Reparameterization invariant distance on the space of curves in the hyperbolic plane Alice Le Brigant∗ , Marc Arnaudon∗ and Frédéric Barbaresco† ∗

Institut Mathématique de Bordeaux, UMR 5251 Université de Bordeaux and CNRS, France † Thales Air System, Surface Radar Domain, Technical Directorate Voie Pierre-Gilles de Gennes, 91470 Limours, France Abstract. This paper focuses on the study of time-varying paths in the two-dimensional hyperbolic space, and its aim is to define a reparameterization invariant distance on the space of such paths. We adapt the geodesical distance on the space of parameterized plane curves given by Bauer et al. in [1] to the space Imm([0, 1], H) of parameterized curves in the hyperbolic plane. We present a definition which enables to evaluate the difference between two curves, and show that it satisfies the three properties of a metric. Unlike the distance of Bauer et al., the distance obtained takes into account the positions of the curves, and not only their shapes and parameterizations, by including the distance between their origins. Keywords: curve matching, Riemannian shape analysis, reparameterization group, hyperbolic space PACS: 02.40.Ky

INTRODUCTION The study of oriented paths -or curves- in differential manifolds, can be central to innovative approaches in signal processing, and particularly in radar detection. A key point of such methods is to be able to compute the distance between two curves. Here, we place ourselves in the two-dimensional hyperbolic space and study the curves which lie in that space. It will be our aim in this article, to find a satisfying definition of distance between two open curves in the hyperbolic plane. By satisfying, we mean invariant under reparameterization, i.e. we would like the distance between two curves to be the same whatever the chosen parameterization. In other words, we want to induce a distance on the space of curves modulo reparameterization, which we call the shape space. Bauer et al. suggested such a metric on the space of plane curves in [1]. They mostly look at closed curves, but the results are easily transposable to open curves. Their first step is to define a reparameterization invariant Riemannian metric G on the space Imm(S1 , R2 ) of parameterized plane curves. The fact that it is reparameterization invariant assures that it induces a Riemannian metric on the space of parameterized curves quotiented by the parameterization group, i.e. the shape space S = Imm(S1 , R2 )/Diff(S1 ). The geodesical distances in the space of parametrized curves and in the shape space are then linked by the following property dist(C0 ,C1 ) =

inf

φ ∈Diff(S1 )

dist(c0 , c1 ◦ φ ),

(1)

where C0 = π(c0 ) and C1 = π(c1 ) if π : Imm(S1 , R2 ) → Imm(S1 , R2 )/Diff(S1 ) is the natural projection from the space of parameterized curves onto the shape space. So it all comes down to defining a reparameterization invariant metric on the space of parameterized curves. The simplest example of a reparamerization invariant metric on the space Imm(S1 , R2 ) is the L2 -metric Z

Gc (h, k) =

S1

hh, kids

where c ∈ Imm(S1 , R2 ) is a curve, h, k ∈ Tc Imm(S1 , R2 ) are infinitesimal deformations and we integrate over arc-length ds in order to have the reparamaterization invariance. Unfortunately, the geodesic distance induced by this metric on the shape space vanishes, as was shown in [9]. That is why Bauer et. al look into more complicated metrics called Sobolev-type metrics, which contain derivatives of various orders of the infinitesimal deformations h and k. They study a family of first-order Sobolev metrics in particular, given by Z Ga,b c (h, h) =

S1

a2 hDs h, ni2 + b2 hDs h, vi2 ds.

1 ˙ ˙ where a, b ∈ R+ are constants, Ds h = ||1c|| ˙ h and Ds k = ||c|| ˙ k denote the arc-length derivatives of h and k respectively, and v and n are the unit tangent and normal vectors to the curve c. They show that this metric can be obtained as the pullback of the L2 metric in the space C∞ (S1 , R3 ) of curves in space, by a certain R-transform Ra,b . For more related work of these authors, see [4] and [5]. Very recent developments on these topics can also be found in [6], [7], and [8]. In this article we try to adapt the distance of Bauer et al. to the space of curves in the hyperbolic space, by considering the same transformation. However, the definitions that we suggest take into account the positions of the curves, whereas the distance of Bauer et al. does not distinguish between a curve and its translation with the same parameterization.

