Bertrand Banos

MONGE-AMP ÈRE EQUATIONS AND GENERALIZED COMPLEX GEOMETRY. THE TWO-DIMENSIONAL CASE

We associate an integrable generalized complex structure to each 2dimensional symplectic Monge-Ampère equation of divergent type and, using the Gualtieri ∂ operator, we characterize the conservation laws and the generating function of such equation as generalized holomorphic objects.

Introduction

A general approach to the study of non-linear partial differential equations, which goes back to Sophus Lie, is to see a k-order equation on a n-dimensional manifold N n as a closed subset in the manifold of k-jets J k N . In particular, a second-order differential equation lives in the space J 2 N . Neverthess, as it was noticed by Lychagin in his seminal paper "Contact geometry and non-linear second-order differential equations" ( [START_REF] Lychagin | Contact geometry and non-linear second order differential equations[END_REF]), it is sometimes possible to decrease one dimension and to work on the contact space J 1 N . The idea is to define for any differential form ω ∈ Ω n (J 1 N ), a second order differential operator ∆ ω : C ∞ (N ) → Ω n (N ) acting according to the rule ∆ ω (f ) = j 1 (f ) * ω, where j 1 (f ) : N → J 1 N is the section corresponding to the function f .

The differential equations of the form ∆ ω = 0 are said to be of Monge-Ampère type because of their "hessian -like" non-linearity. Despite its very simple description, this classical class of differential equations attends much interest due to its appearence in different problems of geometry or mathematical physics. We refer to the very rich book Contact geometry and Non-linear Differential Equations ( [START_REF] Kushner | Contact geometry and Non-linear Differential Equations[END_REF]) for a complete exposition of the theory and for numerous examples.

A Monge-Ampère equation ∆ ω = 0 is said to be symplectic if the Monge-Ampère operator ∆ ω is invariant with respect to the Reeb vector field. In other words, the n-form ω lives actually on the cotangent bundle T * N , and symplectic geometry takes place of contact geometry. The Monge-Ampère operator is then defined by

∆ ω (f ) = (df ) * ω.
This partial case is in some sense quite generic because of the beautiful result of Lychagin which says that any Monge-Ampère equation admitting a contact symmetry is equivalent (by a Legendre transform on J 1 N ) to a symplectic one.

We are interested here in symplectic Monge-Ampère equations in two variables. These equations are written as :

(1)

A ∂ 2 f ∂q 2 1 + 2B ∂ 2 f ∂q 1 ∂q 2 + C ∂ 2 f ∂q 2 2 + D ∂ 2 f ∂q 2 1 ∂ 2 f ∂q 2 2 - ∂ 2 f ∂q 1 ∂q 2 2 + E = 0, 1
with A, B, C, D and E smooth functions of (q, ∂f ∂q ). These equations correspond to 2-form on T * R 2 , or equivalently to tensors on T * R 2 using the correspondence

ω(•, •) = Ω(A•, •),
Ω being the symplectic form on T * N . In the non-degenerate case, the traceless part of this tensor A defines either an almost complex structure or an almost product structure and it is integrable if and only the corresponding Monge-Ampère equation is equivalent to the Laplace equation or the wave equation. This elegant result of Lychagin and Roubtsov ([13]) is quite frustrating: which kind of integrable geometry could we define for more general Monge-Ampère equations ?

It has been noticed in [START_REF] Crainic | Generalized complex structures and Lie brackets[END_REF] that such a pair of forms (ω, Ω) defines an almost generalized complex structure, a very rich concept defined recently by Hitchin ([8]) and developed by Gualtieri ([6]), which interpolates between complex and symplectic geometry. It is easy to see that this almost generalized complex structure is integrable for a very large class of 2D-Monge-Ampère equations, the equations of divergent type. This observation is the starting point for the approach proposed in this paper: the aim is to present these differential equations as "generalized Laplace equations".

In the first part, we write down this correspondence between Monge-Ampère equations in two variable and 4-dimensional generalized complex geometry.

In the second part we study the ∂-operator associated with a Monge-Ampère equation of divergent type and we show how the corresponding conservation laws and generating functions can be seen as "holomorphic objects".

Monge-Ampère equations and Hitchin pairs

In what follows M is the smooth symplectic space T * R 2 endowed with the canonical symplectic form Ω. Our point of view is local (in particulary we do not make any distinction between closed and exact forms) but most of the results presented here have a global version.

A primitive 2-form is a differential form ω ∈ Ω 2 (M ) such that ω ∧ Ω = 0. We denote by ⊥ : Ω k (M ) → Ω k-2 (M ) the operator θ → ι XΩ (θ), the bivector X Ω being the bivecor dual to Ω. It is straightforward to check that in dimension 4, a 2-form ω is primitive if and only if ⊥ω = 0.

1.1. Monge-Ampère operators.

DEFINITION. Let ω be a 2-form on M . A 2-dimensional submanifold L is a generalized solution of the equation ∆ ω = 0 if it is bilagrangian with respect to Ω and ω.

Note that a lagrangian submanifold of T * R 2 which projects isomorphically on R 2 is a graph of a closed 1-form df : R 2 → T * R 2 . A generalized solution can be thought as a smooth patching of classical solutions of the Monge-Ampère equation ∆ ω = 0 on R 2 . EXAMPLE 1 (Laplace equation). Consider the 2D-Laplace equation

f q1q1 + f q2q2 = 0.
It corresponds to the form ω = dq 1 ∧ dp 2 -dq 2 ∧ dp 1 , while the symplectic form is Ω = dq 1 ∧ dp 1 + dq 2 ∧ dp 2 . Introducing the complex coordinates z 1 = q 1 + iq 2 and z 2 = p 2 + ip 1 , we get ω + iΩ = dz 1 ∧ dz 2 . Generalized solution of the 2D-Laplace equation appear then as the complex curves of C 2 .

The following theorem (so called Hodge-Lepage-Lychagin, see [START_REF] Lychagin | Contact geometry and non-linear second order differential equations[END_REF]) establishes the 1 -1 correspondence between Monge-Ampère operators and primitive 2-forms: THEOREM. i) Any 2-form admits the unique decomposition ω = ω 0 + λω, with ω 0 primitive. ii) If two primitive forms vanish on the same lagrangian subspaces, then there are proportional.

Remark. A Monge-Ampère operator ∆ ω is therefore uniquely defined by the primitive part ω 0 of ω, since λΩ vanish on any lagrangian submanifold. The function λ can be arbitrarily chosen.

Let ω = ω 0 + λΩ be a 2-form. We define the tensor A by ω = Ω(A•, •). One has A = A 0 + λId and A 2 0 = -pf(ω 0 )Id, where the function pf(ω 0 ) is the pfaffian of ω 0 defined by

ω 0 ∧ ω 0 = pf(ω 0 )Ω ∧ Ω.

Therefore,

A 2 = 2λA -(λ 2 + pf(ω 0 ))Id. The equation ∆ ω = 0 is said to be elliptic if pf(ω 0 ) > 0, hyperbolic if pf(ω 0 ) < 0, parabolic if pf(ω 0 ) = 0. In the elliptic/hyperbolic case, one can define the tensor

J 0 = A 0 | pf(ω 0 )|
which is either an almost complex structure or an almost product structure.

THEOREM (Lychagin -Roubtsov [START_REF] Lychagin | A classification of Monge-Ampère equations[END_REF]). The following assertions are equivalent i) The tensor J 0 is integrable.

ii) The form

ω 0 / | pf(ω 0 )| is closed.
iii) The Monge-Ampère equation ∆ ω = 0 is equivalent (with respect to the action of local symplectomorphisms) to the (elliptic) Laplace equation f q1q1 + f q2q2 = 0 or the (hyperbolic) wave equation f q1q1 -f q2q2 = 0.

Let us introduce now the Euler operator and the notion of Monge-Ampère equation of divergent type (see [START_REF] Lychagin | Contact geometry and non-linear second order differential equations[END_REF]).

DEFINITION. The Euler operator is the second order differential operator

E : Ω 2 (M ) → Ω 2 (M ) defined by E(ω) = d⊥dω. A Monge-Ampère equation ∆ ω = 0 is said to be of divergent type if E(ω) = 0. EXAMPLE 2 (Born-Infeld Equation). The Born-Infeld equation is (1 -f t ) 2 f xx + 2f t f x f tx -(1 + f 2 x )f tt = 0. The corresponding primitive form is ω 0 = (1 -p 2 1 )dq 1 ∧ dp 2 + p 1 p 2 (dq 1 ∧ dp 1 ) + (1 + p 2 2 )dq 2 ∧ dp 1 . with q 1 = t and q 2 = x. A direct computation gives dω 0 = 3(p 1 dp 2 -p 2 dp 1 ) ∧ Ω,
and then the Born -Infeld equation is not of divergent type.

