Ad}}_{{\phi}_{t0}}^{\ast}({M}_{0})$, then

$$\text{Supp}({M}_{t})={\phi}_{0t}^{v}(\text{Supp}({M}_{0}))$$

Indeed, assume that *M*_{0} vanishes Ω′ Ω. Let *w* have its support included in _{0}* _{t}* (Ω′). Then (

As a first example, consider the case when *M*_{0} is finitely supported, and more precisely a sum of Dirac measures. This is legitimate since our hypotheses on *L* imply that Dirac measures belong to ^{*}, therefore have the form *Lv*_{0} for some *v*_{0} . So, we assume that

$$({M}_{0},w)=\sum _{i}{\langle {a}_{i},w({x}_{i})\rangle}_{{\mathbb{R}}^{d}},$$

(22)

where (*x _{i}*)

$$\begin{array}{l}{\delta}_{x}^{\ast}a:\text{g}\to \mathbb{R}\\ w\mapsto \langle a,w(x)\rangle \end{array}$$

(23)

Denoting *x _{i}* (

$${M}_{t}=\sum _{i=1}^{n}{\delta}_{{x}_{i}(t)}^{\ast}{a}_{i}(t)$$

(24)

with

$${a}_{i}(t)={({d}_{{x}_{i}(t)}{\phi}_{t0})}^{\ast}{a}_{i}$$

(25)

so that the momentum remains a sum of Dirac measures. This is a special case of the property considered in the next section.

Measure-based momenta are given by

$$(M,w)={\int}_{\mathrm{\Omega}}\langle {\nu}_{0},w\rangle d{\mu}_{0}$$

(26)

where *μ*_{0} is a measure on Ω and *ν*_{0} is measurable and *μ*_{0}-integrable. They generate a large class of geodesic evolutions, and have the attractive property that the momentum *Lv _{t}* can be explicitly computed from the momentum at the origin.

Assume that (*M*_{0}, *w*) = *∫*_{Ω}*ν*_{0}, *w**dμ*_{0} then the linear momentum functional evolves according to

$$\begin{array}{l}({M}_{t},w)={\int}_{\mathrm{\Omega}}{\langle {\nu}_{t},w\rangle}_{{\mathbb{R}}^{d}}d{\mu}_{t}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\text{where}\\ {\nu}_{t}(x)={({d}_{x}{\phi}_{t0})}^{\ast}{\nu}_{0}\circ {\phi}_{t0}(x),{\mu}_{t}\doteq \mu \circ {\phi}_{t0},\end{array}$$

(27)

i.e. *μ _{t}* (

The statement follows straightforwardly from the substitutions

$$\begin{array}{l}({M}_{t},w)={\int}_{\mathrm{\Omega}}{\langle {\nu}_{0}(x),({(d{\phi}_{0t})}^{-1}w\circ {\phi}_{0t})(x)\rangle}_{{\mathcal{R}}^{d}}d\mu (x)\\ ={\int}_{\mathrm{\Omega}}{\langle {d}_{{\phi}_{0t}}(x){\phi}_{t0}^{\ast}{\nu}_{0}(x),w\circ {\phi}_{0t}(x)\rangle}_{{\mathcal{R}}^{d}}d{\mu}_{0}(x)\\ ={\int}_{\mathrm{\Omega}}{\langle {\nu}_{t}(x),w(x)\rangle}_{{\mathcal{R}}^{d}}d{\mu}_{t}(x)\end{array}$$

(28)

Point-supported momentum evolution considered in the previous section, clearly is a particular case of this statement. As another illustration, consider the case of measures which are supported by submanifolds of Ω. In this case, the initial momentum is concentrated along the boundary Σ_{0} of a *k*-dimensional *C* ^{1} sub-manifold in Ω * ^{d}*.

Let *σ*_{0} by the surface measure (given as the induced volume form on the sub-manifold) and let *μ*_{0} be supported by Σ_{0}, such that for any smooth function on Ω, *∫*_{Ω}*f dμ*_{0} = *∫*_{Σ0}*f α*_{0}*dσ*_{0} for some density *α*_{0} (not necessary positive) on the surface. Let *ν*_{0}: Ω → * ^{d}* (the values of

$$(L{v}_{0},w)={\int}_{{\mathrm{\sum}}_{0}}{\langle {\nu}_{0},w\rangle}_{{\mathbb{R}}^{d}}{\alpha}_{0}d{\sigma}_{0}.$$

(29)

Using Statements 3 and 4, we get that the tranported measure *μ _{t}* is located on the transported sub-manifold Σ

$$(L{v}_{t},w)={\int}_{{\mathrm{\sum}}_{t}}\langle {\nu}_{t},w\rangle {\alpha}_{t}d{\sigma}_{t}(y).$$

(30)

In the case where the sub-manifold is Ω itself, then *σ _{t}* =

The last property transported by geodesic evolution which is considered here is the normality with respect to a smooth submanifold of Ω. Since normality will be extensively studied in the next section, we here provide an illustration in a particular case.

