Optimal control for reconstruction of curves without cusps

Ugo Boscain, Remco Duits, Francesco Rossi, Yuri Sachkov

Abstract— We consider the problem of minimizing∫ `

0

√ξ2 +K2(s) ds for a planar curve having fixed initial

and final positions and directions. The total length ` is free.Here s is the variable of arclength parametrization, K(s)is the curvature of the curve and ξ > 0 a parameter. Thisproblem comes from a model of geometry of vision due toPetitot, Citti and Sarti.We study existence of local and global minimizers forthis problem. We prove that if for a certain choice ofboundary conditions there is no global minimizer, thenthere is neither a local minimizer nor a stationary curve(geodesic).Our main tool is the construction of the optimal synthesisfor the Reed and Shepp car with quadratic cost.

I. INTRODUCTION

In this paper we are interested in the followingvariational problem1:

(P) Fix (xin, yin, θin), (xfin, yfin, θfin) ∈ R2 × S1.Assume that (xin, yin) 6= (xfin, yfin). On the space of(regular enough) planar curves, parameterized by plane-arclength2 find the solutions of:

γ(0) = (xin, yin), γ(`) = (xfin, yfin),

γ(0) = (cos(θin), sin(θin)),

γ(`) = (cos(θfin), sin(θfin)),∫ `0

√ξ2 +K(s)2 ds→ min (here ` is free.)

Here K = xy−yx(x2+y2)3/2

is the geodesic curvature of theplanar curve γ(·) = (x(·), y(·)). This problem comesfrom a model proposed by Petitot, Citti and Sarti (see[5], [9] and references therein) for the mechanism ofreconstruction of corrupted curves used by the visualcortex V1. The model is explained in Section II.

U. Boscain is with the Center for Applied Mathematics, Ecole Poly-technique, France (ugo.boscain@polytechnique.edu)

R. Duits is with Eindhoven University of Technology, P.O. Box 513,5600 MB, Eindhoven, The Netherlands. Department of Mathematicsand Computer Science (R.Duits@tue.nl)

F. Rossi is with Aix-Marseille Univ, LSIS, 13013, Marseille, France(francesco.rossi@lsis.org)

Y. Sachkov is with Program Systems Institute Pereslavl-Zalessky,Russia (sachkov@sys.botik.ru)

1In this paper by S1 we mean R/ ∼ where θ ∼ θ′ if θ = θ′+2nπ,n ∈ Z. By P 1 we mean R/ ≈ where θ ≈ θ′ if θ = θ′+nπ, n ∈ Z.

2Here by plane-arclength we mean the arclength in R2. Later on, weconsider also parameterizations by arclength on R2×S1 or R2×P 1,that we call sR-arclength.

It is convenient to formulate the problem (P) as aproblem of optimal control, for which the functionalspaces are also more naturally specified.

(Pcurve) Fix (xin, yin, θin), (xfin, yfin, θfin) ∈R2 × S1 and ξ > 0. In the space of integrable controlsv(·) : [0, `]→ R, find the solutions of:

(x(0), y(0), θ(0)) = (xin, yin, θin),

(x(`), y(`), θ(`)) = (xfin, yfin, θfin),(x, y, θ

)= (cos(θ), sin(θ), 0) + v(s) (0, 0, 1)∫ `

0

√ξ2 +K(s)2 ds =

∫ `

0

√ξ2 + v(s)2 ds→ min .

Here ` is free.Since in this problem we are taking v(·) ∈ L1([0, `]),

we have that the curve q(.) = (x(·), y(·), θ(·)) : [0, `]→R2 × S1 is absolutely continuous and the planar curvex(·) := (x(·), y(·)) : [0, `]→ R2 is in W 2,1([0, `]).

The main issues we address in this paper are relatedto existence of minimizers for problem (Pcurve). Moreprecisely, the first question we are interested in for(Pcurve) is the following:

Q1) Is it true that for every initial and final condition,the problem (Pcurve) admits a global minimum?

In [3] it was shown that there are initial and final con-ditions for which (Pcurve) does not admit a minimizer.From the modelization point of view, the non-existenceof global minimizers is not a crucial issue. It is verynatural to assume that the visual cortex looks only forlocal minimizers, since it seems reasonable to expectthat it primarly compares nearby trajectories. A secondproblem we address in this paper is the existence of localminimizers for the problem (Pcurve). More precisely,we answer to the following question:

Q2) Is it true that for every initial and final conditionthe problem (Pcurve) admits a local minimum? If not,what is the set of boundary conditions for which a localminimizer exists?