DISTANCE ON THE SPACE OF PARAMETERIZED CURVES IN H We consider the R-transform as a function of the space Imm([0, 1], H) of curves in the hyperbolic plane  ∞ )  Imm([0, 1], H) −→ C! ([0, 1], T H × R+!! a,b √ R : . v 0  ˙ 1/2 a + 4b2 − a2 c 7−→ ||c|| 0 1 In the case of plane curves, the image of the R-transform is the set of curves with values in a certain cone C a,b . Therefore the distance chosen by Bauer et al. in [1] is simply the pointwise distance between the image curves in that cone Z 2π

dist(c, d) = 0

distC a,b (R(c)(θ ), R(d)(θ ))dθ ,

where c, d ∈ Imm([0, 2π]) and distC a,b is the distance on the cone. In our case, the image of a curve in the hyperbolic space does not lie in one single cone. Instead, for a given curve c ∈ Imm([0, 1], H), the image Ra,b (c)(t) of each point c(t) belongs to a cone placed above c(t),  a,b Cc(t) = (u, r) ∈ Tc(t) H × R+ |a2 r2 = (4b2 − a2 )||u||2 . It is useful, in order to compute distances, to have all the image vectors of a given curve c in the same cone. That is why we bring back each image vector Ra,b (c)(t) ∈ Tc(t) H × R+ to the origin c(0) of the curve by parallel transport. To do so we only have to parallel transport the component in Tc(t) H along the curve c, and leave the component in R+ untouched. We denote by Pct1 →t2 (u) the parallel transport of the vector u ∈ Tc(t1 ) H from c(t1 ) to c(t2 ) along c, and by extension Pct1 →t2 (q) the vector obtained by parallel transporting the component u ∈ Tc(t1 ) H of a vector q = (u, r) of Tc(t1 ) H × R+ . We define the distance between two paths c and d in the hyperbolic plane as the pointwise distance between the image curves R(c) and R(d), once they are sent in the same image cone. For each t ∈ [0, 1], we first send the image vectors R(c)(t) and R(d)(t) of both curves on the image cones based at their respective origins Cc(0) et Cd(0) , and then we parallel transport one of the two vectors from one cone onto the other, for example from Cd(0) onto Cc(0) , along a curve that connects them, as illustrated in figure 1. Our first idea was to use the geodesic that connects them, but the function obtained did not verify the triangular inequality. That is why we use the curve that minimizes the obtained function. Once both vectors R(c)(t) and R(d)(t) are in the same cone, the only thing left to do is to compute the geodesical distance between them in the cone that contains them. This is written dist(c, d) =

inf

dγ (c, d),

γ path of H γ(0) = c(0), γ(1) = d(0)

with dγ2 (c, d) =

Z 1 0

distC2 c(0)

  t→0 1→0 t→0 Pc (R(c)(t)), Pγ ◦ Pd (R(d)(t)) dt,

where distCc(0) refers to the geodesical distance on the cone. It is given by Bauer et al. in the case of a cone in R3 , and it is easy to see that an analogous method leads to the same distance for a cone in T H × R+ . For more details see [1]. This definition can lead to a distance of zero between for example two parallel geodesics of the hyperbolic plane (the arcs of two concentric circles) if their origins are vertically aligned. That is why we add to the above expression the length `(γ) of the path connecting both origins q dist(c, d) = inf (2) dγ2 (c, d) + `2 (γ), γ path of H γ(0) = c(0), γ(1) = d(0)

thus defining a function taking into account the relative positions of the curves. In what follows we will give explicit formulas for parallel transport in the two-dimensional hyperbolic space, and examine whether this expression actually defines a distance function.

Figure 1. Illustration of the parallel transports necessary to compute the distance between two parameterized curves c and d

PARALLEL TRANSPORT OF A VECTOR ALONG A CURVE IN H In order to be able to compute the distance (2), we need to explicit the parallel transport of a vector u ∈ Tc(t1 ) H along a curve c = (x, y) in the two-dimensional hyperbolic space. We define a parallel vector field v along c such that v(t1 ) = u. We want Pct1 →t0 (u) = v(t0 ). In the Poincaré half-plane. In the Poincaré half-plane representation, this is written ∇c˙v = 0, that is v˙ = Av

  1 y˙ x˙ with A = . y −x˙ y˙ 

 0 1 A is of the form aI + bK where I is the identity matrix and K = . The set −1 0 {aI + bK|a, b ∈ K} of these matrices is an abelian Lie algebra. The solution can then be written as R  v(t) = exp tt0 A(τ) dτ v(t0 ) ! R t x(τ) ˙ y(t) log y(t dτ t0 y(τ) 0) . i.e. v(t) = exp B(t) · v(t0 ), with B(t) = R t x(τ) ˙ y(t) − t0 y(τ) dτ log y(t ) 0