Assume that *ν*_{0}, in Eq. (26) can be expressed as

$${\nu}_{0}=\sum _{i=1}^{r}{b}_{0}^{i}\nabla {f}_{0}^{i}$$

(31)

where
${({b}_{0}^{i})}_{1\le i\le r}$ and (
${f}_{0}^{i}$) are two families of functions on Ω and 1 ≤ *r* ≤ *d*. Then, we get from Statement 4 that

$${\nu}_{t}=\sum _{i=1}^{r}{b}_{t}^{i}\nabla {f}_{t}^{i}$$

(32)

where ${b}_{t}^{i}={b}_{0}^{i}\circ {\phi}_{t,0}$ and ${f}_{t}^{i}={f}_{0}^{i}\circ {\phi}_{t,0}$. Equation (32) can be interpreted as a normality property of the geodesic motion under initial condition (31). Indeed, let

$${\mathrm{\sum}}_{0}=\underset{i=1}{\overset{r}{\cap}}{({f}_{0}^{i})}^{-1}(\{0\}).$$

Assume that Σ_{0} is not empty and denotes
${N}_{0}(x)=\text{Span}\{\nabla {f}_{0}^{i}(x)\mid 1\le i\le r\}$ for any *x* Σ_{0}. Under appropriate transversality conditions, mainly

$$r\equiv dim\phantom{\rule{0.16667em}{0ex}}{N}_{0},$$

Σ_{0} can be equipped with a structure of (*d* − *r*)-dimensional *C*^{1} manifold and *N*_{0}(*x*) is exactly the space of vectors normal to Σ at location *x*.

We then deduce easily that

$${\mathrm{\sum}}_{t}=\underset{i=1}{\overset{r}{\cap}}{({f}_{t}^{i})}^{-1}(\{0\})$$

and equality (32) implies for any *x* Σ_{t}

$${\nu}_{t}(x)\in {N}_{t}(x)$$

(33)

where
${N}_{t}(x)=\text{Span}\{\nabla {f}_{t}^{i}(x)\mid 1\le i\le r\}$ is the set of normal vectors to Σ* _{t}* at location

We deduce that if the momentum is normal to some *k*-dimensional sub-manifold Σ_{0}, this normality property is preserved by coadjoint transport along a geodesic.

In the case of *r* = 1 and
${f}_{0}^{1}={f}_{0}$, Σ_{0} is exactly the level set for threshold value 0 of *f*_{0} and the normality of the initial momentum to the level sets is preserved under geodesic motion. Since the threshold value is arbitrary, we deduce that the property is true for all level sets.

The conservation of momentum is a general property of geodesics in a group of diffeomorphisms with a right invariant metric. More can be said in the situation when diffeomorphisms are associated to deformations of geometric structures or images, which is the situation of interest for our applications. In this setting we are still looking for curves with shortest length in *G*, but we partially relax the fixed end-point condition by the constraint that the initial template is correctly mapped to the target: because there is a whole range of diffeomorphisms which deform one given structure into another, this condition is weaker than the fixed end-point condition, which means that there are more degrees of freedom for the optimization, and therefore more constraints on the minimum. For image matching, these additional constraints may essentially be summarized by the statement *the momentum along the geodesic path is at all times normal to the level sets of the image*. This is what we call the *normal momentum constraint*, which is described in this section.

We start with a heuristic analysis, for which *I*_{0}, the image, is a smooth function defined on Ω. Let *I*_{1} be in the orbit of *I*_{0} for the group *G* of diffeomorphisms: there exists *ψ* *G* such that *I*_{0} *ψ*^{−1} = *I _{t}*. By compactness and semi-continuity arguments, one can prove the existence of a geodesic path = (

$${d}_{G}(\text{Id},{\phi}_{1})=\underset{\varphi \in G}{inf}{d}_{G}(\text{Id},\varphi )\phantom{\rule{0.38889em}{0ex}}\text{and}\phantom{\rule{0.38889em}{0ex}}{I}_{0}={I}_{1}\circ \varphi .$$

(34)

Let
$\phi =({\phi}_{t}^{v})$ be such a solution and consider a first order expansion around *t* = 0, * _{t}*(

$$\begin{array}{l}({\mathcal{P}}_{0}):\underset{u\in \text{g}}{inf}\mid u{\mid}_{L}\phantom{\rule{0.38889em}{0ex}}\text{subject}\phantom{\rule{0.16667em}{0ex}}\text{to}:\\ {\langle ({v}_{0}-u)(x),\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\nabla {I}_{0}(x)\rangle}_{{\mathbb{R}}^{d}}=0,\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\forall x\in \mathrm{\Omega}\end{array}$$

(35)

Since (
${\phi}_{t}^{v}$) is a geodesic path minimizing the deformation cost from *I*_{0} to *I*_{1}, it minimizes also the deformation cost from *I*_{0} to *I _{t}* yielding

$${v}_{0}={p}_{{I}_{0}}({v}_{0}).$$

(36)

