GEODESIC BOUNDARY VALUE PROBLEMS WITH SYMMETRY. Colin J. Cotter. Darryl D. Holm. (Communicated by the associate editor name)

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1 Manuscript submitte to AIMS Journals Volume X, Number 0X, XX 200X Website: pp. X XX GEODESIC BOUNDARY VALUE PROBLEMS WITH SYMMETRY Colin J. Cotter Department of Aeronautics Imperial College Lonon Lonon SW7 2AZ, Unite Kingom Darryl D. Holm Department of Mathematics Imperial College Lonon Lonon SW7 2AZ, Unite Kingom (Communicate by the associate eitor name) Abstract. This paper shows how commuting left an right actions of Lie groups on a manifol may be use to complement one another in a variational reformulation of optimal control problems as geoesic bounary value problems with symmetry. In such problems, the enpoint bounary conition is only specifie up to the right action of a symmetry group. In this paper we show how to reformulate the problem by introucing extra egrees of freeom so that the enpoint conition specifies a single point on the manifol. We prove an equivalence theorem to this effect an illustrate it with several examples. In finite-imensions, we iscuss geoesic flows on the Lie groups SO(3) an SE(3) uner the left an right actions of their respective Lie algebras. In an infinite-imensional example, we iscuss optimal large-eformation matching of one close oriente curve to another embee in the same plane. In the curve-matching example, the manifol Emb(S 1, R 2 ) comprises the space S 1 embee in the plane R 2. The iffeomorphic left action Diff(R 2 ) eforms the curve by a smooth invertible time-epenent transformation of the coorinate system in which it is embee, while leaving the parameterisation of the curve invariant. The iffeomorphic right action Diff(S 1 ) correspons to a smooth invertible reparameterisation of the S 1 omain coorinates of the curve. As we show, this right action unlocks an important egree of freeom for geoesically matching the curve shapes using an equivalent fixe bounary value problem, without being constraine to match corresponing points along the template an target curves at the enpoint in time. 1. Introuction. In this paper we are concerne with fining geoesics between points on manifols. The construction of geoesics is useful for stuying problems on manifols since they can escribe the relationship between two points. Within a coorinate patch on a manifol, any point can escribe relative to a reference point by specifying a irection an a length along the geoesic in that irection. This becomes useful for performing statistics on the coorinate patch, for example. In this paper we consier problems in which the enpoint of the trajectory is only fixe up to the orbit of a Lie group. In low imensional cases (an we shall escribe 2000 Mathematics Subject Classification. Primary: 70H33, 58E50; Seconary: 92C55. Key wors an phrases. Shape space, geoesics on manifols, symmetry reuction. DH is partially supporte by Royal Society of Lonon Wolfson Awar. 1

2 2 COLIN J. COTTER AND DARRYL D. HOLM some examples of these) it is often easy to solve these problems by constructing reuce coorinates which o not change uner the action of the Lie group. However, in many cases it is ifficult to construct such coorinates, especially if the problem is to be iscretise an solve numerically. In this paper we provie a framework that allows one to work with full unreuce coorinates on the manifol, by transforming to an equivalent problem which has the enpoint of the trajectory fixe exactly. There are many examples of problems where this framework can be applie, but we are motivate by the problem of obtaining iffeomorphisms on R 2 which map one embee oriente curve Γ A into another embee oriente curve Γ B (with the same orientation), an which minimise a given metric so that they are geoesics in the iffeomorphism group. The aim is to fin a characterisation of curve Γ B with respect to curve Γ A that is inepenent of parameterisations of the curves. This means that we o not specify a priori the point on Γ A which gets matche to each specific point on Γ B, an so the minimisation is performe over all parameterisations of the curves. In practise the computation is performe using a particular parameterise curve q Emb(S, R 2 ) (where S is the embee space, for example, the circle for simple close curves). In computing the equations of motion, a conjugate momentum p q T q Emb(S, R 2 ) is constructe, an the flow taking the initial curve Γ A to the final curve Γ B can be characterise entirely by the initial conitions p q t=0 for the conjugate momentum. In fact, it turns out that p q t=0 is normal to the curve, so the flow can be characterise by a one-imensional signal. Since T q Emb(S, R 2 ) is a linear space, linear statistics can be compute on p q t=0. For example, this may allow one to test the hypothesis that there is a statistical correlation between between the shape of the surface of a biological organ, obtaine from a meical scan, an future evelopment of isease. To iscuss the issues further, we formulate the curve matching problem escribe above, which may be regare as an control problem in the sense of the problems iscusse in [?,?,?]: Definition 1.1 (Curve matching problem). Let q(s; t) be a one-parameter family of parameterise simple close curves in R 2, with s [0, 1] being the curve parameter an t [0, 1] being the parameter for the family. Let u(x; t) be a one-parameter family of vector fiels on R 2. Let η be a iffeomorphism of S 1. We seek q an u which satisfy 1 min u,η 0 2 u 2 V t subject to the constraints [Reconstruction relation] q(s; t) = u(q(s; t), t), (1) t Initial state (Template)] q(s; 0) = q A (s), (2) [Final state (Target)] q(s; 1) = q B (η(s)), (3) where V is the chosen norm which efines the space of vector fiels V. The solution of this problem escribes a geoesic in the iffeomorphism group which takes the simple close curve Γ A parameterise by q A to the simple close curve Γ B parameterise by q B. We represent the shapes of simple close curves as