EXAMPLE 3 (Tricomi equation). The Tricomi equation is

v xx xv yy + αv x + βv y + γ(x, y).
The corresponding primitive form is

ω 0 = (αp 1 + βp 2 + γ(q))dq 1 ∧ dq 2 + dq 1 ∧ dp 2 -q 2 dq 2 ∧ dp 1 ,
with x = q 1 and y = q 2 . Since

dω 0 = (-αdq 2 + βdq 1 ) ∧ Ω,
we conclude that the Tricomi equation is of divergent type.

LEMMA. A Monge-Ampère equation ∆ ω = 0 is of divergent type if and only if it exists a function µ on M such that the form ω + µΩ is closed.

Proof. Since the exterior product by Ω is an isomorphism from Ω 1 (M ) to Ω 3 (M ), for any 2-form ω, there exists a 1-form α ω such that

dω = α ω ∧ Ω. Since ⊥(α ω ∧ Ω) = α ω we deduce that E(ω) = 0 if and only if dα ω = 0, that is d(ω + µΩ) = 0 with dµ = -α ω .
Hence, if ∆ ω = 0 is of divergent type, one can choose ω being closed. The point is that it is not primitive in general .

1.2. Hitchin pairs. Let us denote by T the tangent bundle of M and by T * its cotangent bundle. The natural indefinite interior product on T ⊕ T * is

(X + ξ, Y + η) = 1 2 (ξ(Y ) + η(X)),
and the Courant bracket on sections of

T ⊕ T * is [X + ξ, Y + η] = [X, Y ] + L X η -L Y ξ - 1 2 d(ι X η -ι Y ξ).
DEFINITION (Hitchin [8]). An almost generalized complex structure is a bundle map J :

T ⊕ T * → T ⊕ T * satisfying J 2 = -1, and
(J•, •) = -(•, J•).
Such an almost generalized complex structure is said to be integrable if the spaces of sections of its two eigenspaces are closed under the Courant bracket.

The standard examples are

J 1 = J 0 0 -J * and J 2 = 0 Ω -1
-Ω 0 with J a complex structure and Ω a symplectic form.

LEMMA (Crainic [4]). Let Ω be a symplectic form and ω any 2-form. Define the tensor A by ω = Ω(A•, •) and the form ω by ω = -Ω(1

+ A 2 •, •).
The almost generalized complex structure

(2) J = A Ω -1 ω -A *
is integrable if and only if ω is closed. Such a pair (ω, Ω) with dω = 0 is called a Hitchin pair

We get then immediatly the following:

PROPOSITION 1. To any 2-dimensional symplectic Monge-Ampère equation of divergent type ∆ ω = 0 corresponds a Hitchin pair (ω, Ω) and therefore a 4-dimensional generalized complex structure.

Remark. Let L 2 ⊂ M 4 be a 2-dimensional submanifold. Let T L ⊂ T be its tangent bundle and T 0 L ⊂ T * its annihilator. L is a generalized complex submanifold (according to the terminology of [START_REF] Gualtieri | Generalized complex geometry[END_REF]) or a generalized lagrangian submanifold (according to the terminology of [START_REF] Ben-Bassat | Submanifolds of generalized complex manifolds[END_REF]) if T L ⊕ T 0 L is closed under J. When J is defined by [START_REF] Ben-Bassat | Submanifolds of generalized complex manifolds[END_REF], this is equivalent to saying that L is lagrangian with respect to Ω and closed under A, that is, L is a generalized solution of ∆ ω = 0. 1.3. Systems of first order partial differential equations. On 2n-dimensional manifold, a generalized complex structure write as

J = A π σ -A *
with different relations detailed in [START_REF] Crainic | Generalized complex structures and Lie brackets[END_REF] between the tensor A, the bivector π and the 2-form σ. The most oustanding being [π, π] = 0, that is π is a Poisson bivector.

In [START_REF] Crainic | Generalized complex structures and Lie brackets[END_REF], a generalized complex structures is said to be non-degenerate if the Poisson bivector π is non-degenerate, that is, if the two eigenspaces E = Ker(J -i) and E = Ker(J + i) are transverse to T * . This leads to our symplectic form Ω = π -1 and to our 2-form ω = Ω(A•, •).

One could also take the dual point of view and study generalized complex structure transverse to T . In this situation, the eigenspace E writes as

E = ξ + ι ξ P, ξ ∈ T * ⊗ C ,
with P = π + iΠ a complex bivector. This space defines a generalized complex structure if and only if it is a Dirac subbundle of (T ⊕ T * ) ⊗ C and if it is transverse to its conjugate E. According to the Maurer-Cartan type equation described in the famous paper Manin Triple for Lie bialgebroids ( [START_REF] Liu | Manin triples for Lie bialgebroids[END_REF], the first condition is

[π + iΠ, π + iΠ] = 0.
The second condition says that Π is non-degenerate.

Hence, we obtain some analog of the Crainic's result:

DEFINITION. A Hitchin pair of bivectors is a pair consisting of two bivectors π and Π, Π being non-degenerate, and satisfying

(3) [Π, Π] = [π, π] [Π, π] = 0. PROPOSITION 2.
There is a 1-1 correspondence between Generalized complex structure J = A π A σ -A * with σ non degenerate and Hitchin pairs of bivector (π, Π). In this correspondence, we have

     σ = Π -1 A = π • Π -1 π A = -(1 + A 2 )Π EXAMPLE 4.
If π + iΠ is non-degenerate, it defines a 2-form ω + iΩ which is necessarily closed (this is the complex version of the classical result which says that a non-degenerate Poisson bivector is actually symplectic). We find again an Hitchin pair. So new examples occur only in the degenerate case. Note that π +iΠ = (A+i)Π, so det(π+iΠ) = 0 if and only if -i is an eigenvalue for A. In dimension 4, this implies that A 2 = -1 but this is not any more true in greater dimensions (see for example the classification of pair of 2-forms on 6-dimensional manifolds in [START_REF] Lychagin | A classification of Monge-Ampère equations[END_REF]). Nevertheless, the case A 2 = -1 is interesting by itself. It corresponds to generalized complex structure of the form

J = J 0 σ -J *
with J an integrable complex structure and σ a 2-form satisfying J * σ = -σ and

dσ J = dσ(J•, •, •) + dσ(•, J•, •) + dσ(•, •, J•).
where σ J = σ(J•, •) (see [START_REF] Crainic | Generalized complex structures and Lie brackets[END_REF]). Or equivalently σ + iσ J is a (2, 0)-form satisfying

∂(σ + iσ J ) = 0.
One typical example of such geometry is the so called HyperKähler geometry with torsion which is an elegant generalization of HyperKähler geometry ( [START_REF] Grantcharov | Geometry of hyper-Khler connections with torsion[END_REF]). Unlike the HyperKäler case, such geometry are always generated by potentials ( [START_REF] Banos | Potentials for hyper-Kähler metrics with torsion[END_REF]).

Let us consider now an Hitchin pair of bivectors (π, Π) in dimension 4. Since Π is non-degenerate, it defines two 2-forms ω and Ω, which are not necessarily closed, and related by the tensor A. A generalized lagrangian surface is a surface closed under A, or equivalently, bilagrangian: ω| L = Ω| L = 0. Locally, L is defined by two functions u and v satisfying a first order system

a + b ∂u ∂x + c ∂u ∂y + d ∂v ∂x + e ∂v ∂y + f det J u,v A + B ∂u ∂x + C ∂u ∂y + D ∂v ∂x + E ∂v ∂y + E det J u,v with J u,v =
∂u ∂x ∂u ∂y ∂v ∂x ∂v ∂y Such a system generalizes both Monge-Ampère equations and Cauchy-Riemann systems and is called Jacobi-system (see [START_REF] Kushner | Contact geometry and Non-linear Differential Equations[END_REF]).

With the help of Hitchin's formalism, we understand now the integrability condition (3) as a "divergent type" condition for Jacobi equations.

The ∂-operator

Let us fix now a 2D-symplectic Monge-Ampère equation of divergent type ∆ ω = 0, the 2-form ω = ω 0 + λΩ being closed. We still denote by A = A 0 + λ the associated tensor.

LEMMA. For any 1-form α, the following relation holds:

(4) α ∧ ω -B * α ∧ Ω = 0 with B = λ -A 0 .
Proof. Let α = ι X Ω be a 1-form. Since ω 0 is primitive, we get

0 = ι X (ω 0 ∧ Ω) = (ι X ω 0 ) ∧ Ω + (ι X Ω) ∧ ω 0 = A * 0 α ∧ Ω + α ∧ ω 0 . Therefore, α ∧ ω = α ∧ ω 0 + λα ∧ Ω = (-A 0 + λ) * α ∧ Ω.
We denote by J the generalized complex structure associated with the Hitchin pair (ω, Ω). We also define Θ = ω -iΩ and Φ = exp(Θ) = 1 + Θ + Θ 2 2 .

2.1. Decomposition of forms. Using the tensor J, Gualtieri defines a decomposition

Λ * (T * ) ⊗ C = U 2 ⊕ U -1 ⊕ U 0 ⊕ U 1 ⊕ U 2
which generalizes the Dolbeault decomposition for a complex structure ( [START_REF] Gualtieri | Generalized complex geometry[END_REF]).

