The Square Root Velocity Framework for Curves in a Homogeneous Space

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1 The Square Root Velocit Framework for Curves in a Homogeneous Space Zhe Su Eric Klassen Martin Bauer Department of Mathematics Florida State Universit, Tallahassee, FL su@math.fsu.edu, klassen@math.fsu.edu, bauer@math.fsu.edu Abstract In this paper we stud the shape space of curves with values in a homogeneous space M = G/K, where G is a Lie group and K is a compact Lie subgroup. We generalie the square root velocit framework to obtain a reparametriation invariant metric on the space of curves in M. B identifing curves in M with their horiontal lifts in G, geodesics then can be computed. We can also mod out b reparametriations and b rigid motions of M. In each of these quotient spaces, we can compute Karcher means, geodesics, and perform principal component analsis. We present numerical eamples including the analsis of a set of hurricane paths.. Introduction The field of shape analsis is concerned with the mathematical description, comparison and analsis of geometric shapes. This has applications in a variet of fields in pure and applied mathematics. Eamples include computational anatom, medical imaging, computer vision and functional data analsis. Although the notion of shape varies widel depending on the specific application, man shape spaces share two common difficulties: these spaces are usuall non-linear and often infinite-dimensional. To deal with the resulting challenges the methods of (infinite-dimensional) Riemannian geometr have proved to be a successful approach. The notion of shape space that we adopt for the purpose of this article is the space of unparametried curves with values in a homogeneous space M. Before we describe the contributions of the current work, we want to summarie previous work in this area. We start b considering the case of smooth, open, regular curves with values in some Euclidean space R d : Imm([,],R d ) := { c C ([,],R d ) : c }. () Eric Klassen gratefull acknowledges the support of the Simons Foundation (Grant # 37865). In the field of shape and functional data analsis one is usuall not interested in the actual parametriation of the curves, but onl in their geometric shape. Mathematicall the space of unparametried curves (shapes) can be modeled as the quotient space S([,],R d ) = Imm([,],R d )/Diff + ([,]), () wherediff + ([,]) denotes the group of smooth orientation preserving diffeomorphisms of the interval[, ] onto itself. For applications in shape analsis we want to define a distance or similarit measure on the space of unparametried curves. Towards this aim one can equip the space of all parametried curves with a Diff + ([,])- invariant metric and induce a Riemannian metric on the quotient space b requiring the quotient map π to be a Riemannian submersion. B an invariant Riemannian metric on Imm([,],R d ), we mean a Riemannian metric G with the propert that the re-parametriation group Diff + ([,]) acts b isometries, i.e., G c ϕ (u ϕ,u ϕ) = G c (u,u). ϕ Diff + ([,]). (3) Given an invariant metric G on Imm([,],R d ), the induced metric ons([,],r d ) is then defined as G π(c) (u,u) = inf π c(h)=u G c(h,h). (4) Thus the stud of Riemannian metrics on the shape space S([,],R d ) is reduced to the stud of invariant metrics on the simpler space of parametried curves. It came as a big surprise that the simplest such metric the reparametriation invariantl -metric induces vanishing geodesic distance, which renders it useless for applications in shape analsis, see [,, ]. To overcome this degenerac several modifications of the L -metric have been introduced: Michor and Mumford [] introduced metrics The invariance propert is a necessar but not a sufficient condition to induce a Riemannian metric on the quotient space. In the contet of invariant metrics on spaces of curves, however, it has been shown that the indeed induce a smooth Riemannian metric on the quotient space.