Introduce the set _{I0} = {*h* | *I*_{0}(*x*), *h*(*x*)_{d} = 0, ∀*x* Ω}: the constraints in can be restated as *u* − *v*_{0} _{I0} so that *p _{I}*

$$\begin{array}{l}\forall h\in \text{g}\phantom{\rule{0.16667em}{0ex}}\text{such}\phantom{\rule{0.16667em}{0ex}}\text{that}\phantom{\rule{0.16667em}{0ex}}\text{for}\phantom{\rule{0.16667em}{0ex}}\text{all}\phantom{\rule{0.16667em}{0ex}}x\in \mathrm{\Omega},\langle \nabla {I}_{0}(x),h(x)\rangle =0,\\ \phantom{\rule{0.16667em}{0ex}}\text{we}\phantom{\rule{0.16667em}{0ex}}\text{have}\phantom{\rule{0.16667em}{0ex}}{\langle {v}_{0},h\rangle}_{L}=0.\end{array}$$

(37)

Now, the fact that *I*_{0}(*x*), *h*(*x*) 0 means that *h* is a vector field which is tangent to the level sets of *I*_{0}, and since *v*_{0}, *h** _{L}* = (

We now pass to a rigorous derivation of this property. Since it will be interesting to also consider images which are not smooth, we provide a new definition of the set _{I0}. Obviously, when *I*_{0} is smooth, *h* _{I0} is equivalent to the fact that, for any function *f* which is *C*^{1} on Ω, one has

$${\int}_{\mathrm{\Omega}}{\langle {\nabla}_{x}{I}_{0},h(x)\rangle}_{{\mathbb{R}}^{k}}f(x)dx=0.$$

Applying the divergence theorem (we assume that Ω is smooth enough and take advantage on the fact that elements of vanish on Ω), we get

$$-{\int}_{\mathrm{\Omega}}{I}_{0}(x)\text{div}(hf)(x)dx=0.$$

Since this has a meaning when *I*_{0} *L*^{2}(Ω), we now define

When *I* *L*^{2}(Ω), we denote

$${\text{g}}_{I}=\{h\in \text{g}:{\langle I,\phantom{\rule{0.38889em}{0ex}}\text{div}(hf)\rangle}_{{L}^{2}}=0\phantom{\rule{0.16667em}{0ex}}\text{for}\phantom{\rule{0.16667em}{0ex}}\text{all}\phantom{\rule{0.16667em}{0ex}}f\in {C}^{1}(\mathrm{\Omega})\}.$$

We still denote by *p _{I}* the orthogonal projection on
${\text{g}}_{I}^{\perp}$. The group

Assume that I_{0} L^{2}(Ω) and let
$\phi =({\phi}_{t}^{v})$ be a geodesic path solution of (34). Then, for almost all t [0, 1]

$${v}_{t}\in {\text{g}}_{{I}_{t}}^{\perp}.$$

The proof is given in Appendix A.

Consider again the case of smooth *I*, so that the condition *h* * _{I}* is equivalent to

$${\text{g}}_{I}^{\perp}=\overline{\text{Span}\{{\delta}_{x}^{\ast}(\nabla I(x))\mid x\in \mathrm{\Omega}\}}.$$

(38)

Vector fields *v* such that

$$(Lv,u)={\int}_{\mathrm{\Omega}}{\langle v(x),u(x)\rangle}_{{\mathbb{R}}^{d}}d\mu (x),$$

(39)

where *ν* is normal^{2} to the level sets of *I* belong to
${\text{g}}_{I}^{\perp}$ and it can be shown that they form a dense subset.

For non-smooth *I*, we can similarly introduce *ω _{f}* (

$${\text{g}}_{I}^{\perp}=\overline{\text{Span}\{{\omega}_{f}(I)\mid f\in {C}^{1}(\mathrm{\Omega},\mathbb{R})\}}.$$

(40)

This implies that any element *v* * _{I}* is such that

$$(Lv,u)=\underset{N\to \infty}{lim}{\int}_{\mathrm{\Omega}}I(x)\text{div}(u{f}_{N})(x)dx$$

where *f _{N}*

$$\begin{array}{l}{\int}_{\mathrm{\Omega}}I(x)\text{div}(uf)(x)dx={\int}_{B}\text{div}(uf)(x)dx\\ ={\int}_{\partial B}f(x){\langle u(x),\nu (x)\rangle}_{{\mathbb{R}}^{d}}d{\sigma}_{B}(x).\end{array}$$

In this case, we obtain a dense subspace of * _{I}* by considering elements

$$(Lv,u)={\int}_{\partial B}{\langle v(x),u(x)\rangle}_{{\mathbb{R}}^{d}}d\mu (x).$$

(41)

for some measure *μ* on *B* (the boundary of the shape).

We close this section with a technical, but important, remark. We have called *normal momentum constraint* the property that *v _{t}*

$$(Lv,u)={\int}_{\mathrm{\Omega}}\langle \xi (x),u(x)\rangle d\mu (x)$$

for some measure *μ* on Ω and some vector field on *ξ* Ω, then *ξ* must be (*μ*-almost everywhere) orthogonal to the level sets of *I*. From a practical point of view smoothness of level sets may easily be obtained using algorithms such as mean curvature motion ([27]).

Here, we give a brief account of situations in which proofs of conservation of momentum and the normality property can be carried on in a well-defined context, and retrieve the evolutions described in the previous section.