3 GEODESIC BOUNDARY VALUE PROBLEMS WITH SYMMETRY 3 elements of Emb(S 1, R 2 )/ Diff + (S 1 ), where Diff(S 1 ) is the group of iffeomorphisms of S 1 an the subscript + inicates the proper subgroup of Diff(S 1 ) that is connecte to the ientity (i.e. the orientation preserving iffeomorphisms). However, we o not want to calculate on this space; instea, we want to calculate on the full space Emb(S 1, R 2 ) by minimising over all reparameterisations η(s) Diff + (S 1 ). We require that q A an q B have the same topology, i.e. that there exists a iffeomorphism g Diff + (R 2 ) (with smoothness prescribe by the metric V ) such that q A = g q B. In particular, if we restrict to simple close curves, this requires that q A an q B both traverse the curves in the same irection (i.e. both clockwise or both anti-clockwise). There are two general strategies for solving such problems. The first strategy, use for example in [?], is to use a graient metho (i.e. a moification of the steepest escent metho such as the nonlinear conjugate graient metho [?, an references therein]) to minimise the action integral over paths q(s, t) which satisfy the ynamical constraint (this constraint was enforce softly via a penalty term in [?]). An alternative metho, referre to in [?] as the Hamiltonian metho, is to introuce Lagrange multipliers p(s, t) which enforce the ynamical constraint, an to erive Hamilton s canonical equations for q an p q, following the general erivation escribe in [?, for example]. Minimisation over the reparameterisation η, together with a conservation law obtaine from Noether s theorem, results in the conition that the tangential component of p q vanishes. The aim of the Hamiltonian metho is to turn an optimisation problem into an algebraic equation given by the time-1 flow map of Hamilton s canonical equations. One then solves a shooting problem to fin initial conitions for the normal component p q which generate solutions to Hamilton s equations that satisfy the bounary conition (3). The ifficulty in solving this problem numerically lies in fining a goo numerical iscretisation of the target constraint conition (3). Various functionals have been propose which vanish when the constraint conition is satisfie. In [?] a functional was propose base on singular ensities (measures), an in [?] a functional was propose base on singular vector fiels (currents). An alternative spatial iscretisation for the current functional base on particle-mesh methos was propose in [?]. There are several ifficulties with these functionals: one is that after numerical iscretisation the functionals o not vanish at the minima, an the bounary conition must be replace by a functional minimising conition. It is also ifficult to express the probability istribution of the functional given the istribution of measurement errors; this is important for statistical moelling. In this paper we consier a transformation of problems of the above type, which results in an alternative formulation that removes the reparameterisation variable η from the target constraint, thereby resulting in a stanar two point bounary value problem on T Emb(S, R 2 ) (with a constraint on the initial conitions plus an aitional parameter). This transformation can be applie to a very general class of problems; so we present it in the general case of Lie group actions on a manifol. The rest of this paper is organise as follows. In Section 2, we formulate the optimal control problem, then transform to the geoesic problem with symmetry an prove that the two problems are equivalent. In Section 3 we give some examples an iscuss the application to matching curves an surfaces. Section 4 is the summary an outlook.

4 4 COLIN J. COTTER AND DARRYL D. HOLM 2. Reparameterise geoesic bounary value problems with symmetry. In this section we escribe a general framework for geoesic bounary value problems with symmetry. We efine the following Optimal Control Problem. Definition 2.1 (Geoesic bounary value problem with symmetry). Let Q be a manifol, let G be a Lie group acting on Q from the left, an let H be a (possibly ifferent) Lie group acting on Q from the right that commutes with the left action of G on Q, with corresponing Lie algebras g an h, an corresponing Lie algebra actions X G an X H respectively. Furthermore, let A : g g be a positive-efinite self-ajoint operator an let, g : g g R be a nonegenerate pairing which efines an inner prouct on g. We seek a one parameter family q of points on Q parameterise by t [0, 1], a one parameter family ξ of elements of g for t [0, 1], an η H, which minimise subject to the constraints ξ, Aξ g t, [Reconstruction relation] t q = X ξ G q, (4) [Initial state (Template)] q t=0 = q A, (5) [Final state (Target)] q t=1 = R η q B, (6) where q A, q B are chosen points on Q relate by some element g of G so that q A = gq b, an where R η is the right-action of η on Q. This problem is an example of a Clebsch optimal control problem, as stuie from a geometric point of view in [?]. In this case we seek the shortest path in Q from q A to any point q B η, η H. This means we are seeking the shortest path in Q/H, but are performing the computation on Q. In many cases it is much easier to compute on Q, for example when Q is a vector space. We refer to this process of solving a problem on Q/H by calculating on Q as un-reuction. One approach to solving this problem is to erive equations of motion for q, ξ an an optimal conition for η an then solving a shooting problem to fin η an the initial conitions for ξ which allow equation (6) to be satisfie. We can erive the equations of motion by enforcing the reconstruction relation (4) as a constraint using Lagrange multipliers p q T q Q. This approach leas to the following variational principle. Definition 2.2 (Variational principle for geoesic bounary value problem with symmetry). We seek (p, q) an ξ g for t [0, 1], an η H, which satisfy 1 δs = δ 0 2 ξ, Aξ g + p q, t q X ξ G q t = 0, (7) subject to where we allow p q, q, ξ an η to vary. q t=0 = q A, q t=1 = R η q B, (8) From this variational principle we can erive the equations of motion, which can be use in solving the shooting problem. Before we o this, we recall the efinition of the cotangent-lifte momentum map:

