Thierry Hargé

Borel summation of the small time expansion of the heat kernel. The scalar potential case * †

Keywords: heat kernel, quantum mechanics, semi-classical, asymptotic expansion, Wigner-Kirkwood expansion, Borel summation, Poisson formula, Wiener formula, Feynman formula; A.M.S. subject classification: 35K08, 30E15, 35C20, 81Q30

Let pt be the heat kernel associated to the operator -∆+V (x) defined on Ê ν . We prove, under rather strong assumptions on V , that the small time expansion of pt is Borel summable. If V is defined on the torus, we prove a Poisson formula.

Introduction

Let ν ∈ AE * and V be a regular square matrix-valued function on Ê ν . Denote ∂ 2

x := ∂ 2 x1 + • • • + ∂ 2 xν and (x -y) 2 := (x 1 -y 1 ) 2 + • • • + (x ν -y ν ) 2 for x ∈ Ê ν and y ∈ Ê ν . Let p t (x, y) be the heat kernel associated to the operator -∂ 2

x + V (x). Let p conj t (x, y) be the conjugate heat kernel defined by p t (x, y)=(4πt) -ν 2 exp -(x -y) 2 4t p conj t (x, y).

(1.1)

Then the Minakshisundaram-Pleijel asymptotic expansion holds:

p conj t (x, y) =½ + a 1 (x, y)t + • • • + a r-1 (x, y)t r-1 + t r O t→0 + (1). (1.2)
Here ½ denotes the identity matrix. The expansion in (1.2) is not convergent in general. A goal of this paper is to prove, under rather strong assumptions on V , that this expansion is Borel summable and that its Borel sum is equal to p conj t (x, y) (see definition 2.4). Borel summability allows to recover p conj t (x, y) with the help of the knowledge of the coefficients a 1 (x, y), a 2 (x, y), . . . For instance, if these coefficients vanish then p conj t (x, y) =½. Assume now that V is defined on the torus (Ê / ) ν with values in a space of d × d Hermitian matrices. Let λ 1 λ 2 • • • λ n • • • , λ n → +∞ be the eigenvalues of the operator-∂ 2

x + V (x) acting on periodic d -valued functions. The trace of the heat kernel has an asymptotic expansion +∞ n=1 e -λnt = (4πt) -ν 2 d + a 1 t + • • • + a r-1 t r-1 + t r O t→0 + (1) .

(1.3)

We shall prove the Poisson formula: for t ∈ , Ret > 0,

+∞ n=1 e -λnt = (4πt) -ν 2 q∈ ν e -q 2 4t u q (t), (1.4) with u q (t) = d + a 1,q t + • • • + a r-1,q t r-1 + • • • , q ∈ ν .

(1.5)

In (1.5), each expansion is Borel summable and u q denotes the Borel sum of such an expansion. Since a 1,0 = a 1 , a 2,0 = a 2 , . . ., the expansion in (1.3) is Borel summable but (1.4) shows us that the knowledge of the coefficients a 1 , a 2 . . . does not allow one to recover the left hand side of (1.3) by Borel summation.

Let us now state more precisely the assumptions on V . Let α ∈ Ê and let µ be a Borel measure on Ê ν with values in some complex finite dimensional space of square matrices, such that, for some ε > 0 Ê ν exp(εξ 2 )d|µ|(ξ) < +∞.

(1.6)

We suppose that V (x) = αx 2 -c(x) 1 where c(x) = Ê ν exp(ix • ξ)dµ(ξ). c must be viewed as a perturbation. Assuming c is complex valued instead of matrix valued does not simplify the method and does not change the results. In particular, (1.6) implies that V is analytic on ν and c is bounded on Ê ν . In fact, our results hold if αx 2 is replaced by an arbitrary quadratic form on Ê ν .

For the sake of simplicity, we choose not to write the proofs in this case (see [START_REF] Hargé | A deformation formula for the heat kernel[END_REF] for a generalization of the deformation formula). Note also that functions c such that c(x) := exp(ix 1 ) or c(x) := exp(-x 2 ) satisfy our assumptions.