It is hard to make rigorous, in full generality, the variational argument we have used in the proof of Eq. (15). Notice that the well-definiteness of the conservation of momentum Eq. (17) is an issue by itself, since, when *w* , and * _{t}* is the diffeomorphism generated by a geodesic, there is a priori no reason for (

However, there is a setting in which such a fact is true and easy to obtain: it is when the search for the geodesic is done with an approximation of the target, with some *L*^{2} penalty term added to control the error. We summarize this setting in the case of landmark matching, shape matching and image matching. In these three situations, we will retrieve the coadjoint transport of measure-based momenta. In all cases, results in [11, 35] ensure the existence of minimizers of the variational problem.

In this section, we assume that a measured space (, *ρ*), together with two measurable functions *x*, *y*: → Ω are given. The diffeomorphism is searched to minimize

$$U(\phi )=E(\phi )+\frac{1}{{\sigma}^{2}}{\int}_{\mathcal{J}}\mid {y}_{i}-\phi ({x}_{i}){\mid}^{2}d\rho (i)$$

When *ρ* is discrete, this relates to point-based matching, *x* representing the landmark original positions and *y* giving the landmark target positions. If we express *U* in function of *v*, this requires the minimization of

$$\stackrel{\sim}{U}(v)={\int}_{0}^{1}{\left|\right|{v}_{t}\left|\right|}_{L}^{2}dt+\frac{1}{{\sigma}^{2}}{\int}_{\mathcal{J}}\mid {y}_{i}-{\phi}_{01}^{v}({x}_{i}){\mid}^{2}d\rho (i)$$

The main point here is to notice that the optimal solution *v* generates a geodesic in *G* between id and
${\phi}_{1}^{v}$.

Denote ${\delta}_{x}^{\ast}a$ the linear form on such that $({\delta}_{x}^{\ast}a,v)=\langle a,v(x)\rangle $. Let v be a minimizer of Ũ. Then, letting ${x}_{i}(t)={\phi}_{0t}^{v}({x}_{i})$

$$L{v}_{t}=-\frac{1}{{\sigma}^{2}}{\int}_{\mathcal{J}}{\delta}_{{x}_{i}(t)}^{\ast}[{({d}_{{x}_{i}(t)}{\phi}_{t1}^{v})}^{\ast}({y}_{i}-{x}_{i}(1))]d\rho (i)$$

(42)

The proof of this result is a direct consequence of the identity, valid for *s*, *t* [0, 1], *v*, *h* *L*^{2}([0, 1], ),

$${\frac{d}{d\epsilon}{\phi}_{st}^{v+\epsilon h}(x)|}_{\epsilon =0}={\int}_{s}^{t}{d}_{{\phi}_{su}^{v}}{\phi}_{ut}^{v}({h}_{u}\circ {\phi}_{su}^{v})du.$$

(43)

the proof of which being carried on with usual ODE arguments and being omitted here. It is then straightforward to obtain (42), using the definition of the linear forms
${\delta}_{x}^{\ast}a$, for *x* Ω and *a* * ^{d}*.

Equation (42) is a conservation of momentum equation for

$$L{v}_{0}=-\frac{1}{{\sigma}^{2}}{\int}_{\mathcal{J}}{\delta}_{{x}_{i}}^{\ast}[{({d}_{{x}_{i}}{\phi}_{01}^{v})}^{\ast}({y}_{i}-{x}_{i}(1))]d\rho (i).$$

When is finite, this is equation (31) with ${a}_{i}=-{\scriptstyle \frac{1}{{\sigma}^{2}}}{({d}_{{x}_{i}}{\phi}_{01}^{v})}^{\ast}({y}_{i}-{x}_{i}(1))$. Equation (42) now is exactly (24), since

$$\begin{array}{l}{a}_{i}(t)={({d}_{{x}_{i}(t)}{\phi}_{t0}^{v})}^{\ast}{a}_{i}\\ =-\frac{1}{{\sigma}^{2}}{[{d}_{{x}_{i}(0)}{\phi}_{01}^{v}{d}_{{x}_{i}(t)}{\phi}_{t,0}^{v}]}^{\ast}({y}_{i}-{x}_{i}(1))\\ =-\frac{1}{{\sigma}^{2}}{({d}_{{x}_{i}(t)}{\phi}_{t1}^{v})}^{\ast}({y}_{i}-{x}_{i}(1))\end{array}$$

We now consider the comparison of a binary set-indicator function, *I*_{0} = **1**_{Ω0} (_{0} Ω having smooth boundaries) and a smooth function *I*_{1}, through the minimization of

$$U(\phi )=E(\phi )+\frac{1}{{\sigma}^{2}}{\int}_{\mathrm{\Omega}}\mid {\mathbf{1}}_{{\mathrm{\Omega}}_{0}}\circ {\phi}^{-1}(x)-{I}_{1}(x){\mid}^{2}dx$$

over *G _{L}*. We have

Let v be a minimizer of
$U({\phi}_{01}^{v})$ over L^{2}([0, 1], ). Then

$$L{v}_{t}=\frac{1}{{\sigma}^{2}}{\int}_{\partial {\mathrm{\Omega}}_{1}}\left(\frac{1}{2}-{I}_{1}\right){\delta}_{{\phi}_{1t}^{v}(x)}^{\ast}[{({d}_{{\phi}_{1t}^{v}}{\phi}_{t1}^{v})}^{\ast}{\nu}_{1}]d{\sigma}_{1}(x)$$

(44)

where
${\mathrm{\Omega}}_{1}={\phi}_{01}^{v}({\mathrm{\Omega}}_{0})$, ν_{1} is the outward normal to Ω_{1} and σ_{1} is the volume measure on Ω_{1}.