5 GEODESIC BOUNDARY VALUE PROBLEMS WITH SYMMETRY 5 Definition 2.3. Given an action of a Lie algebra g on Q, the cotangent-lifte momentum map J : g is efine from the formula J(p q ), ζ g = p q, X ζ q (9) for all ζ g. Since we have two Lie algebra actions, we shall write J G for the cotangent-lifte momentum map corresponing to the left action X G of g on Q, an J H for the cotangent-lifte momentum map corresponing to the right action X H of h on Q. Lemma 2.4 (Equations of motion for geoesic problem). At the optimum, the following equations are satisfie (weakly, for appropriate pairings): Furthermore, Proof. The proof is a irect calculation. δs = δξ, Aξ g + δp q, q t = 0 0 t q X ξ G q = 0, (10) t p q + ( ( T q X G ξ q )) pq = 0, (11) Aξ J G (p q ) = 0. (12) δξ, Aξ J G (p q ) g + J H (p q ) t=1 = 0. (13) δp q, q pq t + ( ( T q X G ξ q )) pq, δq + p q, δ q t X ξq t δ(x ξq) t + t [ p q, δq Since δp, δq an δξ are all arbitrary, stationarity δs = 0 implies equations (10-12) an their appropriate pairings. The bounary term becomes [ ] 1 p, δq = p, T η (R η q) δη = p, Xγ H q t=1 = J t=0 t=1 H (p q ), γ h, t=1 where γ is the generator of δη, i.e. δη = exp(ɛγ)η. Hence, we also obtain ɛ=0 equation (13) an its appropriate pairing. Lemma 2.4 reformulates the geoesic calculation as a shooting problem in which one seeks initial conitions for p q such that q t=1 = R η q B where η is fixe by the conition (13) which arises from minimising over η an ensures we have the shortest path over Q/H. Next we show conservation of the momentum map J H ; this will enable us to transfer conition (13) from t = 1 to t = 0. Lemma 2.5 (Vanishing momentum). The system of equations (11-13) has conserve momentum which satisfies J H (p q ) = 0 for all 0 t 1. Proof. The problem in Definition 2.2 is invariant uner transformations q R α q, α H, which are generate by γ h, an hence the corresponing momentum map J H (p q ) is conserve, by Noether s theorem. Since it vanishes for t = 1, by equation (13), it must vanish for all times. ɛ ] 1 t=0.

6 6 COLIN J. COTTER AND DARRYL D. HOLM Lemma 2.5 states that solutions of the optimal control problem all have vanishing right action momentum map J H (p q ) = 0. This is what facilitates the unreuction. Namely, we can compute on Q instea of Q/H by keeping J H (p q ) = 0. To obtain the shortest path between two points in Q/H by solving in Q, select a point q Q which is a member of the equivalence class which is the initial point in Q/H, an fin initial conitions for p q such that J H (p q ) = 0; so that the solution to equations (10-12) satisfies q t=1 = R η q for some η H. Computationally, there are reasons why solving the problem in this form may be ifficult. In Section 3.3, we shall escribe how the ifficulty arises for the curve matching problem specifie in the Introuction. In this paper, we shall introuce a reformulation of the problem for which there is a single fixe value of q t=1. Before introucing the reformulation, we briefly iscuss the reuce equation for the Lie algebra variable ξ g. The latter is the Euler-Poincaré equation for Hamilton s principle with Lagrangian given by the energy ξ, Aξ g /2, where A : g g is the positive-efinite self-ajoint operator in Definition 2.1 of the geoesic matching problem. Lemma 2.6 (Reuce equation for geoesic problem). The Lie algebra variable ξ for the geoesic matching problem state in Definition 2.1 satisfies t γ, Aξ + a ξ Aξ g = 0, γ g, (14) where we efine the operation a : g g g an its ual a : g g g as a ω γ = [ω, γ] = ωγ γω, a ω µ, γ g = µ, a ω γ g, ω, γ g, Proof. The proof follows from irect computation of the time-erivative of γ, Aξ g an substitution of equations (10-12). See for example, [?,?]. We will next efine a moification of the problem state in Definition 2.1, which has the avantage that the enpoint conitions o not contain a free reparameterisation variable. This reformulation is more amenable when solving the curve matching problem numerically, for example. We shall procee to show that solutions of the moifie problem can be transforme into solutions of the problem state in Definition 2.1. Definition 2.7 (Reparameterise geoesic problem with symmetry). Let Q be a manifol, let G be a Lie group acting on Q from the left, an let H be a (possibly ifferent) Lie group acting on Q from the right that commutes with the left action of G, with corresponing Lie algebras g an h, an corresponing Lie algebra actions X G an X H respectively. Furthermore, let A : g g be a positive-efinite selfajoint operator. We seek a one parameter family q of points on Q for t [0, 1], a one parameter family ξ of elements of g for t [0, 1], an ν h, which minimise ξ, Aξ g t, where, g is the usual inner prouct on g, subject to the constraints [Reconstruction relation] t q = X ξ G q + Xν H q, (15) [Initial state (Template)] q t=0 = q A, (16) an q A, q B are chosen points on Q. [Final state (Target)] q t=1 = q B, (17)