2 that are weighted b the curvature of the foot point curve c and Shah [8] and Mennucci and Yei [33] studied length weighted L -metrics. While overcoming the degenerac of the geodesic distance, the eistence of length minimiing curves for these metrics remains a delicate problem. It turned out to be a more promising approach to include (arclength) derivatives of the tangent vector in the definition of the metric, ielding the class of Sobolev metrics. These metrics have received rigorous theoretical analsis and, in particular, there eist analtic results on local and global eistence of geodesics [, 8, 7]. From an application point of view, a certain famil of first order Sobolev metrics proved advantageous, as there eist isometric transformations to flat spaces allowing for eplicit calculations of geodesics and geodesic distance [3, 3, 3]. This famil of metrics, also called elastic metrics, can be written as: G c (h,h) = a D s h N +b D s h T ds; (5) here D s and ds denote differentiation and integration with respect to arc-length and letd s h N (resp. D s h T ) denote the components ofd s h which are normal (resp. tangent) to the tangent vectorċof the curve. Fora = b and curves with values in R, Younes et al. [35] introduced the basic mapping to represent this metric; for the space of curves with values in general Euclidean space R d Srivastava et al. [3] developed the SRV transform to represent the metric with a = and b =. These transformations have been generalied to arbitrar parameters a, b in []. Using the SRV, efficient numerical calculations of geodesics have been developed and it also has given rise to rigorous results on the metric completion and the eistence of minimiing reparametriations [7]. In particular it has been shown that the metric completion of Imm([,],R d ) is the space of absolutel continuous functionsac([,],r d ) [8] and that in the case of PL curves [5] and C curves [6], optimal reparametriations eist, leading to length-minimiing paths in shape space S([,],R d ). Recentl, there has been an effort to generalie these metrics (and in particular the SRV transform) for curves with values in a general Riemannian manifold M. Su et al. [3] introduced the TSRVF (transported square root velocit function), in which all SRVFs are parallel transported along geodesics to the tangent space at a single reference point M. This method is computationall effective, but it has the disadvantage of introducing distortions for curves that venture far awa from, and the metric depends on the chosen reference point. Zhang et al. [36] introduced a different adaptation of the SRV, in which each path α : [,] M is represented b a path in the tangent space at its own initial point α(); the velocit vectors are parallel translated along the path α itself to this initial point. The paths are then compared using a metric on the total space of the tangent bundle TM. This method avoids the distortion and arbitrariness of the TSRVF resulting from the choice of a reference point; however, the computations are much more difficult. Le Brigant [7] introduced a more intrinsic metric on curves, defined pointwise along the curve. This method also avoids the arbitrariness and distortion of the TSRVF, but at a greater computational cost. In [8], Celledoni et al. adapted the SRV framework to the analsis of curves in a Lie group with a right-invariant metric. The basic idea is to use right translation to identif all tangent vectors to elements of the Lie algebra. The approach taken in the current paper is a generaliation of this idea to curves in homogenous spaces. For the space Imm(N,M) of immersions between two possibl higher dimensional manifolds much less is known. Sobolev metrics thereon have been introduced in [5] and in the case of surfaces inr d certain generaliations of the SRV framework have been studied in [7, 9, 3, 4,, ]. For more details on Riemannian metrics on spaces of curves and surfaces we refer to [34, 9, 3, 4]. Contributions of this paper: We introduce a new generaliation of the SRV transform for curves with values in a homogeneous space M = G/K, where G is a Lie group andk is a compact Lie subgroup. Man of the Riemannian manifolds that arise in applications can be