Taking a variation *v* + *εh*, the main issue is to compute the derivative of

$$\frac{1}{2}{\int}_{\mathrm{\Omega}}\mid {\mathbf{1}}_{{\mathrm{\Omega}}_{0}}\circ {\phi}_{10}^{v+\epsilon h}(x)-{I}_{1}(x){\mid}^{2}dx$$

This integral can be rewritten

$${\int}_{{\mathrm{\Omega}}_{01}^{v+\epsilon h}({\mathrm{\Omega}}_{0})}\left(\frac{1}{2}-{I}_{1}(x)\right)dx+\frac{1}{2}{\int}_{\mathrm{\Omega}}\mid {I}_{1}(x){\mid}^{2}dx$$

Since the last term is constant, we see that the problem boils down to the computation of the derivative of the first term, which can be written, after a change of variable and letting *f*_{1} = 1/2 − *I*_{1},

$${\int}_{{\mathrm{\Omega}}_{0}}{f}_{1}\circ {\phi}_{01}^{v+\epsilon h}(x)\mid {d}_{x}{\phi}_{01}^{v+\epsilon h}\mid dx$$

Define *u* by
$u\circ {\phi}_{01}^{v}(x)={\scriptstyle \frac{d}{d\epsilon}}{\phi}_{01}^{v+\epsilon h}(x)$. Simple computations, which can be, for example, found in [10], yields the fact that

$$\frac{d}{d\epsilon}{\int}_{{\mathrm{\Omega}}_{0}}{f}_{1}\circ {\phi}_{01}^{v+\epsilon h}(x)\mid {d}_{x}{\phi}_{01}^{v+\epsilon h}\mid dx={\int}_{{\mathrm{\Omega}}_{1}}\text{div}({f}_{1}u)dx$$

Now the conclusion is a direct consequence of Eq. (43) and of the divergence theorem.

Here again, one straighforwardly checks that the conservation of momentum is satisfied. We have in particular

$$(L{v}_{0},w)=\frac{1}{{\sigma}^{2}}{\int}_{\partial {\mathrm{\Omega}}_{1}}\left(\frac{1}{2}-{I}_{1}\right)\langle {({d}_{{\phi}_{10}^{v}}{\phi}_{01}^{v})}^{\ast}{\nu}_{1},\phantom{\rule{0.38889em}{0ex}}w\circ {\phi}_{10}^{v}(x)\rangle d{\sigma}_{1}(x)$$

Letting ${\mu}_{0}={\phi}_{01}^{v}({\sigma}_{1})$, and ${\nu}_{0}(x)={({d}_{x}{\phi}_{01}^{v})}^{\ast}{\nu}_{1}\circ {\phi}_{10}^{v}(x)$ this can be rewritten

$$(L{v}_{0},w)=\frac{1}{{\sigma}^{2}}{\int}_{\partial {\mathrm{\Omega}}_{0}}\left(\frac{1}{2}-{I}_{1}\circ {\phi}_{01}^{v}(x)\right)\langle {v}_{0},w\rangle d{\mu}_{0}(x)$$

which is under the general form of a measure-based momentum.

In this section, we let *I*_{0} and *I*_{1} be two smooth enough (say *C*^{1}) functions defined on Ω (images). We consider the image matching problem which corresponds to minimizing, over *G*,

$$U(\phi )=E(\phi )+\frac{1}{{\sigma}^{2}}{\int}_{\mathrm{\Omega}}\mid {I}_{0}\circ {\phi}^{-1}(x)-{I}_{1}(x){\mid}^{2}dx$$

This problem is equivalent to minimizing

$${\int}_{0}^{1}{\left|\right|{v}_{t}\left|\right|}_{L}^{2}dt+\frac{1}{{\sigma}^{2}}{\int}_{\mathrm{\Omega}}\mid {I}_{0}\circ {({\phi}_{01}^{v})}^{-1}(x)-{I}_{1}(x){\mid}^{2}dx$$

This matching problem has been studied, in particular in [4], to show that the optimal solution should satisfy, at each time *t*,

$$L{v}_{t}=-\frac{1}{{\sigma}^{2}}\mid d{\phi}_{t,1}^{v}\mid ({I}_{0t}^{v}-{I}_{1t}^{v})\nabla {I}_{0t}^{v}$$

(45)

in which we have introduced the notation:
${I}_{0t}^{v}={I}_{0}\circ {\phi}_{t0}^{v},{I}_{1t}^{v}={I}_{1}\circ {\mathrm{\Phi}}_{t1}^{v}$, and |*d*| for the Jacobian of . This equation is in fact an equation of conservation of momentum, with

$$L{v}_{0}=-\frac{1}{{\sigma}^{2}}\mid d{\phi}_{01}^{v}\mid ({I}_{0}-{I}_{10}^{v})\nabla {I}_{0}$$

as can be deduced from Eq. (30), with
$\alpha =-{\scriptstyle \frac{1}{{\sigma}^{2}}}\mid d{\phi}_{01}^{v}\mid ({I}_{0}-{I}_{10}^{v})$. Moreover, we can check also the normality property (31) which holds here with with *r* = 1 and *f*_{0} = *I*_{0}. This allows us to conclude that for the geodesic path in the image space generated by inexact matching, the lifting of the path in *G* defines a geodesic for which the momentum stays normal to the level sets of the current image *I*_{0} _{t}_{,0} at time *t*.