7 GEODESIC BOUNDARY VALUE PROBLEMS WITH SYMMETRY 7 Note that in this moifie efinition, we o specify the final bounary conition for q without allowing arbitrary symmetry transformations using H. However, we also introuce an aitional variable ν which moves q in the irection of symmetries generate by h. We shall erive the equations of motion associate with this moifie problem, an the associate conservation laws. These will lea us to conclue that it possible to construct solutions of the problem in Definition 2.1 out of solutions of the problem in Definition 2.7, an the latter can be solve as a shooting problem in which the bounary conitions are explicitly specifie, rather than as an algebraic conition. As before, we can erive the equations of motion for q, ξ an the conition for ν by enforcing the reconstruction relation (15) as a constraint using Lagrange multipliers p q T q Q. We then use variational calculus to obtain the equations of motion, as escribe in the following lemma. Lemma 2.8 (Equations of motion for reparameterise geoesic problem). At the optimum, the following equations are satisfie in the sense of appropriate pairings: Furthermore, t q X ξ G q Xν H q = 0, (18) t p ( q + T q X G ξ q + Xν H q ) pq = 0, (19) Aξ J G (p q ) = 0. (20) t=0 Proof. Similar to the proof of Lemma 2.4. J H (p q ) t = 0. (21) Proceeing as before, we can transform (21) into an initial conition by making use of the conservation of the right-action momentum map, J H. Lemma 2.9 (Vanishing momentum for reparameterise geoesic problem). The system of equations (19-20) has conserve momentum J H = 0. Proof. The problem in Definition 2.7 is invariant uner transformations q qα, α H, which are generate by γ = h, an hence J H is conserve by Noether s theorem. Lemma 2.8 gives 0 = 0 J Ht = J H. Next we note that ξ obtaine from Definition 2.7 satisfies the same reuce Euler-Poincaré equation as ξ obtaine from Definition 2.1. Lemma 2.10 (Reuce equation for geoesic problem). The Lie algebra variable ξ obeys equation (14) weakly, i.e., for an appropriate pairing. Proof. Similar to the proof of Lemma 14, noting that the actions X G an X H commute. This means that we can show that the two problems prouce equivalent solutions provie that the initial conitions for ξ are the same in both cases. The following theorem establishes this result. Theorem Let q, p q, ν, ξ be obtaine from the solution of equations (18-21), an efine ψ = exp(νt) H.

8 8 COLIN J. COTTER AND DARRYL D. HOLM Then the transforme variables constructe from q = R ψ 1q, p q = T q ψ ( p q ), ξ = ξ, t [0, 1], (22) ( i.e. the cotangent lift of ψ) satisfy equations (10-12) together with the bounary conitions (5,6,13), with η = ψt=1 1. Hence, q, η an ξ form a (local) extremum for the problem in Definition 2.2. Proof. First we take ξ an ξ from equations (12) an (20) respectively, an show that ξ = ξ. Since the left an right actions commute, we have that γ, Aξ g = p q, X G γ q T M = p q, X G γ R ψ q T M = p q, T q R ψ X G γ q T M = T q R ψ (p q ), X G γ q T M = p q, X G γ q T M = γ, Aξ g, an hence ξ = ξ. Next we verify the equations for q an p q. Taking the time erivative of q, we have an so t q = t (R ψq) = T q R ψ t q + X ν H (R ψ q), ( ) t q = (T qr ψ ) 1 t q X ν H q = (T q R ψ ) 1 X G q = X G q = X G ξ ξ ξ q, as require. To check the time evolution equation for p q, we take the inner prouct with an arbitrary tangent vector v T q Q, to fin: t p q, v = T t q R ψ (p q ), v = pq, (T q R ψ ) v t = t p q, (T q R ψ ) v + p q, t (T qr ψ ) v ( ) = Tq X G q + X H ξ ν q p q, (T q R ψ ) v + ( p q, T q X H ν q ) (T q R ψ ) v ) = p q, T q (X Gξ q (T q R ψ ) v ) = Tq R ψ (p q ), T q (X Gξ q v ) = p q, T q (X Gξ q v T T ( ) = X Gξ q Q p q, v, as require. It remains to check the bounary conitions. Trivially, q t=0 = q A, q t=1 = q t=1 ψ 1 t=1 = qb η, as require. Finally, we nee to check the en conition (13). From Corollary 2.9, we have J H (p q ) t=0 = J H (p q ) t=0 = 0, an Lemma 2.5 implies that J H (p q ) t=1 = 0. Hence the bounary conitions are satisfie. 3. Examples. In this section we escribe examples of the reparameterise geoesic problem with symmetry, an iscuss its applications to the characterisation of the shape of curves an surfaces.