The last question is interesting, since one could com-pare the limit boundary conditions for which a math-ematical reconstruction occurs with the limit boundaryconditions for which a reconstruction in human percep-tion experiments is observed. Indeed, it is well known

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from perception experiments that the visual cortex V1does not connect all initial and final conditions.

The main result of this paper is the following.Theorem 1: Fix an initial and a final condition qin =

(xin, yin, θin) and qfin = (xfin, yfin, θfin) in R2×S1.The only two following cases are possible.1) There exists a solution (global minimizer) for(Pcurve) from qin to qfin.2) Problem (Pcurve) from qin to qfin does not admitneither a global nor a local minimum nor a geodesic.Both cases occur, depending on boundary conditions.We recall that, given an optimal control problem, apair trajectory-control (q(.), u(.)) is a local minimizer ifthere exists a neighborood of u(.) in which all controlshave a bigger cost than (q(.), u(.)).

We also recall that a curve q(.) is a geodesics if forevery sufficiently small interval [t1, t2] ⊂ Dom(q(.)),the curve q(.)|[t1,t2] is a minimizer between q(t1) andq(t2).

It is interesting to observe that for (Pcurve) theLavrentiev phenomenon occurs, i.e. there exist mini-mizing curves that are absolutely continuous but notLipschitz. See Section V-A. This is a quite exoticphenomenon, rarely observed. See [12] for more details.

For the second sentence of Q2, we have computednumerically the configurations for which a solutionexists, see Figure 1.

Fig. 1. Configurations for which we have existence of minimizerswith ξ = 1, starting from qin = (0, 0, 0). The negative y-semiaxiscan be recovered by symmetry.

We prove these result by introducing the followingauxiliary mechanical problem:

(PMEC) Fix (xin, yin, θin)(xfin, yfin, θfin) ∈ R2×S1 and ξ > 0. In the space of L∞ controls u(.), v(.) :[0, `]→ R, find the solutions of:

(x, y, θ) = (cos(θ), sin(θ), 0) + v(t)(0, 0, 1),

(x(0), y(0), θ(0)) = (xin, yin, θin),

(x(`), y(`), θ(`)) = (xfin, yfin, θfin),∫ `0

√ξ2u(s)2 + v(s)2 ds→ min (here ` is free)

This problem (which cannot be interpreted as a prob-lem of reconstruction of planar curves, as explained in[4]) has been completely solved in a series of papers byone of the authors (see [8], [10], [11]). He developeda software for finiding numerical solutions to problem(PMEC) for arbitrary boudary conditions.

Remark 2: Observe that all the problems we treatdepend on a parameter ξ > 0. It is easy to reduce ourstudy to the case ξ = 1, by dilation of the plane. Seee.g. [3]. For this reason, we fix ξ = 1 from now on.

The structure of the paper is the following. In Sec-tion II we present the model by Petitot-Citti-Sarti. InSection III we recall definitions and properties for 3Dcontact problems, that we will use to study (PMEC).Section IV is devoted to the structure of geodesicsfor (PMEC), while Section V presents the connectionsbetween (PMEC) and (Pcurve). Finally, Section VI isdevote to the proof of Theorem 1.

II. THE MODEL BY PETITOT-CITTI-SARTI FOR V1

In this section, we describe a model of the mechanismused by the visual cortex V1 to reconstruct curves whichare partially hidden or corrupted. The goal is to explainthe connection between reconstruction of curves and theproblem (Pcurve) studied in this paper.

The model we present was initially due to Petitot (see[9] and references therein), then refined by Citti-Sarti[5], and by authors of this paper in [4], [6], [7].