Let us introduce some notations to understand this decomposition. The space T ⊕ T * acts on Λ * (T * ) by

ρ(X + ξ)(θ) = ι X θ + ξ ∧ θ,
and this action extends to an isomorphism (the standard spin representation) between the Clifford algebra CL(T ⊕T * ) and the space of linear endomorphisms End(Λ * (T * )).

Remark. With these notations, the eigenspace E = Ker(J -i) is also defined by

E = X + ξ ∈ T ⊕ T * , ρ(X + ξ)(Φ) = 0 , DEFINITION. The space U k is defined by U k = ρ Λ 2-k E Φ . Note that J identifyed with the 2-form (J•, •) lives in Λ 2 (T ⊕ T * ) ⊂ CL(T ⊕ T * ).
We get then an infinitesimal action of J on Λ * (T * ).

LEMMA (Gualtieri). U k is the ik-eigenspace of J.

Remark. We see then immediatly that U -k = U k , since J is a real tensor.

PROPOSITION 3. i) U 2 = CΦ. ii) U 1 = α ∧ Φ, α ∈ Λ 1 (T * ) ⊗ C . iii) U 0 = (θ -i 2 ⊥θ) ∧ Φ, θ ∈ Λ 2 (T * ) ⊗ C . Proof. The eigenspace E is E = X -ι X Θ, X ∈ T ⊗ C . Now, ρ(X -ι X Θ)(Φ) = ι X Θ + ι X Θ ∧ Θ -ι X Θ -ι X Θ ∧ Θ = ι X (Θ -Θ) ∧ (1 + Θ).
Since Θ -Θ = -2iΩ and X → ι X Ω is an isomorphism between T and T * , we get then the description of U 1 .

Choose now two complex vectors X and Y and define α = ι X Ω and

β = ι Y Ω: ρ (X -ι X Θ) ∧ (Y -ι Y Θ) Φ = ρ X -ι X Θ -2iβ ∧ Φ = -2iρ X -ι X Θ β + β ∧ Θ = -2i β(X)(1 + Θ) -β ∧ ι X Θ -ι X Θ ∧ β -ι X Θ ∧ β ∧ Θ = -2i β(X)(1 + Θ) + ι X (Θ -Θ) ∧ β ∧ (1 + Θ) -ι X Θ ∧ β ∧ Θ = -2i β(X)(1 + Θ) -2iα ∧ β ∧ (1 + Θ) + β ∧ ι X Θ 2 2 Moreover, since β ∧ Θ 2 = 0, we have β(X)Θ 2 = β ∧ ι X Θ 2 and then ρ (X -ι X Θ) ∧ (Y -ι Y Θ) Φ = -2i(β(X) -2iα ∧ β) ∧ Φ. But ⊥(α ∧ β) = -β(X) = α(Y ).
We obtain then the description of U 0 .

The next proposition describes the space U R 0 of real forms in U 0 . It is a direct consequence of the proposition above. PROPOSITION 4. Let Λ 2 0 be the space of (real) primitive 2-forms.

Then U R 0 = [θ + a(iΩ + 1)] ∧ Φ, θ ∈ Λ 2 0 and a ∈ R .
Remark. We have actually

(Λ 1 ⊕ Λ 3 ) ⊗ C = U -1 ⊕ U 1 and (Λ 0 ⊕ Λ 2 ⊕ Λ 4 ) ⊗ C = U -2 ⊕ U 0 ⊕ U 2 .
For example, the decomposition of a 1-form α ∈ Λ 1 (T * ) is

α = α -iBα 2 ∧ Φ + α + iBα 2 ∧ Φ.
This decomposition is a pointwise decomposition. Denote now by U k the space of smooth sections of the bundle U k . The Gualtieri decomposition is now

Ω * (M ) ⊗ C = U -2 ⊕ U -1 ⊕ U 0 ⊕ U 1 ⊕ U 2 .

DEFINITION. The operator

∂ : U k → U k+1 is simply ∂ = π k+1 • d
The next theorem is completely analogous to the corresponding statement involving an almost complex structure and the Dolbeault operator ∂.

THEOREM (Gualtieri [6]). The almost generalized complex structure J is integrable if and only if

d = ∂ + ∂. EXAMPLE 5. Let α ∈ Ω 1 (M ) be a 1-form. From d(α ∧ Φ) = dα ∧ Φ we get ∂(α ∧ Φ) = i 2 (⊥dα)Φ ∂(α ∧ Φ) = (dα -i 2 ⊥dα) ∧ Φ.
It is worth mentionning that one can also define the real differential operator d J = [d, J], or equivalently (see [START_REF] Cavalcanti | New aspects of the dd c -lemma[END_REF])

d J = -i(∂ -∂).
Remark. Cavalcanty establishes in [START_REF] Cavalcanti | New aspects of the dd c -lemma[END_REF], for the particular case ω = 0, an isomorphism Ξ :

Ω * (M ) ⊗ C → Ω * (M ) ⊗ C satisfying Ξ(dθ) = ∂Ξ(θ), Ξ(δθ) = ∂Ξ(θ)
with δ = [d, ⊥] the symplectic codifferential. Since dδ is the Euler operator, Monge-Ampère equations of divergent type write as ∆ ω = 0 with Ξ(ω) pluriharmonic on the generalized complex manifold M 4 , exp(iΩ) .

Conservation laws and Generating functions. The notion of conservation laws is a natural generalization to partial differential equations of the notion of first integrals.

A 1-form α is a conservation law for the equation ∆ ω = 0 if the restriction of α to any generalized solution is closed. Note that conservations laws are actually well defined up closed forms. EXAMPLE 6. Let us consider the Laplace equation and the complex structure J associated with. The 2-form dα vanish on any complex curve if and only if [dα] 1,1 = 0, that is ∂α 1,0 + ∂α 0,1 = 0 or equivalently ∂α 1,0 = ∂∂ψ for some real function ψ. (Here ∂ is the usual Dolbeault operator defined by the integrable complex structure J.) We deduce that α

-dψ = β 1,0 + β 0,1 with β 1,0 = α 1,0 -∂ψ is a holomorphic (1, 0)-form.
Hence, the conservation laws of the 2D-Laplace equation are (up exact forms) real part of (1, 0)-holomorphic forms.

According to the Hodge-Lepage-Lychagin theorem, α is a conservation law if and only if there exist two functions f and g such that dα = f ω + gΩ. The function f is called a generating function of the Monge-Ampère equation ∆ ω = 0. By analogy with the Laplace equation, we will say that the function g is the conjugate function to the generating function f .

LEMMA. A function f is a generating function if and only if

dBdf = 0.
Proof. f is a generating function if and only if there exists a function g such that

0 = d(f ω + gΩ) = df ∧ ω + dg ∧ Ω = (dg + Bdf ) ∧ Ω,
and therefore g exists if and only if dBdf = 0.

COROLLARY. If f is a generating function and g is its conjugate then for any c ∈ C, L c = (f + ig) -1 (c) is a generalized solution of the Monge-Ampère equation ∆ ω = 0.

Proof. The tangent space T a L c is generated by the hamiltonian vector fields X f and X g . Since

Ω(BX f , Y ) = Ω(X f , BY ) = df (BY ) = Bdf (Y ) = dg(Y ),
we deduce that X g = BX f and therefore L c is closed under B = λ -A 0 . L c is then closed under A 0 and so bilagrangian with respect to Ω and ω. EXAMPLE 7. A generating function of the 2D-Laplace equation satisfies dJdf = 0, and hence it is the real part of a holomorphic function.

The above lemma has a nice interpretation in the Hitchin/Gualtieri formalism: PROPOSITION 5. A function f is a generating function of the Monge-Ampère equation ∆ ω = 0 if and only if f is a pluriharmonic function on the generalized complex manifold (M 4 , exp(ω -iΩ)), that is ∂∂f = 0.

Proof. The spaces U 1 and U -1 are respectively the i and -i eigenspaces for the infinitesimal action of J. So

Jdf = J df -iBdf 2 ∧ Φ + df + iBdf 2 ∧ Φ = i df -iBdf 2 ∧ Φ - df + iBdf 2 ∧ Φ = Bdf + (B 2 + 1)df ∧ Ω.
Moreover,

d (B 2 + 1)df ∧ Ω = d(B 2 df ∧ Ω) = d(Bdf ∧ ω) = (dBdf ) ∧ ω.
We deduce that dJdf = 0 if and only if dBdf = 0. Since dJdf = 2i∂∂f , the proposition is proved.

Decompose the function f as f = f -2 + f 0 + f 2 . Since ∂f -2 = 0 and ∂f 2 = 0, f is pluriharmonic if and only if f 0 is so. Assume that the ∂∂-lemma holds (see [START_REF] Cavalcanti | New aspects of the dd c -lemma[END_REF] and [START_REF] Gualtieri | Generalized geometry and the Hodge decomposition[END_REF]). Then it exists ψ ∈ U 1 such that ∂f 0 = ∂∂ψ.