Quantum mechanics gives many examples of divergent expansions. We focus on semi-classical expansions related to the Schrödinger equation since they present a lot of similarities with the small time expansion of the heat kernel. The semi-classical viewpoint allows to expand a quantum quantity in terms of powers of h. The coefficients of this expansion can be viewed as classical quantities. Giving a meaning to the sum of this expansion by using only the coefficients allows one to recover the quantum quantity by means of classical quantities [B-B, V1, V2]. The same interpretation holds for the expansion of the heat kernel for small t (the parameter t may be viewed as β, the inverse of the temperature). The heat kernel of an operator can be viewed as a quantum quantity: for instance, its trace, if it exists, gives the partition function of the spectrum of the operator. The coefficients of the expansion have a classical interpretation: for instance in (1.1), the main term in the exponential is the square of the distance between x and y, which is a classical quantity. See also Remark 8.4 in [START_REF] Hargé | Noyau de la chaleur en dimension quelconque avec potentiel en temps petit[END_REF].

Recovering quantum quantities with the help of the coefficients of their semiclassical expansion is a question considered by Voros [V1, V2] and Delabaere, Dillinger, Pham [D-D-P] for the one dimensional Schrödinger equation. Their use of Borel summation is not elementary as ours. Their assumptions, which allows to consider tunnelling for instance, involve to deal with ramified singularities in the Borel plane (Ecalle's alien calculus). In a following paper [START_REF] Hargé | Borel summation of the semi-classical expansion of the partition function associated to the Schrödinger equation with a one-well potential[END_REF], we prove that the h expansion of the partition function of the Schrödinger operator is Borel summable in all dimensions but with restrictive assumptions on the potential allowing a simple Borel summation process.

Concerning the expansion of the heat kernel for small t, we are not aware of references using Borel summation. However, Colin de Verdière [START_REF] De Verdière | Spectre du Laplacien et longueurs des géodésiques périodiques.I[END_REF][START_REF] De Verdière | Spectre du Laplacien et longueurs des géodésiques périodiques[END_REF] gives a Poisson formula in the case of a smooth Riemannian manifold. The setting of his work is much more general than ours, but the result is asymptotic and does not give an exact formula. The exact formula on the torus is a direct consequence of the result on Ê ν : it is just putting together independent expansions (one more time, we do not need alien calculus). The case of the torus is very simple but singular.

It is convenient to write the potential c as the Fourier transform of a Borel measure. This point of view is used by many authors [It, Ga, A-H], working with a rigorous definition of Wiener and Feynman integrals. The assumptions on c and the method allow to consider in a natural way p t (x, y) with x, y ∈ ν , t ∈ , Ret 0 instead of x, y ∈ Ê ν , t ∈ Ê ∪ iÊ.

Our proofs use a so-called deformation formula and a so-called deformation matrix. In the free case (i.e V (x) = αx 2 -c(x) with α = 0), this formula can be found in [It,Formula (77)]. In this reference, the deformation matrix is not given explicitly. One can also give a formula for the solution of the heat or Schrödinger equation with an arbitrary initial condition (of course no factorization occurs contrary to the heat kernel case), see [Ga, A-H] and more precisely [START_REF] Gawedzki | Construction of quantum-mechanical dynamics by means of path integrals in phase space[END_REF]Eq.(27)] and Eq.(3.12),(5.16)]. In these references, the deformation formula is a mean to study Feynman integral; the complex viewpoint, which is straightforward in the free case, is not considered. In a heuristic way, this formula is well known [On] and can even be considered as a particular case of a more general one [Ge-Ya]. In this setting, there is no reason to consider c as the Fourier transform of a Borel measure. See also [Fu-Os-Wi] and [START_REF] Hargé | Noyau de la chaleur en dimension quelconque avec potentiel en temps petit[END_REF]. We strongly advise the reader to look at the known heuristic proof of this formula given in the Appendix. This proof, which uses Wiener representation of the heat kernel and Wick's theorem, gives an explanation of the shape of the formula and an interpretation of a so-called deformation matrix (see also [It]) which is important in our method. However, there is a serious drawback in thinking of the deformation formula as a consequence of the existence of Wiener or Feynman integrals, in particular from a rigorous point of view. The deformation formula is elementary: it does not use sophisticated notions about infinite-dimensional spaces. Another heuristic proof, avoiding Wiener representation, can be found in [On]. Here Wick's theorem also plays a central role. In Section 4.2, we give a rigorous proof of the deformation formula, which avoids Wiener representation but does not explain in a satisfactory way the shape of the formula. We call it deformation formula because we want to emphasize its perturbative nature. It is therefore not surprising that the assumptions (4.17) (the same as those used for the mathematical foundation of the Feynman integrals [A-H]) or (4.24) on the function c are highly restrictive. For instance, the heat kernel associated to the operator ∂ 2