9 GEODESIC BOUNDARY VALUE PROBLEMS WITH SYMMETRY Example: SO(3). We illustrate our results with the case of the action of SO(3) on itself which gives rise to the equations of a rotating rigi boy. We consier the problem in which the en point bounary conition is only etermine up to a rotation of the rigi boy about its z-axis. Of course, this problem can also be solve by picking reuce coorinates, but we use it as here as a simple example. Definition 3.1 (Optimal control of a symmetric rigi boy). For a given symmetric matrix I, we seek Q(t) which is a one-parameter family of matrices SO(3), ω(t) which is a one-parameter family of matrices in so(3), an R θ which is a rotation in the z-axis by an angle θ, which satisfy subject to the constraints min ω,θ 0 1 ω, Iω 2 so(3) so(3) t [Reconstruction relation] Q(t) = ω(t)q(t), (23) [Initial (Template)] Q(0) = Q 0, (24) [Final (Target)] Q(1) = Q 1 R θ. (25) This problem is an example of the optimal control problem in Definition 2.1, with the manifol Q being SO(3), the group G being SO(3) acting from the left, an the group H being SO(2) acting from the right. We ientify q Q, ω ξ, R θ η, A I an p q P, where P T SO(3) is the conjugate momentum to Q. Application of Lemmas 2.4 an 2.5 gives the following equations: Q ωq = 0, P + ω T P = 0, Iω (P QT QP T ) = 0, Tr ( P T Qw ) = 0, w = 1 0 0, (26) which are the equations for a rotating rigi boy with vanishing z-component of the angular momentum. For this problem, obtaining a solution is simple, since one can efine coorinates on T (SO(3)), an remove the coorinates associate with the R θ irection an the corresponing vanishing conserve momentum, an solve a two-part bounary problem for the remaining coorinates. However, we wish to evelop a methoology for numerical iscretisations of infinite-imensional problems where it is less clear how to o this. Hence, we efine the following formulation which makes use of a time-varying relabelling transformation in the R θ irection. Theorem 2.11 states that to obtain solutions to equations (26), we can alternatively solve the following moifie problem: Definition 3.2 (Reparameterise optimal control of a symmetric rigi boy). We seek Q(t) which is a one-parameter family of matrices in SO(3), ω(t) which is a one-parameter family of matrices in so(3) an ν which is a generator of rotations about the z-axis, which satisfy min ω,ν 0 1 ω, Iω 2 t so(3) so(3)

10 10 COLIN J. COTTER AND DARRYL D. HOLM subject to the constraints [Reconstruction relation] Q(t) = ω(t)q(t) + Q(t)ν, (27) [Initial (Template)] Q(0) = Q 0, (28) where I is a chosen symmetric matrix. [Final (Target)] Q(1) = Q 1, (29) Lemmas 2.8 an 2.9 states that the solution to this problem satisfies the following equations: Q ωq + Qν = 0, P + ω T P P ν T = 0, Iω (P QT Q P T ) = 0, Tr ( P T Qw ) = 0. (30) Hence, to obtain a solution to equations (26) with bounary conitions (24) an (25) we solve the two-point bounary value problem given by equations (30) with bounary conitions (28-29). This can be formulate as a shooting problem, in which we seek ν an initial conitions for P with vanishing z-component of angular momentum, such that the en point bounary conition (29) is satisfie. We then construct the reparameterisation matrix R(t) from R(t) = exp(νt), an use equation (22) to reconstruct the solution in the form: Q(t) = Q(t)R(t) an P (t) = Q(t)R T (t), implying ω(t) = ω(t) Example: SE(3). We next escribe the example of the action of SE(3) on itself from the left, with SO(2) acting from the right. This example coul escribe a ocking problem of a spacecraft onto a space station. The spacecraft can apply torque to rotate about a central point, or can apply thrust to move itself in the irection in which it is pointing, an we wish to ock the spacecraft using minimal energy. In the language of image registration, this is known as rigi registration. We consier the problem in which the en point bounary conition is only etermine up to a rotation of the rigi boy about its z-axis. In the spacecraft analogy, this correspons to a ocking proceure which oes not require the spacecraft to have any particular orientation about the z-axis when ocking. As in the previous example, this problem can also be solve by picking reuce coorinates, but it serves as a prototype for infinite imensional problems. Following the notation of [?], we represent the elements q SE(3), p q Tq SE(3), ξ se(3) as 4 4 matrices q ( Q r 0 1 ), p q ( ) P p, ξ 0 0 ( ) ω v, 0 0 where ω is an antisymmetric matrix, an v R 3. The energy cost (metric functional) an reconstruction relation are then given by S = E t = 1 ( ) ξ, Aξ t=0 2 se(3) t, q = Xξ G ωq ωr + v q = ξq =, t=0 0 0 ( ) I b A = b T, b R 3, c R. c The start an en conitions for the orientation an poistioin are specifie as ( ) ( ) Q q t=0 = A r A Q, q 0 1 t=1 = B R θ r B, 0 1