Define then G 0 ∈ U 0 by G 0 = i(∂ψ -∂ψ). We obtain

∂(f 0 + iG 0 ) = 0
and f 0 appears as the real part of an "holomorphic object". Nevertheless, this assumption is not really clear. Does the ∂∂-lemma always hold locally ?

The following proposition gives an alternative "holomorphic object" when the closed form ω is primitive (that is λ = 0). PROPOSITION 6. Assume that the closed form ω is primitive and consider the real forms U = ω ∧ Φ and V = (iΩ + 1) ∧ Φ.

A function f is a generating function of the Monge-Ampère equation ∆ ω = 0 with conjugate function g if and only

∂(f U -igV ) = 0.
Proof. According to proposition 4, the closed forms U and V live in U R 0 . Therefore,

d J (f U ) = -Jd(f U ) and d J (gV ) = -Jd(gV ). Since J 2 = -1 on U -1 ⊕ U 1 , we get 2∂(f U -igV ) = (d -id J )(f U -igV ) = (1 + iJ)(df U -d J gV ). But, df U = df ∧ ω ∧ Φ = df ∧ ω, and
d J gV = -Jdg ∧ V = -J(idg ∧ Ω + dg ∧ Φ) = - 1 2 J(dg ∧ Φ + dg ∧ Φ) = - i 2 (dg ∧ Φ -dg ∧ Φ) = -dg ∧ Ω. We obtain finally 2∂(f U -igV ) = df ∧ ω + dg ∧ Ω. EXAMPLE 8 (Von Karman equation). The 2D-Von Karman equation is v x v xx -v yy = 0.
The corresponding primitive form is

ω = p 1 dq 2 ∧ dp 1 + dq 1 ∧ dp 2 ,
which is obviously closed. The form U and V are

U = p 1 dq 2 ∧ dp 1 + dq 1 ∧ dp 2 + 2p 1 dq 1 ∧ dq 2 ∧ dp 1 ∧ dp 2 V = 1 + p 1 dq 2 ∧ dp 1 + dq 1 ∧ dp 2 + (p 1 -1)dq 1 ∧ dq 2 ∧ dp 1 ∧ dp 2 2.
3. Generalized Kähler partners. Gualtieri has also introduced the notion of Generalized Kähler structure. This is a pair of commuting generalized complex structure such that the symmetric product (J 1 J 2 ) is definite positive. The remarkable fact in this theory is that such a structure gives for free two integrable complex structures and a compatible metric (see [START_REF] Gualtieri | Generalized complex geometry[END_REF]). This theory has been used to construct explicit examples of bihermtian structures on 4-dimensional compact manifolds (see [START_REF] Hitchin | Instantons, Poisson structures and generalized Kähler geometry[END_REF]).

The idea is that the +1-eigenspace V + of J 1 J 2 is closed under J 1 and J 2 and that the restriction of (•, •) to it is definite positive. The complex structures and the metric come then from the natural isomorphism V + → T .

From our point of view, this approach gives us the possibility to associate to a given partial differential equation, natural integrable complex structures and inner products. Nevertheless, at least for hyperbolic equations, such inner product should have a signature, and we have may be to a relax a little bit the definition of generalized Kähler structure: DEFINITION. Let ∆ ω = 0 be a 2D-symplectic Monge-Ampère equation of divergent type and let J be the generalized complex structure associated with. We will say that this Monge-Ampère equation admits a generalized Kähler partner if it exists a generalized complex structure K commuting with J such that the two eigenspaces of JK are transverse to T and T * .

Note that a powerful tool has been done in [START_REF] Hitchin | Instantons, Poisson structures and generalized Kähler geometry[END_REF] to construct such structures: LEMMA (Hitchin). Let exp β 1 and exp β 2 be two complex closed form defining generalized complex struture J 1 and J 2 on 4-dimensional manifold. Suppose that

(β 1 -β 2 ) 2 = 0 = (β 1 -β 2 ) 2
then J 1 and J 2 commute.

Let us see now on a particular case how one can use this tool. Consider an elliptic Monge-Ampère equation ∆ ω = 0 with dω = 0 and Ω ∧ ω = 0. Assume moreover it exists a closed 2-form Θ such that Ω ∧ Θ = ω ∧ Θ = 0 and 4ω = Ω 2 + Θ 2 . Note that exp(ω -iΩ) and exp(-ω -iΘ) satisfy the conditions of the above lemma. We suppose also that Θ 2 = λ 2 Ω with λ a non vanishing function. This implies that ω 2 = µ 2 Ω 2 with With the triple (ω, Ω, Θ) we construct I + and I -defined by

µ = √ 1 + λ 2
I + = 1 2    0 -1 1 0 -1/p 1 0 0 -1/p 1 -(1 + 4p 1 )/p 1 0 0 -1/p 1 0 1 + 4p 1 -1 0     I -= 1 2     0 -1 -1 0 -1/p 1 0 0 1/p 1 (1 + 4p 1 )/p 1 0 0 -1/p 1 0 -(1 + 4p 1 ) -1 0    
It is worth mentioning that I + and I -are well defined for all p 1 = 0. But the metric G is definite positive only for p 1 < -1 4 . Remark. It would be very interesting to understand the behaviour of generating functions and generalized solution of this kind of Monge-Ampère equations with respect to the Gualtieri metric. In particulary, Gualtieri has introduced a scheming generalized Laplacian dd * + d * d (see [START_REF] Gualtieri | Generalized geometry and the Hodge decomposition[END_REF]) and to know if generating functions (which are pluriharmonic as we have seen) are actually harmonic would give important informations on the global nature of the solutions. This will be the object of further investigations.

2 .EXAMPLE 9 .

 29 The triple (ω, Ω, Θ) defines a metric G and an almost hypercomplex structure (I, J, K)such that ω = µG(I•, •), Ω = G(J•, •), Θ = λG(K•, •).Define now the two almost complex structures •, •)we deduce that I + and I -are integrable.LEMMA. A function g is the conjugate of a generating function f of the Monge-Ampère equation ∆ ω = 0 if and only ifdI + dg = -dI -dg.Proof. f is a generating function with conjugate g if and only if0 = df ∧ ω + dg ∧ Ω = (-µKdf + dg) ∧ Ωthat is if and only if d K µ dg = 0. Consider again the Von Karman equationv x v xx -v yy = 0.with corresponding primitive and closed formω = p 1 dq 2 ∧ dp 1 + dq 1 ∧ dp 2 .Define then Θ by Θ = dp 1 ∧ dp 2 + (1 + 4p 1 )dq 1 ∧ dq 2 .


Mohammad Y Abualhoul
email: mohammad.abualhoul@inria.fr

Oyunchimeg Shagdar
email: oyunchimeg.shagdar@inria.fr

Fawzi Nashashibi
email: fawzi.nashashibi@inria.fr

Visible Light Inter-Vehicle Communication for Platooning of Autonomous Vehicles

Keywords: VLC, Platoon, ITS, Road safety applications

In this paper, we study a use of Visible Light Communication (VLC) technology for a platoon of autonomous vehicles. We present a low-cost, low-latency and simple outdoor VLC prototype, which can be installed as a vehicular taillighting system. The architecture of our VLC system is introduced, followed by performance evaluation with an especial attention on the VLC link resilience to ambient noise and communication range. Through the experiments, we observe that a use of proper optical filter stage at the receiver side, together with narrowing the transmitter Field-of-view (FOV), result in an extended communication range and make the VLC system more resilient to the ambient noises. Experimental results show that the system can provide 30 meters of intervehicle communication with 36 ms of latency, yet on sunny day conditions. The benefit of using the VLC system for platooning control is showed using a Simulink system that integrates our VLC platform for inter-communications to simulates the performance of autonomous vehicles platoon.

I. INTRODUCTION

Road safety applications of Intelligent Transportation Systems (ITS) require high communication reliability and low latency. Although the IEEE802.11p technology [1] is standardized to facilitate the V2X communication for such applications, a large number of studies reported that this technology still experiences some drawbacks, such as the channel congestion and the increased delay in highly dense vehicular traffic scenarios [START_REF] Eichler | Performance evaluation of the ieee 802.11p wave communication standard[END_REF]. In recent years, Visible Light Communication (VLC) is emerging as a secure and costeffective technology for data transmission with a broad range of potential applications. In this work, we focus on testing the use of VLC as an integral supportive technology for ITS road safety applications, such as platooning. In this application, a specific number of vehicles form a group and travel together at high speed with relatively small inter-vehicle distances.

Indeed, one of the most efficient ways to increase road capacity and improve safety in highly dense vehicular traffic scenarios; is by driving vehicles in platoons with constant inter-vehicle spaces [START_REF] Varaiya | Smart cars on smart roads: problems of control[END_REF]. According to [START_REF]Vehicle Dynamics and Control[END_REF] and [START_REF] Fernandes | Platooning of autonomous vehicles with intervehicle communications in sumo traffic simulator[END_REF], road capacity can be increased dramatically by the use of tightly spaced platoon formation as expressed by the following formula:

C = v n nl + (n -1)D + d , (1)
where d represents the inter-platoon distance, D the intraplatoon distance, l the vehicle length, v the steady-state speed, and n the number of vehicles in each platoon.