The matrix B(t) is diagonalizable and therefore its exponential is easy to compute. Finally we see that from u we obtain v(t0 ) = exp (−B(t1 )) · u after a rotation of angle

b(t1 ) =

R t1 x(τ) ˙ t0 y(τ)

coupled with a homothety of ratio k(t1 ) = 

Pct1 →t0 (u) = v(t0 ) =

R



˙ t1 x(τ) y(t0 )   t0 y(τ) dτ R ˙ y(t1 ) sin t1 x(τ) t0 y(τ) dτ

cos

y(t1 ) y(t0 ) ,

R  ˙ − sin tt01 x(τ) dτ R y(τ)   u. ˙ cos tt01 x(τ) y(τ) dτ

(3)

In the Poincaré disk. After analogous calculations, we find that in the Poincaré disk R t1 xy−y v(t0 ) is obtained from u after a rotation of angle b(t1 ) = t0 1−(x˙ 2 +yx˙ 2 ) dτ coupled with q 2 0) a homothety of ratio k(t1 ) = 1−r(t , 1−r(t )2 1

s Pct1 →t0 (u) = v(t0 ) = where r =



R



˙ x˙ t1 xy−y 1 − r(t0   t0 1−(x2 +y2 ) dτ R ˙ x˙ 1 − r(t1 )2 sin t1 xy−y t0 1−(x2 +y2 ) dτ

)2

cos

R  xy−y ˙ x˙ − sin tt01 1−(x dτ 2 +y2 ) R   u, ˙ x˙ t1 xy−y cos t0 1−(x2 +y2 ) dτ (4)

p x2 + y2 .

THE PROPERTIES OF A METRIC ARE FULFILLED Let us show that the function that we defined verifies the three required properties of a metric. Identity of indiscernibles. If c and d are two curves in the hyperbolic plane such that dist(c, d) = 0, then it is clear that their origins coincide. Indeed, if they didn’t, at least the term `(γ) would be strictly positive and the distance wouldn’t vanish. Since a loop γ going from and back to c(0) = d(0) would induce a strictly positive term dγ (c, d), we have Z 1
2 d{c(0)} (c, d) = distC2 c(0) Pc−1 (R(c)(t)) , Pd−1 (R(d)(t)) dt = 0. 0

That is, the "raisings" of the images R(c) and R(d) to the cone Cc(0) are the same, and since the R-transform and the parallel transport are bijective transformations, we can conclude that c and d coincide. Symmetry. First of all, let us notice that for any two vectors u, v in a cone Cγ(0) based at the origin of a curve γ, the distance between these vectors computed in the cone Cγ(0) is the same as the distance between their parallel transports along γ, computed in the same cone Cγ(1) based in γ(1), which is also the parallel transport along γ of the cone Cγ(0) : distC a,b (u, v) = distC a,b (Pγ0→1 (u), Pγ0→1 (v)). γ(0)

γ(1)

From there, we can say that dγ2 (c, d)

Z 2π

= 0

Z 2π

= =

  distC2 c(0) Pct→0 (R(c)(t)), Pγ1→0 ◦ Pdt→0 (R(d)(t)) dt
distC2 d(0) Pγ ◦ Pc−1 (R(c)(t)), Pd−1 (R(d)(t)) dt

0 dγ2˜ (d, c),

˜ = γ(1 − t), ∀t ∈ where γ˜ is the same curve as γ but with the opposite orientation : γ(t) [0, 1]. Therefore, q q 2 2 ˜ = dist(d, c), dist(d, c) = inf dγ (c, d) + ` (γ) = dγ2˜ (d, c) + `2 (γ) inf γ˜ path of H ˜ γ(0) = d(0) ˜ γ(1) = c(0)

γ path of H γ(0) = c(0) γ(1) = d(0)

and we have the symmetry. Triangular inequality. Now the triangular inequality. Let us consider the cone C a,b , whatever its base point, as a manifold that we will call M. We equip this manifold with a Riemannian structure, i.e. we define a scalar product h·, ·iM , and its corresponding norm || · ||M , on each tangent space Tq M to M in a vector q. The distance between two vectors q, q¯ of the cone C a,b = M is the length of the geodesic path γ connecting them, Z 1

distM (q, q) ¯ = `M (γ) =

0

˙ ||γ(t)|| M dt.