11 GEODESIC BOUNDARY VALUE PROBLEMS WITH SYMMETRY 11 where R θ is a rotation through any angle θ about the z-axis. If we solve the problem in Definition 2.1, then Lemmas 2.4 an 2.5 give the equations Q = ωq, ṙ = ωr + v, P = ω T P, ṗ = ω T p, δe δω (P QT QP T δe ) = 0, δv + p = 0, Tr ( P T Qw ) = 0. (31) where w is as efine in equation (26). From Lemma 2.5, this quantity vanishes for all t. Theorem 2.11 then states that a solution to these equations can be obtaine by solving the following reparameterise equations: Q = ωq + αqw, ṙ = ω r + v, Ṗ = ωt P αqw T, ṗ = ω T p, δe δω (P QT Q P T ) = 0, δe δv + p = 0, (32) with Tr ( P T Qw ) = 0, α R, Q t=0 = Q A, r t=0 = r A, Q t=1 = Q B, r t=1 = r B. This gives a two-point bounary value problem with a constraint on the initial conitions plus an extra parameter, which can be solve as a shooting problem by fining α an P (subject to the angular momentum constraint) such that Q an r reach their target values Q B an r B. A solution to the problem in Definition 2.1 can then be reconstructe by efining R(t) = exp(αwt), an using the following formulae: Q(t) = Q(t)R(t), P (t) = P (t)r T (t), r(t) = r(t), p(t) = p(t), ω(t) = ω(t), v(t) = v(t) Curve matching. In this section we return to the problem escribe in Definition 1.1, an iscuss a number of practical issues which are aresse by the formulation iscusse in this paper. The aim of solving the problem is to fin a characterisation of the simple close curve Γ B in terms of the reference simple close curve Γ A, together with a scalar perioic function p(s) which specifies the initial conitions for the normal component of the conjugate momentum p(s; t). In this case, Q is the space Emb(S 1, R 2 ) of functions q : S 1 R 2, G is the group Diff + (R 2 ) of iffeomorphisms of R 2 which acts on Q from the left: Φ L (g, q)(s) = g(q(s)), s S 1, an H is the group Diff + (S 1 ) of iffeomorphisms of S 1 which acts on Q from the right: Φ R (g, q)(s) = q(g(s)), s S 1. The left an right actions can be expresse succinctly as GQ = Emb(S 1, G R 2 ), HQ = Emb(H S 1, R 2 ). It is clear from this that the actions of G an H commute with each other. Lemma 2.4 gives the ynamical equations q(s; t) = u(q(s; t), t), t w, Au(, t) X = t p(s; t) = ( u(q(s; t), t))t p(s; t), p(s; t) w(q(s; t), t) s, S 1 where, X is the inner prouct on the vector fiels X(Ω) associate with the norm X, for any test function w X. This equation for u has the weak solution Au = p(t, s)δ(x q(t, s)), (33) which is the singular-solution momentum map, J Sing iscusse in [?]. The en conition is p(s; 1) s q(s; 1) = 0, s S1, (34)

12 12 COLIN J. COTTER AND DARRYL D. HOLM an Lemma 2.5 states that this conserve momentum vanishes for all t. This correspons to p(s; t) being normal to the curve parameterise by q(s; t). Hence, to fin geoesics between Γ A an Γ B, we solve a shooting problem an seek initial conitions p(s; 0) with vanishing tangential component, which fix q(s; 1) = q B (η(s)) for some η Diff + (S 1 ). Having solve this problem, one can escribe Γ B entirely in terms of p(s) = p(s; 0) n(s) where n is the normal to Γ A. The solution to the problem also provies the istance between the two curves. There are a number of ifficulties with solving such a shooting problem numerically. The parameterisation of the curve must necessarily be iscretise numerically, typically by a list of points, as in [?,?], which can be obtaine from a piecewiseconstant representation of q as a function of s [?], or as piecewise linear geometric currents [?]. Having taken the iscretisation, the reparameterisation symmetry is broken (although a remnant of it is left behin, as escribe in [?]) which means that it is ifficult to obtain a iscrete form of the en conition q(s; 1) = q B (η(s)). As escribe in the Introuction, this problem has been approache by proposing various functionals which are minimise when q(s; 1) an q(s) overlap the most. However, these functionals prouce quite a complicate lanscape with local minima, an the case of stuying large eformations we have foun that they can result in o artifacts in the shooting process. Also, the changes in these functionals with respect to measurement error are quite technical to quantify which makes statistical inference more complicate. Another ifficulty is that of aaptivity. As illustrate in Figure 1, constraining p to be normal to the curve means that any local large eformations give rise to large amounts of stretching which then results in loss of accuracy in the approximation of the functional use to enforce the en conition for q. One possible way to avoi this is to aaptively refine the gri point ensity in the initial curve uring the shooting process. These ifficulties are all remove if, instea, one solves the following problem: Definition 3.3 (Reparameterise curve matching problem). Let q(s; t) be a oneparameter family of parameterise curves in R 2, with s [0, 1] being the curve parameter an t [0, 1] being the parameter for the family. Let u(x; t) be a oneparameter family of vector fiels on R 2. Let ν be a vector fiel on S 1. We seek q, u an ν which satisfy 1 min u,ν 0 2 u 2 V t subject to the constraints t q(s; t) = u(q(s; t), t) + ν(s) s q(s; t), q(s; 0) = qa (s), q(s; 1) = q B (s), where V is the chosen norm which efines the space of vector fiels V. Lemmas 2.8 an 2.9 state that extrema of this problem can be obtaine by solving the shooting problem t q(s; t) = u(q(s; t), t) + ν(s) q(s; t), s t p(s; t) = p(s; t) u(q(s; t), t) (ν(s)p(s; t)), s w, u(, t) V = p(s; t) w(q(s; t), t) s, w V, S 1