As the constant spacing model suggest, relatively high dense vehicular scenarios can be managed efficiently using platoon formations. Although using radio technology for intra-platoon communication have been previously studied [START_REF] Biswas | Vehicle-to-vehicle wireless communication protocols for enhancing highway traffic safety[END_REF], [START_REF] Fernandes | Platooning with ivc-enabled autonomous vehicles: Strategies to mitigate communication delays, improve safety and traffic flow[END_REF], the improvement of the traffic capacity can on the other hand dramatically increase channel congestion and delay of the radio channel. Maintaining safety requirements for road applications with the presence of this trade-off between road capacity and channel congestion was the main motivation behind suggesting VLC as Vehicle-to-Vehicle (V2V) supportive communication link between platoon members. The use of vehicular tail-lighting to exchange information between platoon members as depicted in Fig. 1, can play a key role in reducing the channel congestion. VLC technology showing significant progress in last decades by taking advantage of several accumulated knowledge and resources such as infrared [START_REF] Kahn | Wireless infrared communications[END_REF] and the recent advances on high power Light Emitting Diodes (LEDs). Nowadays, LED takes place in every modern lighting system including vehicular lighting and signaling systems [START_REF] Peck | Solid state led lighting technology for hazardous environments; lowering total cost of ownership while improving safety, quality of light and reliability[END_REF], the low cost and high switching speed expected to provide an opportunistic deployment of VLC in the short run.

Also, coast-efficiency and security are few of many other interesting proprieties make VLC more suitable candidate to be deployed as supportive communication technology for the proposed application in this work. While many existing efforts have investigated how VLC can be applied to vehicular applications, very few studies have analyzed and demonstrated the technology in real world conditions [START_REF] Kim | Experimental demonstration of vlc-based vehicle-to-vehicle communications under fog conditions[END_REF].

In this paper, we extend our previous study [START_REF] Abualhoul | Platooning control using visible light communications: A feasibility study[END_REF] for the two key elements necessary to recognize VLC technology in ITS applications: (i) the feasibility of VLC for outdoor conditions and under constraints posed mainly by the ambient noise and day light conditions, (ii) the capability of this technology to satisfy the vehicular platooning application safety requirements. This work makes the following contributions:

• We develop a VLC prototype by taking into account of the followings. 1) Using precise optical filtering techniques at the receiver end, allowing us to increase the prototype robustness against ambient noise. 2) Using off-the-shelf components in our implementation, with the observance of the standard feasible form factor of the vehicular lighting system. • We evaluate the performance of the VLC prototype under daylight conditions and for Line-of-sight (LOS) scenario.

• The performance of platooning control, which uses our VLC platform for inter-vehicle communication is evaluated using a Simulink model. The remainder of this paper is organized as follows. Section II identify the requirements for the platoon as hardsafety application. Section III introduces our VLC system architecture with the details of the prototype transmitter and receiver. In section IV, we report the experimental setup, together with the experimental results. Finally, simulation evaluation of VLC-based platoon using Simulink model is presented in V. Conclusion and future work are given in Section VI.

II. PLATOONING APPLICATION REQUIREMENTS

The standardized vehicular communications technology specifies the vehicle safety services and their performance requirements based on two primary metrics. i) Packet-Delivery-Ratio (PDR), which indicate the ratio between the number of the successfully received packets and the number of transmitted packets. ii) Latency, the time needed of a packet to reach its destination. According to the US department of transportation, 8 out of 75 ITS services were recognized as a hard safety with respect to the safety applications requirements [START_REF]Vehicle safety communications project task 3[END_REF]. As detailed in TABLE.I, the highest priority safety services such as crash avoidance and lane change warning target 100% PDR level and latency of 10 to 100 ms at the utmost. In our study, we consider these hard safety requirements for the VLC-based inter-vehicle communication for platoons.

III. ARCHITECTURE OF VLC PLATFORM

The implementation of VLC platform proposes to exploit the vehicular tail-lights system and reduce both cost and installation complexity. In order to realize the VLC double functionality of both data transmission and vehicular lighting, the transmitter wavelength chosen to be in red color visible spectrum range (635 nm) and an emission power of 3 Watt for each of the 6 LED array, representing the vehicle taillights. As for the receiver end, the main components are the silicon Photo-Diode (PD) with a switchable gain detector PDA100A, and the optical band-pass filter with an operation band match the LED emission wavelength. Fig. 2 shows the VLC overall transmitter and receiver block diagrams. As illustrated in the figure, in adding to the VLC ends, the transmitter and receiver architectures consist RTMaps platforms and Arduino micro-controllers. RTMAps is used to prepare specific packet format and connect the transmitter and receiver component with the micro-controller. For processing simplicity, a low-cost Mini-pro Arduino boards based on ATmega328P micro-controller has been used at both transmitter and receiver ends to perform modulation/demodulation, line coding, signal processing and interfacing the vehicles controller area network (CAN-BUS).

A. VLC Transmitter

The transmitter unit is composed of three main components, LEDs with mounted focusing lenses on the top of each, driving circuit and Arduino micro-controller as depicted in Fig. 4(a). In this implementation, 6 high-power (3Watt) LEDs illuminate 635 nm red-light data beam in which calibrated to represent the vehicle tail-light intensity.

The vehicle CAN-BUS data are packed into a specific packet format, where RTMaps [START_REF]intempora[END_REF] extracts the vehicle status (speed, acceleration and steering angle) accompanied with timestamp and packet counter, then generating a particular packet format as shown in Fig. 3. These data are then adapt Manchester line coding in the micro-controller before modulating the LED array. Manchester coding (also known as phase encoding) was used here based on the fact that the energy component of the encoded signal is not dependent on the data itself. Also, each transmitted bit in this context contains the same amount of energy and results in a constant intensity over transmission, either we send 0's or 1's, which ensure that human eye will never observe any light intensity fluctuation during data transmission.

The transmitter LED's array is then driven by the coded data using high-speed MOSEFET switching circuit, which applies the On-Off-Keying (OOK) intensity modulation.

B. VLC Receiver

The receiver unit depicted in Fig. 4(b &c) composed of four main components: an optical filter, photodiode, tansimpedance amplifier and decision circuit. The optical filter was chosen specifically to operate in a narrow bandwidth (635nm +/-1nm) in order to provide immunity against any ambient noise source outside this filter range, and at the same time match the transmitter LED emission wavelength.

LED Array

PD decision circuit PD + mounted filter The used decision circuit LT1715 has signal threshold voltage comparator with a reference voltage 2.1 V. This stage represents the receiver sensitivity, where the comparison reference value has been obtained empirically based on the impact of the ambient noise levels for many different scenarios and different receiver/transmitter positioning. Here, we adopt the simple Intensity Direct Detection (IDD) method.

C. Channel model

VLC channel characteristics are defined mainly by the LED lambertian emission pattern and both transmitter and receiver physical characteristics [START_REF] Kahn | Wireless infrared communications[END_REF], where the path-loss can be presented as:

P loss = (m+1)A ph 2πd n cos m (ϕ) (ψ) cos (ψ) , 0 < ψ < ψ c 0 elsewhere . (2)
For VLC path-loss theoretical model, the path-loss exponent n is equal to 2, where d is the separation distance between transmitter and receiver, ψ c is the PD field of view (FOV) representing the maximum incidence angle. A ph is the physical area of the PD and m = -ln 2 ln(cos φ ) is lampertian emission order, which is a key parameter specifying the directivity of the transmitter. The LED half-power angle φ has a remarkable influence on the coverage range and pattern shape of the lambertian light source. By choosing narrower φ , we can directly increase the communication range range.

IV. EXPERIMENTAL SETUP AND RESULTS

In our previous work [START_REF] Abualhoul | Platooning control using visible light communications: A feasibility study[END_REF], [START_REF] Abualhoul | Enhancing the field of view limitation of visible light communication-based platoon[END_REF], we have investigated the theoretical VLC channel model and evaluated its feasibility for platoon application on simulation basis, the two studies addressed the technology limitations and proposed a solution to enhance the FOV for sharp curvature scenarios. In this work, we step forward and compare the existing model with empirical results obtained experimentally from our developed platform. An independent power source supplied the transmitter module with constant 24 voltage and 700 mA in order to bias the LED array. Receiver module was also supplied by an independent power source in order to enable the trans-impedance amplifier. Arduino micro-controllers at both transmitter and receiver ends were powered by the 5 Volt USB connection and the height were fixed to 85 cm from the ground level. More parameters regarding the experimental setup are given in TABLE II.