viewed as homogeneous spaces, for eample Euclidean spaces, spheres, Grassmannians, hperbolic spaces, positive definite smmetric matrices, as well as all Lie groups. Compared to previous attempts, our approach has the advantage that it still ields eplicit formulas for geodesics and geodesic distance computing a geodesic on the space of parametried curves is equivalent to () computing a geodesic in G and () performing an optimiation over the compact group K. Our construction is based on first defining the SRV for curves with values in Lie groups [8] and then lifting the curve in M to a horiontal curve in the Lie group G. We compare our metric with the metric that has been considered in [36, 7, 6] and show the effectiveness of our algorithms in numerical eamples using hurricane paths, i.e., curves with values on the homogenous space S. In future work, we plan to generalie results on the eistence of minimiing geodesics and optimal reparametriations that are known to hold for curves in Euclidean spaces, to curves in homogeneous spaces.. The SRV for the space of curves with values in a homogenous space Let M = G/K be a homogeneous space, where G is a Lie group and K is a compact Lie subgroup of G. We will denote the Lie algebras ofgandk bgandkrespectivel. Assume that G is equipped with a left invariant Riemannian metric that is also bi-invariant with respect to K. This

3 metric induces a Riemannian metric on M that is invariant under the left action b G, see e.g. [5]. Furthermore we will denote the set of all absolutel continuous curves with values in a manifold N b AC([,],N) here N will be either M or G and b Γ the group of orientation preserving reparametriations,γ = Diff + ([,])... Curves with values in a Lie group G Following the SRVF (introduced b Srivastava et al. in [3]), we define the map where Q : AC([,],G) G L ([,],g) Q(α) = (α(),q), (6) α (t) L α(t) α (t) q(t) = α (t) α (t) = In this definition, the notation L α(t) refers to left translation applied to elements of G, as well as to tangent vectors. Theq-map here is the same as theq-map defined in [8] (using right translation instead of left). We have the following proposition: Proposition. The map Q : AC([,],G) G L ([,],g), defined above, is a bijection. Proof. Letα AC([,],G) denote the preimage underq of a given (α,q) G L ([,],g). B definition, α is the unique solution of the initial value problem α() = α, and α (t) = L α(t) ( q(t) q(t)). In the case of G = g = R n, eistence and uniqueness of such an α was proved b Robinson in [6]. To present a detailed proof of this result in the case of a Lie group G is outside of the scope of this contribution, and we postpone it to a future etended journal version of this article. Since we have alread givengariemannian metric, and L ([,],g) has its ownl metric, we obtain a product Riemannian metric on G L ([,],g). Furthermore, as Q is a bijection, there eists a smooth structure on AC([, ], G) such that Q is in addition a diffeomorphism. We can then use this diffeomorphism to induce a Riemannian metric (and thus distance function) on AC([, ], G). Note that it has been shown in [6] that the mapping Q is not a diffeomorphism and consequentl does not induce a Riemannian metric on AC([,],G), if the former is equipped with its natural smooth structure. Given α,α AC([,],G), let Q(α ) = (α (),q ) and Q(α ) = (α (),q ). The distance function on AC([,],G) then takes the form: d(α,α ) = ( d (α (),α ())+ q q ) / (7) (8) where the d on the right hand side of this equation is the geodesic distance on G. Consider the action of the reparametriation group Γ on AC([, ], G) b right composition and the action of G on AC([,],G) b left multiplication. Given γ Γ and g G, the corresponding actions of γ and g on the product space G L ([,],g) are as follows: g (α,q) γ = (gα, q γ ) γ, (9) where (α,q) G L ([,],g). G acts b isometries, since the metric on G was chosen to be left-invariant. The proof thatγacts b isometries is the same as in ther n case (see [3]) and we omit it. Hence we have the following proposition. Proposition. The Riemannian metric on AC([, ], G) and the corresponding distance function are preserved b the action of G and b the action of the reparameteriation groupγ. We