13 GEODESIC BOUNDARY VALUE PROBLEMS WITH SYMMETRY 13 Figure 1. Figure illustrating the way in which eformation takes place in the parameterisation-inepenent geoesic equations for embee curves. The initial curve is shown on the left, an the final curve is shown on the right. Since the momentum is constraine to be normal to the curve, an since the velocity is obtaine by applying a smoothing kernel to the momentum, the change in the shape emerges as local stretching of the curve, an the iscrete points efining the shape become separate. with bounary conitions q(s; 0) = q A (s), p(s; 0) s qa = 0, q(s; 1) = q B (s). The aim is to fin ν(s) an normal components of p(s; 0) such that these bounary conitions are satisfie. Note that in this moifie problem, the bounary conition for q(s; 1) is specifie pointwise (i.e. there is no reparameterisation variable η in the bounary conition). This means that the error can be escribe irectly in terms of q( ; 1) q B 2 for some chosen norm, which can be iscusse in terms of measurement errors irectly. Theorem 2.11 then states that a solution to the problem escribe in Definition 1.1 can be reconstructe via q(s; t) = q(η(s; t), t), p(s; t) = p(η(s; t), t) s η(s; t), η(s; t) = exp(ν(s)t). This transformation prouces a equivalent shooting problem in which the en conition for q is now fixe. See figures 2 an 3 for an example of a solution to this problem. The numerical iscretisation an solver methos will be benchmarke an analyse in a future publication.

14 14 COLIN J. COTTER AND DARRYL D. HOLM Figure 2. Results from a matching algorithm base on the reparameterise equations. The aim is to fin the initial conitions for p (with vanishing tangential component) an reparameterisation vector fiel ν which maps the initial curve (in this case a circle) onto a set of points observe from another curve. The variables p n an ν are obtaine by minimising a functional consisting of the Eucliean istance between certain points on the final curve plus a regularising function which enforces smoothness of p an ν namely the L 2 norm of p n an the H 3 norm of ν. The reparameterise equations are solve using a splitting metho in which the left-action part is iscretise using the particle-mesh metho escribe in [?], an the reparameterisation is iscretise using a semi-lagrangian metho. The isplaye plots are: (Top-Left) the template curve an the final curve together with the points from the target curve, (Top-Right) the momentum co-vectors plotte on the template curve, (Bottom-Left) the vector fiel on S 1 use to generate the reparameterisation, an (Bottom-Right) the final reparameterisation η plotte against the ientity map for reference. The numerical solution was obtaine in about three minutes after about 300 iterations of the iterative solver, base on the BFGS algorithm. 4. Summary an outlook. In this paper, we stuie an optimal control problem on a Lie group in which the en bounary conition is specifie only up to a symmetry. We showe how this problem can be transforme into a moifie problem in which the en bounary conition is fixe, but an extra parameter is introuce in the reconstruction relation, an prove that the two problems are equivalent. This

15 GEODESIC BOUNDARY VALUE PROBLEMS WITH SYMMETRY 15 Figure 3. Plots showing the iffeomorphism obtaine from the numerical solution isplaye in Figure 2. On the left, the template curve is shown together with a reference gri (the gri is purely for visualisation an is not part of the numerical metho). On the right, the final curve is shown together with the action of the iffeomorphism on the reference gri. approach is motivate by the problem of computing a geoesic on the iffeomorphism group in the plane which takes one curve to another. The transformation gives rise to a system of equations for a parameterise curve in which the en bounary conition for each value of the parameter is fixe. This means that when a iscrete approximation of the curve is use to solve this problem numerically, the en bounary conition can again be fixe exactly. In particular, when solving a shooting problem, this means that the error between the compute curve an the target curve can be compute simply by measuring the Eucliean istance between points for each value. This metho extens straightforwarly to the problem of fining geoesics in the three-imensional iffeomorphism group which take one parameterise surface to another, with the en bounary conition specifie only up to reparameterisations of the target surface. This problem has many applications in, for example, biomeicine, since it allows topologically equivalent surfaces to be characterise by a momentum fiel istribute on the template surface. Such momentum fiels exist in a linear space an so can be manipulate using linear techniques an still a topologically equivalent surface will always be obtaine. In the stanar approach to planar image registration, the problem of registering a specifie close curve (calle the template) at time t = 0 to another (the target) at time t = 1 amounts to eforming the space R 2 in which the template curve is embee until it matches the fixe target image to within a certain tolerance. Here, we consiere a manifol Q comprising the space of close curves S 1 embee into the plane R 2, written as Q = Emb(S 1, R 2 ). There are two Lie group actions available for manipulating the close curves in this escription. The action G R 2 R 2 of the Lie group G = Diff + (R 2 ) by composition from the left eforms the range space R 2, an thereby rags along a curve embee in it as GQ = Emb(S 1, G R 2 ). This left action prouce the singular-solution momentum map, J Sing in equation