A. Experimental setup

B. Experimental results

In our LOS scenario, the distance between transmitter and receiver ends is much greater than optical detector size A ph . Moreover, the terms cos m (ϕ) and cos(θ ) can set to equal 1 by maintaining aligned direct distance between both transmitter and receiver, where θ = ϕ = 0 . It thus follows that one can simplify the path-loss expression in (2) and write the received power with respect to the transmission power as: Based on the fact that all the influencing parameters except the direct distance between transmitter and receiver were fixed. One can encapsulate the constant parameters over the experiment steps (m+1)A ph 2π P T in the term A 0 , where a linear least-square relationship of the received power is expected over the variation in d as the following:

P R = P Loss P T = (m + 1)A ph 2πd n P T . (3)
P R = A 0 d n . (4)
To estimate the values of the two unknowns A 0 and n, we applied a regression process to the acquired received power as shown in Fig. 6. The results indicate that there is perfect agreement between the theoretical lambertian pathloss model expressed by [START_REF] Eichler | Performance evaluation of the ieee 802.11p wave communication standard[END_REF] and the empirical measurements, where the estimated path-loss exponent n ≈ 2. Acquired data for link range shorter than 3 meters has shown unstable behavior at the receiver end due to the current saturation in the PD. We found out that, to overcome this problem, the applied gain, and the decision circuit threshold level should be dynamically adjusted in relevance to the intervehicle distance d. Based on our findings, and for the exact experiment conditions, we suggest using gain levels of 10-20 dB for ranges less than 3 meter and higher levels for the longer communication ranges.

Figure 7 shows the packet delivery ratio (PDR) results with respect to the inter-vehicle distance d., while receiver gain was set to 40 dB. The results show that 100 % of PDR is achievable for up to 30 meters inter-vehicle distance, which is realistic covering range for platooning application. The link end-to-end delay performance is shown in Fig. 8, the acquired results show that the system can deliver the packets without exceeding 36 ms of delay time, which perfectly satisfies the requirements of the many hard safety applications, including the platoon. Figure 9 represents the throughput performance of the same platform. Here, throughput is the number of successfully received useful information bits during one second, excluding the headers, training bits, and the preambles.

One may think that the obtained throughput of 3.8 Kbps is very low, especially if compared to e.g., 6 Mbps of the IEEE 802.11p technology. However, it should be noted that there is no direct comparison in this context, the radio channel of the IEEE 802.11p is shared medium over approximately 1 km of geographical area, which may introduce hundreds of vehicles sharing the same channel. On the other hand, our proposed VLC channel for platoon is hardly shared by two transmitting vehicles, due to the strict LOS and FOV requirements. Moreover, our system throughput can be further improved, since the main bottleneck of the current system is due to the processing speed, platform implementation, and decision circuit, but not the physical VLC channel itself.

V. PERFORMANCE EVALUATION OF VLC-BASED

PLATOON

Platooning control is traditionally based on perception techniques that use vehicle embedded sensors, such as radars. More specifically, they usually use sensors to measure the distance between vehicles and apply, e.g., a classical proportional integral controller to control the velocity v f (t) with respect to time t of the follower vehicle. For a measured inter-distance d(t) at a time t, the velocity of the follower vehicle is given by:

v f (t) = K p • e(t) + K i • t 0 e(τ) dτ, (5)
where K p and K i are respectively the proportional and the integral gains, the error e(t) is

e(t) = d re f -d(t). (6)
These types of perception-based platooning control, however, suffer from the chain instability problem. The problem can be reduced or even eliminated if vehicles in the platoon can exchange information with the platoon leader. For this reason, a large number of efforts have been made on Cooperative Adaptive Cruise Control (CACC). For example, in the platooning controller designed in [START_REF] Rajamani | Demonstration of integrated longitudinal and lateral control for the operation of automated vehicles in platoons[END_REF], the platoon members receive information from the vehicle in front as well as from the platoon leader.

The main advantage of this platooning control is that the system can be proven to be stable under a constant spacing policy. As illustrated by [START_REF] Fernandes | Platooning with ivc-enabled autonomous vehicles: Strategies to mitigate communication delays, improve safety and traffic flow[END_REF] (the parameters of are described in Table III), in order to ensure both the leader-follower and the predecessor-follower string stabilities, vehicles must compute their desired acceleration ẍdes in parallel and in real time. To do so, each vehicle must receive the acceleration from their immediate predecessor ẍi-1 , and the leader vehicle's speed V l , where platoon leader acceleration ẍl must be available to all platoon members [START_REF] Dunbar | Distributed receding horizon control of vehicle platoons: Stability and string stability[END_REF]. Apparently, the VLC technology can be used for the former type of information exchange, i.e., bumper-to-bumper communication. On the other hand, for the latter type of information exchange, i.e., communication from the leader to the platoon members, multi-hop data delivery based on VLC is maybe an approach, but it is probably more logic to consider the radio communication technology.

ẍdes = (1 -C 1 ) ẍi-1 +C 1 ẍl -(2ξ -C 1 (ξ + ξ 2 -1))ω n εi -(ξ + ξ 2 -1)ω n C 1 (V i -V l ) -ω 2 n ε i . (7)
In a simpler model, the individual platoon members can require information only from their preceding vehicles, i.e., bumper-to-bumper communication. This type of platooning control shows inferior stability performance compared to that presented in [START_REF] Rajamani | Demonstration of integrated longitudinal and lateral control for the operation of automated vehicles in platoons[END_REF]. Nevertheless, the performance should be much better compared to the classical perception-based (which uses only the vehicle local sensors) technology. Since the scope of this paper is VLC, which perfectly fits to bumper-to-bumper communication, we apply our VLC platform to this type of platooning control (requires only bumper-to-bumper communication) and evaluate its performance.

Assuming that a platoon consists of homogeneous autonomous vehicles, we evaluate the performance of platooning control, where individual vehicles use the reference speed of their preceding vehicles. In the evaluations, the Inria's autonomous vehicles, Cycab [START_REF]Cycab autonomous vehicle[END_REF], are modeled in the Simulink, to which our VLC platform is connected by a serial port to realize the bumper-to-bumper communication (see Fig. 10). The performance of the VLC-based platoon is compared to that of perception-based platoon, which does not use communication. The results of the perception-based and VLC-based platoons are depicted in Figs. 11 and12, respectively, which plot the velocities of the individual vehicles with respect to the simulation time. As Fig. 11 shows, in the perception-based platoon, each follower vehicle needs approximately 1 second to adjust its speed to that of the preceding vehicle. In contrast, Fig. 12 shows that when our VLC system is used to allow inter-vehicle communication, the time necessary to adjust the velocity is only 35 ms, clearly indicating the benefits of using the VLC system for intra-platoon communication.

Cycab0 VLC Cycab1 VLC Cycab2 V 1 V 0

VI. CONCLUSIONS

Visible light communications have been lately considered as a supportive candidate for vehicular communications, especially in high dense traffic scenarios. However, very few effort have been made to study and address the technology for specific ITS applications considering the real-world conditions. In this work, we tried to fill the missing gap. We presented the system architecture, implementation, and the evaluation of a practical outdoor VLC platform for autonomous platoons. We performed several tests that intended to measure the performances of the received optical power, packets delivery ratio, link throughput and the end-to-end delay. The measured VLC path-loss characteristics showed a substantial agreement with the theoretical model. The platform operated with an excellent end-to-end communication performances regarding the platoon safety requirements of both PDR and delay. We also evaluated the impact of the actual VLC system on the platoon performance using a Cycab Simulink model and compared the results to those of perception-based platooning control. In our ongoing work, and after integrating the platform with an actual Cycab, we aim to improve the system bit rate and re-perform the experiment under the presence of more challenging weather conditions; such as fog, rain, and snow. A sufficient handover algorithm between VLC and radio communication and/or laser sensing systems is intended to be implemented on a future work basis.

Fig. 1 .

 1 Fig. 1. VLC used for intra-platoon communications.

Fig. 2 .

 2 Fig. 2. The transmitter and receiver diagrams of the VLC platform.

Fig. 3 .

 3 Fig. 3. Packet format for intra-platoon communication.

Fig. 4 .

 4 Fig. 4. VLC transmitter and receiver ends.

Figure 5 Fig. 5 .

 55 Figure5shows the outdoor VLC experimental setup, this setup was dedicated to studying the range and the communication link quality in outdoor environment and for a direct LOS, where more sophisticated scenarios such as maneuvers and Lane-change are intended to be our ongoing work. All tests were performed in a sunny daytime of December the 12th between 12:00 and 14:00, Paris time. Multiple iterations of data recording were performed at the receiver end, in which we record data for each +500 cm increment in d, starting from d = 1 meter.

Fig. 6 .

 6 Fig. 6. Empirical estimation of the VLC link Path-loss exponent, A 50dB = 432.0479, n 50dB = 1.996, A 40dB = 139.4479, n 40dB = 1.99.

Fig. 7 .

 7 Fig. 7. Packet Delivery Ratio (PDR) of the VLC platform for 40 dB gain.

Fig. 8 .

 8 Fig. 8. The end-to-End delay performance of the VLC platform for 40 dB gain.

Fig. 9 .

 9 Fig. 9. The throughput performance of the VLC platform for 40 dB gain.

Fig. 10 .