will now derive the formula for the induced Riemannian metric on AC([,],G). Denote b, G the metric on G. Given α AC([,],G) and u T α AC([,],G), one can compute the differential ofq: Q α : T α AC([,],G) T (α(),q) (G L ([,],g)) Q α u = (u(),q α u), () whereq α : T α AC([,],G) T q L ([,],g) and q α u = α / D s (u) α 3/ D s u,δ l (α) G δ l (α). () Here δ l (α) = α α and D s (v) = α δl α(v). For a proof of this computation we refer to [8]. The metric on AC([,],G) is then obtained as the pullback of the natural product metric ofg L ([,],g) under Q: Proposition 3. Let u, v be smooth tangent vectors with foot point an immersion α. The pullback metric G on AC([,],G) at the smooth immersionαis given b G α (u,v) = Q α u,q α v Q(α) = u(),v() G () + D s u N,D s v N G + 4 D su T,D s v T G ds, where we integrate with respect ( to arclength ) ds = α (t) dt, D s u T = D s u, δl (α) δ l α G (α) α and D s u N = D s u D s u T are the tangential component and the normal component ofd s u respectivel. For G = R d the formula for the metric G reduces to (5), i.e., we obtain the elastic metric as defined in [3]. On Lie 3

4 groups the last two terms form the pullback metric obtained b Celledoni et al. in [8] (using right instead of left trivialiation). However, it is different than the metric introduced b Le Brigant et al. [7] and Zhang et al. [36] for arbitrar Riemannian manifolds. In our method the velocities are transported to the Lie algebra using left translation, while the metric in the above mentioned work is based on parallel transport. Thus these metrics will be different ifgis not an abelian Lie group. In Fig. we show eamples of geodesics for curves in hperbolic space. The resulting geodesics are ver similar to the geodesics obtained in [7, 6]. We plan to further investigate the similarities between these methods in future work... Curves with values in a homogeneous space M In this section, we will anale curves in M = G/K b relating them to their horiontal lifts in G. Note that g = k k, where k denotes the Lie algebra of K and k denotes the orthogonal complement of k in g. Denote b AC ([,],G) the set of all absolutel continuous paths in G which are orthogonal to each coset of K that the meet. Since the metric on G is left invariant, α AC ([,],G) is equivalent to L α α (t) k, which is equivalent to q L ([,],k ), where (α(),q) = Q(α). Therefore, Q restricts to a bijection between AC ([,],G) and G L ([,],k ). Now consider the right action ofk ong L ([,],k ) given b: (α,q) = (α, q), (3) where K,α G and q L ([,],k ). Note that K acts b isometries, where we put the standard L metric on L ([,],k ) and the product metric ong L ([,],k ). Denote b π : G M the quotient map, V p = kerπ p the vertical distribution for p G and H p the orthogonal complement of V p in T p G. For ever p G, T p G = H p V p and π p induces an isomorphism between H p and T π(p) M. Thus, givenβ AC([,],M) andα π (β()), there is a unique lift α AC ([,],G) such that α() = α and β(t) = π(α(t)). Note that the horiontal lift of β to α AC ([,],G) depends onl on the choice of the lift α of the initial pointβ(): I α β G π M = G/K (4) Let α, α be two lifts of β() and α, α be the lifts of β in AC ([,],G) starting at α and α respectivel. Then α = α, where = α α K; also ( α, q) = (α,q), (5) where (α,q) = Q(α) and ( α, q) = Q( α). It follows that Q induces a bijection (G L ([,],k ))/K AC([,],M). (6) SinceK is compact and acts freel ong L ([,],k ), there is an inherited Riemannian metric on (G L ([,],k ))/K. A minimal geodesic in the quotient corresponds to a shortest geodesic between two orbits in G L (I,k ) under the action ofk. We use this bijection to transfer the smooth structure and the Riemannian metric on(g L (I,k ))/K toac([,],m), making the latter into a Riemannian manifold. Suppose β,β are two paths in AC([,],M); let α and α be lifts ofβ, β inac ([,],G). Let Q(α ) = (α (),q ), Q(α ) = (α (),q ). (7) The distance between β and β induced from the distance function onac([,],g) is given b: d(β,β ) ( = inf d (α (),α ())+ q q ) /. (8) K Consider now the right action of Γ and the left action of G on G L ([,],k ). Similar as in the case of AC([,],G), we have the following proposition: Proposition 4. The Riemannian metric on AC([, ], M) and the corresponding distance function are preserved b the action of G and b the action of the reparameteriation groupγ. The formula for the induced pull back metric on AC([,],M) is then simpl given b restricting the metric G to horiontal vector fields. 