x -x 2 can not be viewed as a perturbation of the heat kernel associated to the operator ∂ 2

x by this formula. This formula gives an explicit solution of the heat equation viewed as a partial differential equation which is available for every α ∈ Ê and small t ∈ with Ret > 0. For this purpose some properties of the deformation matrix for these values of t are needed (Section 4.1). At last, let us say a few words about the unicity problem. If α 0, the heat kernel is uniquely defined as the kernel of an analytic semi-group [Pa]. In the case α > 0, see [START_REF] Hargé | Some remarks on the complex heat kernel on ν in the scalar potential case[END_REF].

The paper is organized as follows. In Section 2, we present some notation and recall some classical facts about Borel summation. In Section 3, we state the main results and give their proofs in Sections 4.3 and 4.4.

Acknowlegement: I would like to express my gratitude to Vladimir Georgescu for his pertinent advices. He also suggested many improvements in the redaction.

Preliminaries

For z = |z|e iθ ∈ , θ ∈] -π, π], we denote z 1/2 := |z| 1/2 e iθ/2 . Let T > 0. Let

+ := {z ∈ |Re(z) > 0}, D T := {z ∈ ||z| < T }, D + T := D T ∩ + and DT := z ∈ |Re( 1 z ) > 1 T .
DT is the open disk of center T 2 and radius T 2 .

Figure 2.1:

DT D + T + Ret Imt T 2 Let κ > 0. Let Sκ := z ∈ |d(z, [0, +∞[) < κ and S κ := z ∈ ||Imz 1/2 | 2 < κ = z ∈ |Rez > 1 4κ
Im 2 z -κ . S κ is the interior of a parabola which contains Sκ . Let Ω be an open domain in m and let F be a complex finite dimensional space. We denote by A(Ω) the space of F -valued analytic functions on Ω, if there is no ambiguity on F .

Let B denote the collection of all Borel sets on Ê m . An F -valued measure µ on Ê m is an F -valued measurable function on B satisfying the classical count- able additivity property (cf. [Ru]). Let | • | be a norm on F . We denote by |µ| the positive measure defined by

|µ|(E) = sup ∞ j=1 |µ(E j )|(E ∈ B),
the supremum being taken over all partition {E j } of E. In particular, |µ|(Ê m ) < ∞. Let us remark that dµ = hd|µ| where h is some F -valued function satisfying |h| = 1 |µ|-a.e. If f is an F -valued measurable function on Ê m and λ is a positive measure on Ê m such that Ê m |f |dλ < ∞, one can define an F -valued measure µ by setting dµ = f dλ. Then d|µ| = |f |dλ.

We work with finite dimensional spaces of square matrices. We always consider multiplicative norms on these spaces (|AB| |A B|, for A and B square matrices) and we assume that

|½| = 1. For A = (a i,j ) 1 i,j d with a i,j ∈ , we set A * = (ā j,i ) 1 i,j d . For λ, µ ∈ ν , we denote λ • µ := λ 1 µ 1 + • • • + λ ν µ ν , λ := ( λ1 , . . . , λν ), Imλ := (Imλ 1 , . . . , Imλ ν ), λ 2 := λ • λ, |λ| := (λ • λ) 1/2 (if λ ∈ Ê ν , |λ| = √ λ 2
). In the whole paper, sums indexed by an empty set are, by convention, equal to zero.