 10 Fig. 10. Simulink Model for 3 Cycabs forming VLC-based platoon

Fig. 11 .

 11 Fig. 11. Performance of the perception-based platoon.

Fig. 12 .

 12 Fig. 12. Performance of the VLC-based platoon.

ACKNOWLEDGMENT

The authors would like to acknowledge the support of all RITS team members. Also, grateful to Vicente MILANES for providing the Cycab Simulink model.


A C Sparavigna

R Marazzato

Evaluation of GIMP Retinex Filtering of Images by Means of the Shen++ Max Shannon Entropy Finder

Keywords: Image Processing, Foggy Images, Nighttime Images, Retinex, Shannon Entropy, Generalized Entropies

Retinex filters are used for processing and enhancing images, obtaining very good results when the images have a degraded contrast, such as those recording foggy or nighttime scenes. Among the several existing Retinex filters, one freely available on the web is that provided by GIMP, the GNU Image Manipulation Program. This software has some parameters that can be adjusted to optimize the output image. Here we investigate a criterion for the optimization of filtering parameters, based on the maximization of image entropy, evaluated using the Shannon formalism. For this task a software, the Shen++ Max Shannon Entropy Finder, is proposed. The software is working on a bulk set of images generated after processing the original image with several different values of the filtering parameters, giving the filtered images having the largest and the smallest Shannon entropy.

Introduction

In recent papers [START_REF] Sparavigna | An image processing approach based on Gnu Image Manipulation Program Gimp to the panoramic radiography[END_REF][START_REF] Sparavigna | Gimp Retinex for enhancing images from microscopes[END_REF], we have proposed and discussed the use of the Retinex filter of GIMP, the GNU Image manipulation Program, for enhancing images coming from radiographic devices and from microscopes. In general, a Retinex filter can be used for processing and enhancing images, obtaining very good results when they have a degraded contrast, such as the images recording foggy or nighttime scenes. The fact that a Retinex filters is suitable for processing such images is based on its ability of simulating the human vision. Humans are able of seeing details both in the shadows and in the nearby illuminated areas, whereas a photograph or a digital image of the same observed scene are showing either the shadows as too dark or the bright areas as overexposed [START_REF] Barnard | Investigations into multi-scale Retinex[END_REF]. The fundamental reason of such discrepancies is that human vision has some specific features concerning colours, brightness and contrast of a scene that are quite different from those of recording devices [START_REF] Jobson | A Multi-Scale Retinex for bridging the gap between colour images and the human observation of scenes[END_REF]. Inspired by the human vision biological mechanism to adapt itself to these conditions, Edwin H. Land, an American scientist and inventor, best known as co-founder of Polaroid Corporation [START_REF] Land | An alternative technique for the computation of the designator in the retinex theory of color vision[END_REF][START_REF] Land | Recent advances in Retinex theory and some implications for cortical computations[END_REF][START_REF] Land | Recent advances in Retinex theory[END_REF][START_REF] Land | Experiments in color vision[END_REF][START_REF] Land | Color vision and the natural image[END_REF], devised and proposed the first Retinex method of processing images. Through his research works, Land evolved his model until the last version proposed in 1986. The term "retinex" was coined by Land himself combining the words "retina" and "cortex", and used to indicate the results of his researches, which were showing that the human colour perception is involving all levels of vision processes, from the retina to the cerebral cortex. After the first method proposed by Land, several other Retinex approaches had been developed [START_REF] Jobson | A Multi-Scale Retinex for bridging the gap between colour images and the human observation of scenes[END_REF][START_REF] Land | An alternative technique for the computation of the designator in the retinex theory of color vision[END_REF][START_REF] Land | Recent advances in Retinex theory and some implications for cortical computations[END_REF][START_REF] Land | Recent advances in Retinex theory[END_REF][START_REF] Land | Experiments in color vision[END_REF][START_REF] Land | Color vision and the natural image[END_REF][START_REF] Jobson | Properties and performance of a center/surround Retinex[END_REF]: the single-scale Retinex (SSR), the multiscale Retinex (MSR), and, for colour images, the MultiScale Retinex with Colour Restoration (MSRCR). GIMP Retinex is a freely available tool of this last family, developed by Fabien Pelisson [START_REF] Pelisson | GIMP Retinex[END_REF].

Due to the specific features of the Retinex filtering, this filter seems very effective in enhancing the images concerning foggy and nighttime scenarios [START_REF] Sparavigna | Retinex filtering and thresholding of foggy images[END_REF][START_REF] Marazzato | Retinex filtering of foggy images: generation of a bulk set with selection and ranking[END_REF][START_REF] Wanting | Variable filter retinex algorithm for foggy image enhancement[END_REF][START_REF] Dong | Based on Retinex theory of restoration for degraded foggy images[END_REF][START_REF] Xinlong | Calculation model and algorithm in foggy image enhancement[END_REF][START_REF] Cai | Context enhancement of nighttime surveillance by image fusion[END_REF][START_REF] Yamasaki | Denighting: Enhancement of nighttime images for a surveillance camera[END_REF][START_REF] Lin | Multi-scale retinex improvement for nighttime image enhancement[END_REF]. In a Retinex filtering of images, such as in the GIMP Retinex filtering, the resulting image can be adjusted selecting different parameters and then, to each set of parameters an output image is corresponding, having a different visual rendering. Of course, we can decide to adjust the image in consequence of a qualitative approach based on our personal view; however, if our aim is that of having an automatic choice of the best output image, a decision method based on a quantitative evaluation of the filtering is necessary. Since each image is composed of colour tone having in it a specific frequency, if we apply to any image a filtering, we have consequently that the filtered image possesses a different distribution of frequencies of colour tones. Here we use a quantitative method based on the Shannon entropy. A software, Shen++ Max Shannon Entropy Finder, is also proposed; this software, working on a bulk set of images generated after processing the original image with several different values of filtering parameters, is giving a filtered image having the maximum (and the minimum) of Shannon entropy.

Image entropy

In 1948, Claude Shannon defined the entropy H of a discrete random variable X, as the expected value of its information content, that is, H(X) [START_REF] Shannon | A mathematical theory of communication[END_REF][START_REF] Borda | Fundamentals in information theory and coding[END_REF]. In this expression, I is the information content of X, the probability of i-event is i p and b is the base of the used logarithm (common values are 2 or Euler's number e). Therefore, the entropy is a measure of the information contained in a sample set of possible events. If we consider an image and its grey tones, we can calculate an entropy by using the histogram of the tones, instead of the probability distribution. In this manner, we have the "image entropy". In the calculus of image entropy, the Shannon entropy can be used but also the generalized entropies of Tsallis and Kaniadakis [START_REF] Tsallis | Possible generalization of Boltzmann-Gibbs statistics[END_REF][START_REF] Kaniadakis | Statistical mechanics in the context of special relativity[END_REF][START_REF] Sparavigna | Shannon, Tsallis and Kaniadakis entropies in bi-level image thresholding[END_REF][START_REF] Sparavigna | Tsallis entropy in bi-level and multi-level image thresholding[END_REF][START_REF] Sparavigna | On the generalized additivity of Kaniadakis entropy[END_REF][START_REF] Sparavigna | Relations between Tsallis and Kaniadakis entropic measures and rigorous discussion of conditional Kaniadakis entropy[END_REF]. In the Appendix, definitions and relations between these entropies are shortly discussed for reader convenience. Since each image has its entropy, when we apply a filtering on it, its histogram is modified and consequently the entropy is changed. If the filtering has some adjustable parameters, we have the possibility of optimizing them, according to a suitable best choice. In [START_REF] Marazzato | Retinex filtering of foggy images: generation of a bulk set with selection and ranking[END_REF] for instance, we proposed for the GIMP Retinex a ranking of output images, based on the local variance of grey tones. Here, we simply consider the use of the entropy; in this approach, the best choice of parameters is that corresponding to a filtered output image, which is maximizing entropy. The GIMP Retinex is Multi-Scale Retinex with Colour Restoration (MSRCR) [START_REF] Jobson | Properties and performance of a center/surround Retinex[END_REF], which is a freely available tool developed by Fabien Pelisson [START_REF] Pelisson | GIMP Retinex[END_REF]. The output image of GIMP-Retinex can be adjusted selecting different levels, scales and dynamics. We have three "levels" of filtering. The uniform level is treating both low and high intensity areas in the same manner. The low level enhances the lower intensity areas of the image whereas the high level is favouring the clearer areas. Another parameter of the filter is the "scale", which determines the depth of the Retinex scale. A "scale division" determines the number of iterations in the multiscale Retinex filter. The last parameter is a "dynamic" slider, which allows adjusting color saturation contamination around the new average color. The default values of scale, scale division and dynamic slider are 240, 3 and 1,2 respectively. In the Figure 1 we are showing an example of the use of GIMP Retinex on an image having pixels with grey tones. It is a showing a foggy street. The original image is given in the left-upper panel; the other images had been obtained using GIMP Retinex at uniform, low and high levels. The best level seems to be that obtained using the low level.