3. Computing Geodesics 3.. Comparing Curves in M To compute the geodesic between β and β in AC([,],M), we need to compute the geodesic of minimal length between the orbits of Q(α ) and Q(α ) under the action of K. To do this, we need to find K that minimies d (α (),α ())+ q q. (9) Then the geodesic between (α (),q ) and (α (), q ) will project to a geodesic between β and β, see [4, 9] for more details regarding Riemannian submersions. In practice we will search for the optimal using a gradient descent method. Towards this aim we define the functionalf : K R b F() = d (α (),α ())+ q q, () 4

5 Figure. Eamples of geodesics between parametried curves in -dimensional hperbolic space. We show selected particle paths of the geodesic connecting the boundar curves. which is the square of the distance function between (α (),q ) and(α (), q ). SinceK acts transitivel on (α (),q ) K, we can just calculate the gradient at = I. To simplif the presentation, we assume that G is a matri group and that the inner product on the Lie algebra g is given b, = tr( t ), where t means the transpose of. We will calculate the gradient of the two terms of F separatel. The first term of F can be etended to a function F : G R, defined b the same formula: F () = d (α (),α ()). B left invariance of the metric on G, we can rewrite this as F () = d ( α () α (), ). It is a well-known fact that the gradient of this function at = I is given b I F = Log I (α () α ()) g, where Log denotes the inverse Riemannian eponential function at I G. If Log is multivalued, we will take the value with the smallest norm. Now, if we restrict F to K, then the gradient in k will simpl be the projection of the above epression from g to k. Thus the gradient of the first term of F() is given b Proj k ( LogI (α () α ()) ). Now we turn our attention to the second term of F. B the bi-invariance of the metric under multiplication b elements of K, we have F () = q q = q + q q, q = q + q q, q. () Since the first two terms do not depend on, we just need to calculate the gradient of q, q. We will use the first order approimation I + tv and I tv, where V k. Then we have the directional derivative of q, q ati in the direction V : d dtt= q,(i tv)q (I +tv) = q,q V q,vq = = tr(q t q V)dt tr(q t q V)dt tr(q t Vq)dt tr(q q t V)dt = tr( = (q t q q q t )dtv) (q t q q q t )dt,v () So we get the gradient of the second term at = I ( ) Proj k (q q t qq t )dt. (3) Hence the gradient off at = I is I F =Proj k ( Log I (α () α ()) + (q q t q t q )dt). (4) This ields the following algorithm to obtain the geodesic of minimal length between the orbits of (α (),q ) and (α (),q ): () For (α (),q ),(α (),q ) G L ([,],k ), set the step sieǫand calculate the gradient at = I. () Update (α (),q ) to (α (), q ), where = Ep I ( ǫ F). (3) If the norm of F is small enough, then stop. Otherwise go back to step (). Using this algorithm, we obtain a geodesic which locall minimies the distance between the orbits of (α (),q ) and (α (),q ). If we start with a different orbit representative α () G, the algorithm ma converge to a different geodesic. To increase our chance of finding the minimal geodesic, we act on (α (),q ) b several different elements in K, and use the resulting points as initialiations for the gradient algorithm. 3.. Comparing Curves in M up to Reparameteriation In the following, we are interested in comparing the shape of unparameteried curves. To mathematicall formulate our matching problem we define an equivalence relation on AC([,],M) as follows: Given elements β 5