In a simplified model (see [9, p. 79]), neurons ofV1 are grouped into orientation columns, each of thembeing sensitive to visual stimuli at a given point of theretina and for a given direction on it. The retina ismodeled by the real plane, i.e. each point is representedby (x, y) ∈ R2, while the directions at a given point aremodeled by the projective line, i.e. θ ∈ P 1. Hence, theprimary visual cortex V1 is modeled by the so calledprojective tangent bundle PTR2 := R2 × P 1. From aneurological point of view, orientation columns are inturn grouped into hypercolumns, each of them beingsensitive to stimuli at a given point (x, y) with anydirection. Therefore each hypercolumn is represented bya fiber P 1 of the bundle PTR2. Orientation columnsare connected between them in two different ways. Thefirst kind is given by vertical connections, which connectorientation columns belonging to the same hypercolumnand sensible to similar directions. The second is givenby the horizontal connections, which connect orientationcolumns in different (but not too far) hypercolumns andsensible to the same directions. See Figure 2.

In other words, when V1 detects a (regular enough)planar curve (x(·), y(·)) : [0, T ] → R2 it computes a

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Fig. 2. A scheme of the primary visual cortex V1.

“lifted curve” in PTR2 by adding a new variable θ(·) :[0, T ]→ P 1 which satisfies:

(x, y, θ) = u(t)(cos(θ), sin(θ), 0) + v(t)(0, 0, 1) (1)

The new variable θ(.) plays the role of the direction inP 1 of the tangent vector to the curve, when it is defined.Here it is natural to take u(·), v(·) ∈ L1([0, T ]). Thisspecifies also which regularity we need for the planarcurve to be able to compute its lift: we need a curvein W 2,1 such that it has integrable curvature. In thefollowing we call such a planar curve a liftable curve.

Consider now a liftable curve (x(·), y(·)) : [0, T ] →R2 which is interrupted in an interval ]a, b[⊂ [0, T ]. Letus call (xin, yin) := (x(a), y(a)) and (xfin, yfin) :=(x(b), y(b)). Assume moreover that, after computingits lift, the limits θin := limτ→a− θ(τ) and θfin :=limτ→b+ θ(τ) are well defined. In the model by Petitot,Citti, Sarti and the authors of the present article [4], [5],the visual cortex reconstructs the curve by minimizingthe energy necessary to activate orientation columnswhich are not activated by the curve itself. This ismodeled by the minimization of the functionalJ =

∫ ba

(ξ2u(τ)2 + v(τ)2

)dτ → min (a, b fixed).

Indeed, ξ2u(τ)2 (resp. v(τ)2) represents the (in-finitesimal) energy necessary to activate horizontal(resp. vertical) connections. The parameter ξ > 0is used to fix the relative weight of the horizon-tal and vertical connections, which have different na-tures. Minimization of J is equivalent to the min-imization of the cost (which is invariant by repa-rameterization) L =

∫ ba

√ξ2u(τ)2 + v(τ)2 dτ =

∫ ba‖x(τ)‖

√ξ2 +K(τ)2 dτ . See a proof of such equiv-

alence in [8].We thus define the following problem:

(Pprojective) Fix (xin, yin, θin), (xfin, yfin, θfin) ∈R2 × P 1 and ξ > 0. In the space of integrable controlsu(.), v(.) : [0, `]→ R, find the solutions of:

(x, y, θ) = u(t)(cos(θ), sin(θ), 0) + v(t)(0, 0, 1),

(x(0), y(0), θ(0)) = (xin, yin, θin),

(x(`), y(`), θ(`)) = (xfin, yfin, θfin),

L =∫ `0

√ξ2u(s)2 + v(s)2 ds→ min (` free.)

Observe that here θ ∈ P 1, i.e. angles are con-sidered without orientation. The variational problem(Pprojective) is well defined, and we have remarkedin [4] that a solution always exists. However, its maindrawback is the existence of minimizers with cusps, seee.g. [3]. Roughly speaking, cusps are singular points inwhich velocity changes its sign.

However, the presence of cusps has not been observedin human perception experiments, see e.g. [9]. For thisreason, people started looking for a way to require thatno trajectories with cusps appear as solutions of thevariational problem. In [5], [7] the authors proposed torequire trajectories parameterizated by spatial arclength,i.e. to impose ‖x‖ = u = 1. In this way cusps cannotappear. Notice that assuming u = 1, directions must beconsidered with orientation, since now the direction of xis defined in S1. By fixing u = 1 we get the variationalproblem (Pcurve) on which this paper is focused.

III. 3D CONTACT PROBLEMS

In this section, we state classical results of geometriccontrol theory which hold for the 3D contact case.