Thus far we have concentrated on the evolution of the flow of diffeomorphisms and its conservation of momentum. For all of our image understanding work we use the flow (* _{t}*,

Here we examine the finite dimensional landmark orbit denoted * _{n}*, consisting of

$$L{v}_{t}=\sum _{i=1}^{n}{\delta}_{{x}_{i}(t)}^{\ast}{a}_{i}(t),$$

(46)

where *a _{i}* (

$$\frac{d}{dt}({({d}_{{x}_{i}(t)}{\phi}_{t,0})}^{\ast})=-{d}_{{x}_{i}(t)}{v}_{t}^{\ast}{({d}_{{x}_{i}(t)}{\phi}_{t,0})}^{\ast},$$

(47)

we deduce that ${\scriptstyle \frac{d{a}_{i}}{dt}}(t)+{({d}_{{x}_{i}(t)}{v}_{t})}^{\ast}{a}_{i}(t)=0$. To prove (47), one needs to remark that

$${d}_{{x}_{i}(t)}{\phi}_{t,0}={d}_{{\phi}_{0,t}({x}_{i})}{\phi}_{t,0}={({d}_{{x}_{i}}{\phi}_{0,t})}^{-1}$$

Now,

$$\begin{array}{l}\frac{d}{dt}{({d}_{{x}_{i}}{\phi}_{0,t})}^{-1}=-{({d}_{{x}_{i}}{\phi}_{0,t})}^{-1}\frac{d}{dt}({d}_{{x}_{i}}{\phi}_{0,t}){({d}_{{x}_{i}}{\phi}_{0,t})}^{-1}\\ =-{({d}_{{x}_{i}}{\phi}_{0,t})}^{-1}({d}_{{\phi}_{0,t}({x}_{i})}{v}_{t}{d}_{{x}_{i}},{\phi}_{0,t})\times {({d}_{{x}_{i}}{\phi}_{0,t})}^{-1}\\ ={d}_{{x}_{i}(t)}{\phi}_{t,0}{d}_{{x}_{i}(t)}{v}_{t}\end{array}$$

Hence we get the following geodesic evolution in the orbit of landmarks.

The landmarks are transported along the geodesic according to the following equations with velocity vector field satisfying

$${v}_{t}={L}^{-1}\left(\sum _{i=1}^{n}{\delta}_{{x}_{i}(t)}^{\ast}{a}_{i}(t)\right)=\sum _{i=1}^{n}K({x}_{i}(t),\xb7){a}_{i}(t)$$

where K(x, y) is the Green kernel associated with L:^{3}

$$\{\begin{array}{l}\frac{d{a}_{i}}{dt}(t)+{({d}_{{x}_{i}(t)}{v}_{t})}^{\ast}{a}_{i}(t)=0,\hfill \\ \frac{d{x}_{i}}{dt}(t)-{v}_{t}({x}_{i}(t))=0.\hfill \end{array}$$

(49)

Note that the expression of *v _{t}* from Eq. (48) can be introduced into the system (49), yielding an evolution equation which only depends on the landmarks in the orbit.

We notice the reduction of the vector field to the range space of the Green’s kernels travelling over the landmark trajectories is as in [17].

There is a straightforward extension of this result to *geodesic curve evolution*, in which *x*(0) is a parametrized curve *σ* *x*(*σ*) for *σ* [0, *L*] and

$$L{v}_{0}={\int}_{0}^{L}{\delta}_{x(0,\sigma )}^{\ast}{\nu}_{0}(\sigma )d\sigma $$

where *ν*_{0}(*σ*) is normal to *x*(0). In this case, we have

$$L{v}_{t}={\int}_{0}^{L}{\delta}_{x(t,\sigma )}^{\ast}{\nu}_{t}(\sigma )d\sigma $$

with

- $\frac{\partial {\nu}_{t}}{\partial t}(t,\sigma )+{({d}_{x(t,\sigma )}{v}_{t})}^{\ast}\nu (t,\sigma )=0$,
- $\frac{\partial x}{\partial t}(t,\sigma )-{v}_{t}(x(t,\sigma )=0$.

Assume here that *dμ* = *α* (*x*)*dx* has a density with respect to Lebesgue’s measure on Ω. In this case, *dμ _{t}* = |

$$L{v}_{t}=\mid d{\phi}_{t0}\mid \alpha \circ {\phi}_{t0}\nabla {I}_{t}$$

(50)

From the conservation of momentum in Lagrangian coordinates for image based motion, we get for *Lv*_{0} = *α*_{0} *I*_{0} that *Lv _{t}* =

Since ${\scriptstyle \frac{d}{dt}}({z}_{t}\circ {\phi}_{0,t})={\scriptstyle \frac{d}{dt}}(\mid {d}_{\phi 0,t}{\phi}_{t,0}\mid {\alpha}_{0})=-\text{div}({v}_{t})\circ {\phi}_{0,t}\mid {d}_{\phi 0,t}{\phi}_{t,0}\mid {\alpha}_{0}$ we get ${\scriptstyle \frac{d{z}_{t}}{dt}}+d{z}_{t}({v}_{t})+\text{div}({v}_{t}){z}_{t}$. Moreover, we get easily ${\scriptstyle \frac{\partial {I}_{t}}{\partial t}}+\langle \nabla {I}_{t},{\nu}_{t}\rangle =0$. Hence we get the following geodesic evolution equation in image space.