We then define the set of curves with vector values in the manifold M, M = C∞ ([0, 1], M) = C∞ ([0, 1], C a,b ), and its tangent plane in any vector-valued curve q ∈ M ,  Tq M = u ∈ C∞ ([0, 1], Tq M) : u(t) ∈ Tq(t) M ∀t ∈ [0, 1] . In the same way as we did with M, we give a Riemannian structure to the manifold M by equipping it with the L2 -scalar product ∀u, v ∈ Tq M ,

hu, viM :=

Z 1 0

hu(t), v(t)iTq(t) M dt.

That way, the distance between two vector-valued curves q, q¯ ∈ M is the length of the geodesic path γ connecting them in C . Since geodesics in M are of the form ( [0, 1] × [0, 1] → M γ: (a,t) 7−→ γ(a,t)

where each γ(·,t) for t ∈ [0, 1] is a geodesic of M linking q(t) and q(t), ¯ we have s Z 1

distM (q, q) ¯ = `M (γ) =

0

˙ ||γ(a)|| M da =

Z 1

Z 1

0

0

||γa (a,t)||2M dt

da,

where γa (a,t) denotes the partial derivative of γ according to its first variable. Now let us recall that since γ is a geodesic in M , it has constant speed : γa (a,t) depends only on t. This gives s s 2 Z 1 Z 1 Z 1 2 distM (q, q) ¯ = ||γa (a,t)||M da dt ||γa (0,t)||M dt = 0 0 0 s s Z 1

= 0

`2M (γ(·,t))dt

Z 1

= 0

2 (q(t), q(t))dt. distM ¯

Now, let c, d, e be three curves in the hyperbolic space, and γ1 , γ2 two curves linking c and d and d and e respectively. We denote by γ1ˆγ2 the curve obtained by the a,b concatenation of the curves γ1 and γ2 . This curve then links c to e. If here M = Cc(0) , we have according to what preceeds   −1 −1 dγ1ˆγ2 (c, e) = distM Pc−1 (R(c)), Pγ−1 ◦ P ◦ P (R(e)) . γ2 e 1 Since we have defined distM as the Riemannian distance corresponding to the metric h·, ·iM , it naturally verifies the triangular inequality, so that dγ1ˆγ2 (c, e)     −1 −1 −1 −1 −1 −1 ≤ distM Pc−1 R(c), Pγ−1 ◦ P R(d) + dist P ◦ P R(d), P ◦ P ◦ P R(e) M γ1 γ1 γ2 e d d 1     = distM Pc−1 R(c), Pγ−1 ◦ Pd−1 R(d) + distM Pd−1 R(d), Pγ−1 ◦ Pe−1 R(e) 1 2 = dγ1 (c, d) + dγ2 (d, e). Since we also have `(γ1ˆγ2 ) = `(γ1 )+ `(γ2 ), we arrive, by the same calculations needed to determine the distance d on a product X1 × X2 of two metric spaces (X1 , d1 ) and (X2 , d2 ), to the conclusion that q q q dγ21ˆγ2 (c, e) + `2 (γ1ˆγ2 ) ≤ dγ21 (c, d) + `2 (γ1 ) + dγ22 (d, e) + `2 (γ2 ). And so dist(c, e) ≤

inf γ1 path(c, d) γ2 path(d, e)

=

inf γ path(c, d)

n o dγ21 (c, d) + `2 (γ1 ) + dγ22 (d, e) + `2 (γ2 )

n o dγ2 (c, d) + `2 (γ) +

= dist(c, d) + dist(d, e),

inf γ path(d, e)

n o dγ2 (d, e) + `2 (γ)

which is what we wanted.

CONCLUSION We were able to present a function on the space of curves in the hyperbolic space, which satisfies the properties of a distance function, by adapting the distance between plane curves introduced by Bauer et al. using the "R-transform". The distance on the shape space S = Imm([0, 2π], H)/Diff([0, 2π]) can be deduced from our distance in the following way dist(C0 ,C1 ) = inf dist(c0 , c1 ◦ φ ), φ ∈Diff(S1 )

where c0 and c1 are parameterized curves, and C0 and C1 their respective projections on the shape space. The key point now resides in whether we can equip the space of parameterized curves Imm([0, 2π], H) with a Riemannian structure, i.e. whether the distance presented in this article corresponds to the geodesical distance of a certain scalar product.

ACKNOWLEDGMENTS This research was supported by Thales Air Systems and the french MoD DGA (Direction Générale de l’Armement).

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