Definition 3 (3D contact problem): Let M be a 3Dmanifold and let X1, X2 be two smooth vector fieldssuch that dim(Span{X1, X2, [X1, X2]}(q))=3 for everyq ∈M . The variational problem

q = u1X1 + u2X2, q(0) = q0, q(T ) = q1,(2)∫ T

0

√(u1(τ))2 + (u2(τ))2dτ → min

ui(·) ∈ L1([0, T ],Rm), (3)q(·) ∈W 1,1([0, T ],M) (4)

is called a 3D contact problem. If u1(τ)2 + u2(τ)2 = 1for a.e. τ ∈ [0, T ] we say that the curve is parameterizedby sR-arclength.

Remark 4: In the problem above the final time Tcan be free or fixed since the cost is invariant by timereparameterization. As a consequence the spaces L1 and

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W 1,1 in (4) can be replaced with L∞ and Lip. See [3,Section 2.1.1] for more details.

We now apply the Pontryagin Maximum Principle(PMP in the following, see [2]) to our problem.

Proposition 5 (PMP for 3D contact problems):In the 3D contact case, a curve parameterizedby sR-arclength is a geodesic if and onlyif it is the projection of a solution of theHamiltonian system corresponding to the HamiltonianH(q, p) = 1

2 (〈p,X1(q)〉2 + 〈p,X2(q)〉2), withq ∈M, p ∈ T ∗qM , lying on the level set H = 1/2.This simple form of the PMP follows from absence ofnontrivial abnormal extremals in 3D contact geometry.

In general, geodesics are not optimal for all times.On the contrary, minimizers and local minimizers areby definition geodesics.

In the following we denote by (q(t), p(t)) =

et~H(q0, p0) the unique solution at time t of the Hamil-

tonian system q = ∂pH, p = −∂qH , with initialcondition (q(0), p(0)) = (q0, p0). Moreover we denoteby π : T ∗M →M the canonical projection (q, p) 7→ q.

Definition 6: Let (M, span{X1, X2}) be a 3D con-tact manifold and q0 ∈ M . Let Λq0 := {p0 ∈T ∗q0M |H(q0, p0) = 1/2}. We define the exponentialmap starting from q0 as

Expq0 : Λq0 × R+ →M, Expq0(p0, t) = π(et~H(q0, p0)).

Next, we recall definitions of cut and conjugate time.Definition 7: Let q0 ∈ M and q(.) be a geodesic

parameterized by sR-arclength starting from q0. The cuttime for q(.) is

Tcut(q(.)) = sup{t > 0, | q(.)|[0,t] is optimal}.

The cut locus from q0 is

Cut(q0) = {q(Tcut(q(.))) | q(.) geodesic from q0}.Definition 8: Let q0 ∈ M and q(.) be a geodesic

parameterized by sR-arclength starting from q0 withinitial covector p0. The first conjugate time of q(.) is

Tconj(q(.)) = min{t > 0 | (p0, t) critical point of Expq0}.

The conjugate locus from q0 is the set Con(q0) ={q(Tconj(q(.))) | q(.) geodesic from q0}.A geodesic loses its local optimality at its first conju-gate locus. However a geodesic can lose optimality for“global” reasons, as we will show in Section IV-A. Wealso have (see [1]):

Theorem 9: Let γ be a geodesic starting from q0and let Tcut and Tconj be its cut and conjugate times(possibly +∞). Then Tcut ≤ Tconj . Moreover, γ isglobally optimal from t = 0 to Tcut and it is notglobally optimal from t = 0 to Tcut+ε, for every ε > 0.

It is locally optimal from t = 0 to Tconj and it is notlocally optimal from t = 0 to Tconj+ε, for every ε > 0.

IV. GEODESICS OF (PMEC)

In this section we study the geodesics of (PMEC).This is the main tool that we use to answer to questionsQ1 and Q2.

Following [8], [10], [11],applying the PMP and usefulchange of coordinates, we obtain the following Hamil-tonian system:

ν = c, c = − sin ν, (ν, c) ∈ (2S1ν)× Rc, (5)

x = sin ν2 cos θ, y = sin ν

2 sin θ, θ = − cos ν2 .

The curvature of (x(.), y(.)) is K(.) = − cot(ν(.)/2).We now define cusps for geodesics of (PMEC).Definition 10: Let q(.) = (x(·), y(·), θ(·)) be a

geodesic of (PMEC), parameterized by sR-arclength.We say that Tcusp is a cusp time for q(.) (and q(Tcusp)a cusp point) if u(.) changes its sign at Tcusp. We saythat the restriction of q(·) to an interval [0, T ] has nointernal cusps if no t ∈]0, T [ is a cusp time.