6 Figure. Eamples of geodesics between two curves in AC([,],S ), S AC ([,],S ), AC([,],S )/SO(3), S AC ([,],S )/SO(3). Starting points of the curves are marked with a. and β of AC([,],M), define β β if and onl if their orbits β Γ and β Γ have the same closure with respect to the geodesic distance metric on AC([,],M) we defined before, see [5]. We then define the shape space S AC ([,],M) to be the set of equivalence classes under and we refer to the shape of a curve as its equivalence class in S AC ([,],M). The space S AC ([,],M) is not a manifold, but we can endow S AC ([,],M) with a metric so that it becomes a metric space [5, 6]. B proposition 4, the distance function d on AC([, ], M) is reparametriation invariant. We now consider the quotient space S AC ([,],M) = AC([,],M)/Γ. The induced distance is defined to be d([β ],[β ]) ( = inf d (α (),α ())+ q (q,γ) ) /. K γ Γ (5) To compute the geodesic between [β ] and [β ], we need to find the optimal K andγ Γ to minimie this distance. Since the action ofk and the action ofγong L (I,k ) commute with each other, we use the gradient method and the dnamic programming algorithm [9] to obtain a satisfactor convergence of both andγ. For curves in values in R d, the eistence of optimal reparametriations has been shown for PL curves [5] and C curves [6]. In future work we plan to generalie these results to curves with values in homogenous spaces Comparing Curves in M up to Rigid Motions The groupgacts onm as its group of rigid motions. For certain applications, we might want to calculate distances and geodesics in the quotient space AC([,],M)/G. B proposition 4, the distance function onac([,],m) is invariant under the left action of G. On the quotient space AC([,],M)/G, the distance function is then defined b d([β ],[β ]) ( = inf d (α (),gα ())+ q q ) /. K g G (6) Since g appears onl in the first summand in this formula, the minimum value of this distance can be achieved as follows: First choose K to minimie the second term, and then choose g = α ()α (), which will result in d(α (),gα ()) = and the simplified distance formula: d([β ],[β ]) = inf K q q. (7) As a result, the gradient calculation is much simpler in this case Comparing Curves in AC([,],M) up to Rigid Motions and Reparametriations Now we consider both the action of Γ and of G on AC([,],M) simultanousl. We want to mod out both of these actions and focus on the quotientac([,],m)/(g Γ). The distance function is defined to be d([β ],[β ]) 6

7 ( = inf d (α (),gα ())+ q (q,γ) ) /. K γ Γ,g G (8) Similar as in section 3.3, the distance function can be simplified to d([β ],[β ]) = inf K γ Γ q (q,γ). (9) The optimal K and γ Γ can now be found using the gradient and the dnamic programming algorithm, and we then setg = α ()α (). 4. Curves with values ons n In this section we want to describe the important special case of curves with values ons n in more detail. To view the sphere as a homogenous space we consider the Lie group SO(n) = {A GL(n,R) A t A = AA t = I,det(A) = } (3) with corresponding Lie algebra so(n) = {X M(n,R) X +X t = }. (3) It is well known that S n = SO(n + )/SO(n), where we identif SO(n) ( as a subgroup ) of SO(n + ) using the inclusion A. Let n = (,...,) S A n be the north pole. Then the quotient map π : SO(n+) S n is defined bπ(α) = αn. To use the previousl developed theor we need to define a Riemannian metric on SO(n+) that is bi-invariant with respect toso(n): for an pair of tangent vectorsuand v int g SO(n+) withg SO(n+), we define the inner product u,v g = tr(u t v). It is straightforward to check that this metric is indeed bi-invariant under multiplication b elements of SO(n + ) and thus in particular b multiplication with elements of SO(n) SO(n + ). Using the bi-invariance of the metric, the Riemannian eponential map at the identit is equal to the Lie group eponential [5] and is thus of the form v e v. The inverse Riemannian eponential map at identit is simpl the log function g log(g). The following well-known lemma will be useful in calculating the lift of paths ins n to paths inso(n+): Lemma. Let p,q S n and p q, then the most efficient rotation that takesp q can be epressed as ( ) R p,q = I p+q (p+q)(pt +q t ) (I pp t ). (3) B most efficient, we mean the rotation closest to I with respect to the bi-invariant metric onso(n+). This formula is onl valid if p q, since if p = q there is no unique shortest rotation taking p to q. Using the above lemma we obtain the following algorithm for lifting paths: Let β AC([,],S n ). Then a lift α AC([,],SO(n + )) of β can be computed as follows: () If β() = n, set α() = I n, (33) wherei n is the(n ) (n ) identit matri. For β() n, setα() = R n,β(). () Given α(t), set α(t + t) = R β(t),β(t+ t) α(t) for a chosen step sie t. Proposition 5. Given β AC([,],S n ), b using the lift algorithm described above, the lift α is in AC ([,],SO(n + )), that is, it satisfies the two properties:. β(t) = π(α(t)) for allt I,. α(t) α(t)t I SO(n) for allt I. Proof. Obviousl, the first propert holds. And the discrete form of second also holds, that is, the geodesic between α(t) and α(t + t) is perpendicular to the orbits with respect to these two points. Assume that α(t+ t) = α(t) for SO(n+). B the bi-invariance of the metric, we have the distance d(α(t),α(t + t)) = d(α(t),α(t)) = d(i,). It is eas to see that left translates the orbit α(t) to the orbit α(t + t), which is equivalent to left translating β(t) to β(t + t), that is, β(t) = β(t + t). R β(t),β(t+ t) SO(n + ) is the most efficient rotation such that d(α(t),r β(t),β(t+ t) α(t)) = d(i,r β(t),β(t+ t) ) is smallest, which means the distance between α(t) and R β(t),β(t+ t) α(t) realies the shortest possible distance between all pairs of representatives of these two orbits. 5. Applications to hurricane tracks Finall we want to demonstrate the effectiveness of the proposed framework using real data. We consider 75 hurricane tracks from the Atlantic hurricane database and 75 hurricane tracks from the Northeast and North Central Pacific hurricane database (HURDAT). The data under consideration is depicted in Fig. 3. Each hurricane path is represented as a curve in S, and is discretied as a piecewisegeodesic polgon. Our first step is to calculate the matri of all pairwise distances. For unparametried curves using an The data was obtained from the National Hurricane Center website: 7

8 Figure 3. In S AC ([,],S ) (first row) and in S AC ([,],S )/SO(3) (second row): the Karcher mean (black) of 75 hurricane tracks in Atlantic database, of 75 hurricane tracks in North Central Pacific hurricane database and of the combined 5 hurricane tracks (from left to right). Intel Core i7-45u (.GH) machine, the computation of these 75 boundar value problems took less than two minutes 3. We note that our algorithms are orders of magnitudes faster than the algorithms developed in [36, 7, 6], while at the same time overcoming the disadvantages of the methods used in [3] Figure 4. The distance matri of 5 hurricane tracks visualied using multi-dimensional scaling in two dimensions. Left: distances calculated in S AC ([,],S ). Right: distances calculated in S AC ([,],S )/SO(3). Data points representing hurricanes from the Atlantic are marked with a ; hurricanes from the Pacific region with a. In Figure 4 we visualied the distance matrices using multi-dimensional scaling []. As one might epect there is clear clustering between the hurricane tracks from the Atlantic region and those of the Northeast and North Central Pacific region if we regard them as elements of S AC ([,],S ). However if, in addition, we mod out b rigid motions, then the obtained distance matri does not 3 In the implementation we made use of the one-dimensionalit ofk = SO(), which allowed us to solve the minimiation over K without the gradient method. seem to capture this information anmore. This suggests that the clustering in the previous eperiment was mainl based on location and that the shape of a hurricane path does not possess enough information to allow for a significant statement on its region of origin. Finall we calculate the Karcher mean of all hurricanes and of each of the groups separatel as well. These results are depicted in Fig. 3. Using the Karcher means as a charts, this potentiall allows to linearie the shape space using the corresponding tangent spaces. As an eample, we show geodesics from the Karcher mean in the direction of the first two principal directions in Fig. 5. It seems that the first principal direction encodes the variet in shape, whereas the second direction seems to mainl reflect the change in the length of the hurricanes. Figure 5. The first two principal directions starting from the Karcher mean (black) of 5 hurricane tracks in S AC ([,],S ) (first row) and ins AC ([,],S )/SO(3) (second row). 8

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