16 16 COLIN J. COTTER AND DARRYL D. HOLM (33), which introuces the parameterisation of the close curve by its position an momentum supporte on a elta function efining the curve in R 2. Uner this left action of G, the curve preserves the initial parameterisation of its omain space in S 1, although the current positions of the S 1 labels in the plane R 2 will change as the range space is transforme. Alternatively, the action H S 1 S 1 of the Lie group H = Diff + (S 1 ) by composition from the right transforms coorinates in the omain space S 1 as HQ = Emb(H S 1, R 2 ), while keeping the curve fixe in the range space R 2. The present paper iscusse how the left action of Diff + (R 2 ) an the right action of Diff + (S 1 ) on Q = Emb(S 1, R 2 ) may complement each other in formulating a variational approach for registration of curves uner large eformations. The left action of Diff + (R 2 ) correspons to eforming the curve by a time-epenent transformation of the coorinate system in which it is embee, while leaving the parameterisation of the curve invariant. The ynamics of this eformation is formulate as an Euler-Poincaré equation for J Sing X(R 2 ) that results in Hamilton s canonical equations for the momentum an position variables of the curve that comprise the singular-solution momentum map (33). This solution provies the ynamics for curves that fulfills D Arcy Thompson s vision of shape transformation [?]. This vision unerlies common practice in image registration [?]. The right action of Diff + (S 1 ) correspons to aaptively reparameterising the S 1 omain coorinates of the curve. This reparameterisation of the curve coul be useful, for example, in esigning numerical methos that enhance the resolution of its main features as it eforms in R 2. As we have seen, this right action unlocks the parameterisation in the control problem to allow it more freeom for matching the curve shapes using an equivalent bounary value problem, without being constraine to match corresponing points along the template an target curves at the enpoint in time. As explaine above, the action of Diff + (S 1 ) from the right gives us the momentum map J S : T Emb(S 1, R 2 ) X(S), which we use to ensure that the momentum of the curve has no tangential component. This momentum map for right action is given explicitly as J S = p sq. The two momentum maps may be assemble into a single figure as in [?]: J Sing X(R 2 ) T Emb(S 1, R 2 ) J S X(S 1 ) We use the compatibility of these two momentum maps proven in [?] to ivie the curve matching problem into inepenent registration an reparameterisation problems, leaing to the reformulation of the curve matching problem as an equivalent geoesic bounary value problem. We are currently eveloping numerical algorithms for the transforme equations applie to embee curves an surfaces. As note in [?], applying a piecewise constant representation to the q an p variables in the untransforme equations results in a set of orere points. When this approach is extene to the transforme equations, a finite volume metho is obtaine, with the extra terms taking the form

17 GEODESIC BOUNDARY VALUE PROBLEMS WITH SYMMETRY 17 of an avection term in the q equation, an a continuity term in the p equation, which are very well evelope in the finite volume approach. One can also utilise the fact that the left- an right-actions commute by applying a splitting metho (so that steps of left- an right-action are alternate) an them use a semi-lagrangian metho for the right-action steps, as was use in the numerical examples in this paper. The commuting actions allow great flexibility an one can even perform all the right-action steps first before applying the left-action steps. We will also investigate iscontinuous higher-orer polynomial representations of p an q which lea to iscontinous Galerkin methos. Since the error in the transforme problem can be quantifie in terms of the Eucliean istance between points on the curve for each parameter value, the reparameterise formulation also makes it much easier to perform Bayesian stuies in which one observes points on a curve with observation error from some probability istribution, an then one attempts to estimate the probability istribution for the initial conitions of p (an ν) for which specifie points on the curve match the actual position of the observe points, after applying the time-1 flow map of the transforme geoesic equations for p an q. This approach provies a consierably simplifie observation operator to which algorithms such as the Monte Carlo Markov Chain algorithm can be applie. Acknowlegements. We are grateful to A. Clark, J. Peiro, S. L. Cotter, A. Stuart, D. C. P. Ellis, F. Gay-Balmaz, T. S. Ratiu, A. Trouvé, F.-X. Vialar an L. Younes for valuable iscussions. We thank the Royal Society of Lonon Wolfson Awar Scheme for partial support uring the course of this work. aress: colin.cotter@imperial.ac.uk