Eq. (5) is the pendulum equation. There exist 5 typesof geodesics, corresponding to pendulum trajectories.

1) Type S: stable equilibrium of the pendulum: ν ≡0.The planar trajectory is (x(t), y(t)) ≡ (0, 0).

2) Type U: unstable equilibria of the pendulum: ν ≡±π. The planar trajectories are straight lines.

3) Type R: rotating pendulum. Fig. 3-Left.4) Type O: oscillating pendulum. Fig. 3-Center.5) Type Sep: separating trajectory of the pendulum.

Fig. 3-Right.

Fig. 3. Geodesics of type R, O, Sep.A. Optimality of geodesics

Let q(.) = (x(.), y(.), θ(.)) be a geodesic parameter-ized by sub-Riemannian arclength, and let t ∈ [0, T ], beits arc. Define the mappings of geodesics:

S,T : q(.) 7→ qS(.), qT(.), with q(.) : [0, T ]→ R2×S1

where

θS(t) = θ(T )− θ(T − t),xS(t) = − cos θ(T )∆x(t)− sin θ(T )∆y(t),

yS(t) = − sin θ(T )∆x(t) + cos θ(T )∆y(t),

θT(t) = −θS(t), xT(t) = xS(t),

yT(t) = sin θ(T )∆x(t)− cos θ(T )∆y(t),

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with ∆x(t) = x(T ) − x(T − t), ∆y(t) = y(T ) −y(T − t). The mapping S acts as reflection of the curve(x(.), y(.)) in the middle perpendicular to the segmentthat connects the points (x(0), y(0)) and (x(T ), y(T ));the mapping T acts as reflection in the midpoint of thissegment. See Figure 4.

Fig. 4. Action of S (left) and T (right) on (x(.), y(.)).

A point q(t) of a trajectory q(.) is called a Maxwellpoint corresponding to the reflection S if q(t) = qS(t)and q(·) 6≡ qS(·). The same definition can be given forT. Examples of Maxwell points for the reflections S andT are shown in Figure 5.

Fig. 5. Maxwell point for reflection S (left) and T (right).

The following theorem, proved in [8], [10], [11]describes optimality of geodesics.

Theorem 11: A geodesic q(.) on the interval [0, T ],is optimal if and only if each point q(t), t ∈ (0, T ), isneither a Maxwell points corresponding to S or T, northe limit of a sequence of Maxwell points.

Denote by Tpend the period of motion of the pendu-lum (5). It was proved in [11] that the cut time satisfiesthe following:• Tcut = 1

2Tpend for geodesics of type R,• Tcut ∈ ( 1

2Tpend, Tpend) for geodesics of type O,• Tcut = +∞ = Tpend for types S, U and Sep.Corollary 12: Let q(.) be a geodesic. Let Tcusp, and

Tcut be the first cusp time and the cut time (possibly+∞).Then Tcusp ≤ Tcut.

Proof: For geodesics of types R and O it followsfrom the phase portrait of pendulum (5) that there exists

t ∈ (0, 12Tpend) such that ν(t) = 2πn. This impliesK = +∞, i.e. for such t we have a cusp point. ThenTcusp ≤ 1

2Tpend ≤ Tcut.For geodesics of types S, U and Sep, the inequality

Tcusp ≤ Tcut is obvious since Tcut = +∞.Corollary 13: Let q(.) defined on [0, T ] be a mini-

mizer having an internal cusp. Then any other minimizerbetween q(0) and q(T ) has an internal cusp.Proof It was proved in [11] that for any points q0, q1 ∈R2 × S1, there exist either one or two minimizersconnecting q0 to q1. Moreover, if there are two suchminimizers q(.) and q(·), then q(·) is obtained from q(.)by a reflection S or T. So if q(.) has an internal cusp,then q(·) has an internal cusp as well.