The image is transported along the geodesic according to the following equations: with vector field v_{t} = L^{−1}(z_{t} I_{t}):

- $\frac{d{z}_{t}}{dt}+d{z}_{t}({v}_{t})+\text{div}({v}_{t}){z}_{t}=0$,
- $\frac{\partial {I}_{t}}{\partial t}+\langle \nabla {I}_{t},{v}_{t}\rangle =0$.

Notice that these equations appear as a limit case of the evolution equations which have been studied in [34] for image comparison.

As illustrated above, the pair (*I*_{0}, *μ*) provides a device for modeling deformations. In the cases we have studied, *I*_{0} was representing some geometrical structure (a curve, an image), which evolved with time accordi to the generated deformation, and *μ*, essentially quantifies the speed and direction of the deformation.

We get from this a natural way to represent the deformation of a template. Using Grenander’s original terminology, *I*_{0} would precisely be the template and *μ* is the generator of the deformation. Thus, fixing *I*_{0}, and letting *μ* vary, we get a model which represents perturbations of the template.

An example of deformations of an image is provided in Fig. 2. The images have been obtained by solving Eq. (50) from an initial image *g* of a slice of macaque brain, and taking

$$\alpha (x)=\frac{X(x)}{1+\mid {\nabla}_{x}{I}_{0}\mid}$$

where *X* is a Gaussian process.

The following results illustrate the computation of the momentum *Lv*_{0} = *α*_{0}*I*_{0} (as described in Section 6.2) from geodesic paths between two images. These geodesics are computed using F. Beg’s implementation of image matching, as described in [4]. In these experiments, the operator *L* is ( + *c* Identity),^{2} implemented via fast Fourier transform. The shooting algorithm solves the equation provided in Proposition 4 with initial condition (*I*_{0}, *z*_{0} = *α*_{0}).

Figure 3 shows the three objects studied, a smooth Gaussian bump for shift, circles for scale, and two mitochondria examin *g* both forward and inverse shooting.

Shown in Fig. 4–6 are examples illustrating the image based momentum and the diffeomorphisms generated via geodesic shooting. Figure 4 shows the results of the translation experiment. Panels 1 and 2 show *I*_{0} and *I*_{1}; panel 3 shows the diffeomorphism generated via geodesic shooting applied to *I*_{0}, and illustrates how solving the conservation of momentum equation allows to recover *I*_{1} from *I*_{0} and *Lv*_{0}.

Rows 1, 2: Results from translation experiment. Panels 1, 2, and 3 show *I*_{0}, *I*_{1}, *I*_{0} _{10}. Panels 4, 5 and 6 show *α*_{0}, *Lv*_{0} versus *α*_{0}*I*_{0}, and *V*_{0} and *L*^{−1}*α* *I*_{0}.

Results from mitochondria 2. Panels 1, 2, and 3 show *I*_{1}, *I*_{0}, *I*_{1} _{10}. Panels 4, 5 and 6 show *α*_{0}, *LV*_{0} versus *α*_{0} *I*_{0}, and *V*_{0}.

Shown in panel 4 is the density *α* showing the concentration near the boundary of *I*_{0}. Superposed in panel 5 are the predicted directions of the momentum, given by *α*_{0}*I*_{0}, and the value *Lv*_{0} obtained from Beg’s algorithm. In almost all case they appear as one line indicating a good accuracy of the algorithm. Panel 6 indicates that the vector field *V*_{0} demonstrating that while *α* and the momentum *Lv*_{0} are highly localized, the velocity of motion extends over the entire object.

Shown in Fig. 5 are similar results for the scale experiment.

Rows 1, 2: Results from scale experiment. Panels 1, 2, and 3 show *I*_{0}, *I*_{1}, *I*_{0} ; _{10}. Panels 4, 5 and 6 show *α*_{0}, *Lv*_{0} versus *α*_{0}*I*_{0}, and *V*_{0} and *L*^{−1}*α* *I*_{0}.

Shown in Fig. 6 are two sets of results for the geodesic shooting of the mitochondria. The organization of the results are the same as for the translation and scale experiments. Shown in Fig. 6 are two sets of results for the geodesic shooting of the mitochondria. The organization of the results are the same as for the translation and scale experiments.

Michael I. Miller was supported by grants from the National Institute of Wealth numbers P41-RR15241-01A1, U24-RR0211382-01, P20-MH071616-01.