∑ = i i i ) )I(x p(x ∑ - = i i b i ) p(x log ) p(x
Let us stress that GIMP Retinex, like any Retinex filter had been developed for being used on images having colour tones. An example is proposed in the Figure 2, where we observe a nightime scene. The original image is given in the left-upper panel, and, as in the case of the Figure 1, the other images are the outputs of GIMP Retinex filtering at uniform, low and high levels.

Evaluating entropy with the Shen++ Max Shannon Entropy Finder

Let us consider a grey tone image and calculate the Shannon entropy image and the Kaniadakis entropy, in the range of its entropic parameter [0,0.1] (see the appendix for formulas). The Shannon entropy is the limit value of Kaniadakis entropy when the entropic index approaches zero. Let us start from four images given in the Figure 1. We can see the original image and the three images obtained after filtering with GIMP Retinex setting the default values, in the case of the three levels, uniform, low and high. Looking at Figure 1 it seems that it is the low level the possible best choice, and in fact, as we can see in the plot of Figure 3, the image obtained with the filter at low level has the highest value of entropy. Of course, besides the choice of level, we have also to determine the values of the other three parameters. Let us consider the dynamic slider for instance. In the Figure 4, we have four images obtained from filtering in low level, with the default values of scale and scale division, but having different values of dynamic slider. In this case, the choice of the best image is not so evident. If we consider their entropies, we have the results given in the Figure 4. In this figure, it is also plotted the entropy of the original image as reference. Note that, in this case, two of the filtered image are worse than the original. Figures 3 and4 are showing the entropy based on the grey tones of the image. These examples and other given in [START_REF] Sparavigna | Effects of GIMP Retinex Filtering Evaluated by the Image Entropy[END_REF] are showing that the entropy is sensitive in distinguishing the role of parameters. However, these examples are also telling that, when we decide to adjust all the parameters, it is necessary a calculation of the entropy on a bulk set of images, such as that considered in [START_REF] Marazzato | Retinex filtering of foggy images: generation of a bulk set with selection and ranking[END_REF]. For the approach described in [START_REF] Marazzato | Retinex filtering of foggy images: generation of a bulk set with selection and ranking[END_REF], we developed a simple script in Script-Fu, including loops used to generate the bulk set of images processed with GIMP Retinex, with all different values of its input parameters (s,n,d,l) in their range Ω. Parameters s, n, d and l, are the scale, scale division, dynamics and level parameters respectively. To evaluate the images having the largest or the lowest entropies in a bulk set we have developed the Shen++ Max Shannon Entropy Finder, a software implemented in Visual Basic. The software is working on the histogram of equivalent colour tones, that is (R+G+B)/3, where R,G and B are the colour tones of pixels. In the Figure 5 we can see the snapshots of the software in two cases, when it is evaluating the entropy of a single image (maximum and minimum of entropy are coincident), and when it is working on the bulk set of images generated with GIMP Retinex. The values of the corresponding parameters are encoded in the name of the output image.

Discussion

The criterion of considering the image having the maximum entropy as the best output of the GIMP Retinex filtering, was preliminary proposed in Ref. [START_REF] Sparavigna | Effects of GIMP Retinex Filtering Evaluated by the Image Entropy[END_REF], and discussed there with an analysis of a few images. Here, using the Shen++ Max Shannon Entropy Finder , we had the possibility to work on a large set of images. As a result we have observed that, in the examined cases from which we have proposed the examples given in the Figures 567, the output having the maximum of entropy is giving also a good visual effect. It happens, in these cases, that the entropy of the selected Retinex filtered image is about 25% larger than the entropy of the original image. This percentage value could seem small. However we have to consider that the Shannon entropy of any image is limited by a precise value. Let us consider the entropy in the case of a discrete distribution of probability which is uniform. It is immediate to see that, if we have N values, the entropy is equal to H

∑ = = - = - = N i N N N N 1 ln ) / 1 ln( ) / 1 ln( ) / 1 ( .
If N is the number of the colour tones, N=255, the entropy in the case of an uniform distribution of colour tones is equal to 5.54. The entropy of the uniform distribution is also the largest entropy we can find from an image. Then, since the entropy of the original image is quite large, the entropy of the GIMP Retinex processed image can increase of a relatively small amount. However, there are also cases for which a large variation of entropy is observed. An example is given in the Figure 8. The case proposed in this figure has the original image with a histogram possessing two peaks at black and white tones. The entropy of the original image is 1.09, then quite smaller of the limit value of entropy, 5.54. In the bulk set of the filtered images we find a maximum entropy of 4.68. This entropy is more than four times the entropy of the original image. The large increase of entropy is produced by the GIMP Retinex which is introducing several other colour tones in the black and white regions. The output image corresponding the maximum of entropy, given in the Fig. 8, has new colour tones and a histogram which has increased its richness and consequently the entropy. In this manner, from the simple example of Fig. 8, we can see that a Retinex software can highlight the information contained in the image, by adding new color tones in the local environment. In fact, after our analysis with the Shen++ Max Shannon Entropy Finder, we can agree to the conclusion of [START_REF] Seok | A novel method of determining parameters of CLAHE based on image entropy[END_REF] about entropy: "although entropy does not directly represent the enhancement of image quality, it can depict the richness of details. In addition, when average and root mean square contrast do not change, entropy is a useful factor to understand the local or global contrast variation in the given image". In fact, we are working for preparing a new version of the software which is also considering other statistical parameters, such as average intensity and contrast, for investigating other features of the GIMP Retinex filtering of images.

Figure 1 :

 1 Figure 1: The original image is given in the left-upper panel, (Courtesy: Ian W. Fleggen, Wikipedia, 20880313, Foggy Street). The other images had been obtained using GIMP Retinex at uniform, low and high levels. The best level seems to be the low one.

Figure 2 :

 2 Figure2: The original image is given in the left-upper panel, (Courtesy: Melari, Wikipedia, Curchby night in Monza). The other images had been obtained using GIMP Retinex at uniform, low and high levels. As in the Figure1, the best level seems to be the low one.

Figure 3 :

 3 Figure 3: Of the images given in Fig.1, we can calculate the entropies. The plot shows the Kaniadakis entropy as a function of its entropic index. When this index approaches zero, Kaniadakis entropy becomes the Shannon entropy. The entropy is maximized in the case of the low-level filtered image.

Figure 4 :

 4 Figure 4: Considering low-level filtered images, we can adjust the value of the dynamic slider. The default value is 1,2. In the panel on the right, we can find the image entropies of the four images on the left.

Figure 5 :

 5 Figure 5: To find the images having the largest and the smallest entropy in a bulk set we have developed the Shen++ Max Shannon Entropy Finder, working on the histogram of the equivalent colour tones. Here we can see the snapshots of two cases. In the upper part, we see the evaluation of entropy on the single, original colour image, from which the grey tone image of Figure 1 had been obtained. Note that, since we have just one image, the maximum and minimum of entropy are coincident. In the lower part, we can see the images having the maximum and minimum of entropy from the bulk set of images generated with GIMP Retinex. The values of the corresponding parameters are encoded in the name of the output image.

Figure 6 :

 6 Figure 6: Two snapshots Shen++ Max Shannon Entropy Finder. In the upper part, we see the evaluation of entropy on the single, original image of Figure 2. In the lower part, we can see the images having the maximum and minimum of entropy from the bulk set of images generated with GIMP Retinex.

Figure 7 :

 7 Figure 7: Two snapshots Shen++ Max Shannon Entropy Finder. In the upper part, we see the evaluation of entropy on the single, original image used for Ref. 13. In the lower part, we can see the images having the maximum and minimum of entropy from the bulk set of images generated with GIMP Retinex.

.

Figure 8 :

 8 Figure 8: Two snapshots of the Shen++ Max Shannon Entropy Finder. In the upper part, we can see that the entropy of the black and white original image is 1.09 . In the lower part, we can see the images having the maximum and minimum of entropy. The large entropy (4.68) of the output image is obtained because GIMP Retinex software, for the given parameters corresponding to the output image, is adding several colour tones that are enriching the histogram.

Appendix

In the following formulas, we can see how the entropies of Shannon, Tsallis and Kaniadakis are defined. Tsallis and Kaniadakis entropies are generalized entropies depending on and entropic index. To apply these formulas to images, we have to consider p i as the frequency of grey tones. The index I is giving the specific grey tone. We have:

In (2A),(A3) we have entropic indices q and κ. Note that

If we have two independent subsystems A,B, the joint entropies H(A,B) are given by:

S(B) S(A) B) S(A, (A4) + = q)T(A)T(B) (1 T(B) T(A) B) T(A, (A5)

Note that for the generalized additivity of κ-entropy, we need another function containing probabilities (see [START_REF] Sparavigna | On the generalized additivity of Kaniadakis entropy[END_REF] and references therein). When we have a small value of the entropic index, function ℑ is equal to 1. For relations existing between Kaniadakis and Tsallis entropies, see please Ref. 27.