V. CONNECTIONS BETWEEN (PMEC) AND (Pcurve)

In this section, we state precisely the connectionsbetween minimizers of problems (Pcurve) and (PMEC)defined above. We first prove that a minimizer of(PMEC) without cusps is also a minimizer of (Pcurve).Indeed, take a minimizer q(.) of (PMEC) withoutcusps, thus with ‖x(τ)‖ > 0 for τ ∈ [0, T ]. Then,reparametrize the time to have u = ‖x‖ ≡ 1 (thisis possible exactly because it has no cusps). Thisnew parametrization of q(.) satisfies the dynamics for(Pcurve) and the boundary conditions. Assume now bycontradiction that there exists a curve q(.) satisfying thedynamics for (Pcurve) and the boundary conditions witha cost that is smaller that the cost for q(.). Then q(.)also satisfies the dynamics for (PMEC) and boundaryconditions, with a smaller cost, hence q(.) is not aminimizer. Contradiction.

Observe now that each curve Γ(.) = (x(.), y(.), θ(.))satisfying the dynamics for (Pcurve) with a certaincontrol v(.), also satisfies the dynamics for (PMEC)with controls u(.) ≡ 1 and v(.). For simplicity ofnotation, we give the following definition.

Definition 14: Let Γ(.) = (x(.), y(.), θ(.)) be a curvein SE(2) satisfying the dynamics for (Pcurve) witha certain control v(.). We define the correspondingcurve q(.) for (PMEC) as the same parametrized curve(x(.), y(.), θ(.)), and the corresponding pair as the pairtrajectory-control (q(.), (u(.), v(.))) with u(.) ≡ 1.

A key propostion, that can be proved by explicitcomputations, is the following.

Lemma 15: If Γ(.) is a geodesic for (Pcurve), thenthe corresponding curve is a geodesic for (PMEC).

A. (Pcurve) shows the Lavrentiev phenomenon

We now show that the problem (Pcurve) exhibitsthe Lavrentiev phenomenon, meaning that there existabsolutely continuous minimizers that are not Lipschitz.

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Consider a geodesic of (PMEC) defined on [0, T ]having no internal cusp and corresponding to controlsu(·) and v(·). From Corollary 12 it follows that it isoptimal. Assume now that this geodesic has a cusp atT . Then limt→T u(t) = 0 and K(t) = v(t)/u(t)→∞.Notice however that

√1 + |K(τ)|2 =

√u(τ)2 + v(τ)2

is integrable on [0, T ], since its integral is exactly thecost of the geodesic, that is finite, see e.g. [8]. Thismeans that the corresponding trajectory for (Pcurve)(for which v(s) = K(s)) has unbounded control andit is not Lipschitz.

This is the so-called the Lavrentiev phenomenon. Inthis case, direct application of standard techniques forthe computation of local minimizers, such as the PMP,would provide local minimizers in the “too small” setof controls L∞([0, T ],R).

VI. PROOF OF THEOREM 1

In this section we prove the main results of this paper,answering to question Q1 and Q2. We show that theset of boundary conditions for which a solution for(Pcurve) exists coincides with the set of boundary con-ditions for which a local minimizer exists. Moreover,such set coincides with the set of boundary conditionsfor which a geodesics joining them exists.

Proof: Given boundary conditions qin, qfin, let q(.)be a solution of (PMEC) with such boundary condi-tions. The two cases in Theorem 1 correspond to theexistence/non-existence of cusps for q(.).

We prove the first case. If q(.) has no internal cusps,then, following Section V, one can prove that q(.) isalso a minimizer for (Pcurve).

We prove the second case by contradiction. We firststudy the non-existence of a geodesic. Let q(.) have aninternal cusp. Let now Γ(.) be a geodesic for (Pcurve).Applying Lemma 15, we have that the correspondingcurve q(.) is a geodesic for (PMEC). Since Γ(.) hasno cusps, then q(.) has no cusps, thus it is optimal(Corollary 12). Since both q(.) and q(.) are optimal,one can recover one from the other via S or T. Sinceq(.) has a cusp, then q(.) has one too.

We now study the existence of a minimizer or alocal minimizer. Let Γ(.) be either a minimizer or alocal minimizer for (Pcurve). In both cases, Γ(.) is ageodesic, so one can repeat the previous proof to havethe non-existence of such Γ(.).

Acknowledgements: This research has been sup-ported by the European Research Council, ERC StG2009 “GeCoMethods”, contract n. 239748, by the ANR“GCM”, program “Blanc–CSD” project number NT09-504490, and by the DIGITEO project “CONGEO”.

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