Since
${\text{g}}_{{I}_{t}}^{\perp}$ is closed, we have to show that for almost all *t*, *v _{t}* =

The proof essentially consists in showing that, for all 0 ≤ *t* ≤ 1

$${I}_{0}\circ {\phi}_{t0}={I}_{0}\circ {\phi}_{t0}^{\epsilon}.$$

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Indeed, assume that this result is proved. Considering *ε* = 1 and *t* = 1, we deduce that
${I}_{1}={I}_{0}\circ {\phi}_{1,0}^{1}$. However, since *h _{t}*,

to *I*_{1}, we deduce that
${\int}_{0}^{1}\mid {h}_{t}{\mid}_{L}^{2}dt=0$ and the proof is ended.

We now return to Eq. (51). Using the formula

$$\frac{d{\phi}_{t0}(x)}{dt}=-{d}_{x}{\phi}_{t0}{v}_{t}(x)$$

and letting ${q}_{t}^{\epsilon}(x)={\phi}_{t0}\circ {\phi}_{0t}^{\epsilon}(x)$, we obtain

$$\begin{array}{l}\frac{\partial {q}_{t}^{\epsilon}(x)}{\partial t}=-{d}_{{\phi}_{0t}^{\epsilon}(x)}{\phi}_{t0}{v}_{t}\circ {\phi}_{0t}^{\epsilon}(x)+{d}_{{\phi}_{0t}^{\epsilon}(x)}{\phi}_{t0}({v}_{t}\circ {\phi}_{0t}^{\epsilon}(x)+\epsilon {h}_{t}\circ {\phi}_{0t}^{\epsilon}(x))\\ =\epsilon {d}_{{\phi}_{0t}^{\epsilon}(x)}{\phi}_{t0}ht\circ {\phi}_{0t}^{\epsilon}(x)\end{array}$$

We first prove Eq. (51) under the assumption that *I*_{0} is *C*^{1}. From the computation above, we have

$$\begin{array}{l}\frac{\partial}{\partial t}({I}_{0}\circ {q}_{t}^{\epsilon})(x)=\epsilon \langle {\nabla}_{{\phi}_{t0({\phi}_{0t}^{\epsilon}(x))}}{I}_{0},{d}_{{\phi}_{0t}^{\epsilon}(x)}{\phi}_{t0}{h}_{t}\circ {\phi}_{0t}^{\epsilon}{(x)\rangle}_{{\mathbb{R}}^{d}}\\ =\epsilon \langle {\nabla}_{{\phi}_{0t}^{\epsilon}(x)}{I}_{t},{h}_{t}({\phi}_{t0}^{\epsilon}(x))){\rangle}_{{\mathbb{R}}^{d}}=0\end{array}$$

since by definition of the projection *p _{It}* (

$${\langle {h}_{t}(x),\nabla {I}_{t}(x)\rangle}_{{\mathbb{R}}^{d}}={\langle {v}_{t}(x),\nabla {I}_{t}(x)\rangle}_{{\mathbb{R}}^{d}}-{\langle {p}_{{I}_{t}}({v}_{t}),\nabla {I}_{t}(x)\rangle}_{{\mathbb{R}}^{d}}=0.$$

This implies
${I}_{0}\circ {\phi}_{t,0}\circ {\phi}_{0,t}^{\epsilon}={I}_{0}$ which yields Eq. (51) in this case. When *I*_{0} is not smooth, the proof goes by showing that

$$\frac{\partial}{\partial t}{\int}_{\mathrm{\Omega}}{I}_{0}\circ {q}_{t}^{\epsilon}(x)f(x)dx={\int}_{\mathrm{\Omega}}{I}_{t}(x)\text{div}({h}_{t}{g}_{t}^{\epsilon})(x)dx$$

for smooth *f* on Ω and
${g}_{t}^{\epsilon}\circ {\phi}_{t}^{\epsilon}\mid d{\phi}_{t}^{\epsilon}\mid =f$ which can be done either by a direct (heavy) computation, or by using a density argument, based on the fact that, by the divergence theorem, this is true for smooth *I*_{0} (we skip the details).

^{1}These arguments are purely formal since *h _{t}* includes the Lie bracket which cannot be guaranteed to belong to (in which case our variation would not be justified).

^{2}When *I* has smooth level sets, we say that a vector field *ν* is normal to its level sets when, denoting by Ω* _{i}* the set {

^{3}The explicit form for *L*^{−1} depending on the kernel *K* is defined as follows. For any *x*, *y* Ω, the bilinear form *K _{x,y}* on

$${K}_{xy}(a,b)\doteq {\langle {\delta}_{x}^{\ast}a,{\delta}_{y}^{\ast}b\rangle}_{{\text{g}}^{\ast}}=({\delta}_{y}^{\ast}b,{L}^{-1}({\delta}_{x}^{\ast}a))={\langle {L}^{-1}({\delta}_{x}^{\ast}a)(y),b\rangle}_{{\mathbb{R}}^{d}}$$

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MICHAEL I. MILLER, Center of Imaging Science & Department of Biomedical Engineering, The John Hopkins University, 301 Clark Hall, Baltimore, MD 21218, USA.

ALAIN TROUVÉ, CMLA, ENS de Cachan, 61 Avenue du Président Wilson, 94235 Cachan CEDEX, France.

LAURENT YOUNES, Center for Imaging Science & Department of Applied Math and Statistics, The Johns Hopkin, 3245 Clark Hall, Baltimore, MD 21218, USA.

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