Here is an improved version of a theorem of Watson. This is in fact a theorem of Nevanlinna, rediscovered by Sokal [So]. It gives a concise presentation of what we need about Borel summation. In what follows, functions are defined on some subset of and take their values in a complex finite dimensional space F . First, we need: Definition 2.1 Let κ > 0 and T > 0. We say that

• A function f satisfies P κ,T if and only if f is analytic on DT (fig. 2.1)
and there exist a 0 , a 1 , . . . ∈ F , R 0 , R 1 , . . . analytic functions on DT such that, for every r 02 and t ∈ DT ,

f (t) = a 0 + • • • + a r-1 t r-1 + R r (t),
and for every κ < κ, T < T , there exists K > 0 such that, for every r 0

and t ∈ D T , |R r (t)| K r! κr |t| r • A function 3 f satisfies Q κ,T if and only if f is analytic on Sκ (fig. 2.2)
and for every κ < κ, T < T , there exists K > 0 such that, for every

τ ∈ Sκ | f (τ )| Ke |τ | T . (2.1)
This definition is justified by the following theorem and remark. Note that, if f satisfies P κ,T , the coefficients a n are uniquely determined.

Theorem 2.2 Let κ > 0 and T > 0.

• If f verifies P κ,T , then

f (τ ) := ∞ r=0 a r r! τ r (2.2)
admits an analytic continuation on Sκ which verifies Q κ,T .

• If f verifies Q κ,T , then f (t) := +∞ 0 f (τ )e -τ t dτ t (2.3) verifies P κ,T .
• f given by (2.2) is called the Borel transform of f . f given by (2.3) is called the Laplace transform of f . These two transforms are inverse each to other.

Remark 2.3 If P κ,T holds, the knowledge of the coefficients a 0 , a 1 , . . . allows one to recover f : f is the Laplace transform of its Borel transform which only depends on a 0 , a 1 , . . . by (2.2). The shape of the domain DT is crucial. For instance, the conclusion of this remark may fail if DT is replaced by a truncated cone like C T,α := {t = re iθ ∈ ||θ| < α, r < T } with T > 0, α < π 2 . Under our assumptions on V , the conjugate heat kernel will verify at least P κ,T for some κ > 0 and T > 0. We do not prove a resummation estimate on a truncated cone C T,α with T > 0, α > π 2 and in general, the conjugate heat kernel will not verify the assumptions of Watson's theorem (cf. [So]).

Definition 2.4 Let ã1 , . . . , ãr , . . . ∈ F . One says that the formal power series f = r 0 ãr t r is Borel summable if there exist κ, T > 0 and a function f satisfying P κ,T such that, for every r 0, a r = ãr . f is called the Borel sum of f .

Main results

The following theorem concernes the free case. In this case, it is possible to give precise properties of the Borel transform of the conjugate heat kernel. In particular, this Borel transform is analytic on the complex plane and is exponentially dominated by the square root of the Borel variable on parabolic domains which are symmetric with respect to the positive real axis.

Theorem 3.1 Let ε > 0. Let µ be a measure on Ê ν with values in a complex finite dimensional space of square matrices verifying Ê ν exp(εξ 2 )d|µ|(ξ) < ∞.

Let c(x) = exp(ix • ξ)dµ(ξ) and let u be the solution of

   ∂ t u= ∂ 2 x u + c(x)u u| t=0 + = δ x=y ½ (3.1)
Let v be defined by u=(4πt) -ν/2 e -(x-y) 2 4t v. Then v admits a Borel transform v (with respect to t) which is analytic on 1+2ν . Let κ, R > 0 and let

C := 2 Ê ν exp 2κ ε + ε 2 ξ 2 + R|ξ| d|µ|(ξ) 1/2 . Then, for every (τ, x, y) ∈ S κ × 2ν such that |Imx| < R and |Imy| < R, |v(τ, x, y)| exp C|τ | 1/2 . (3.2)
Remark 3.2 S κ is the interior of a parabola which contains Sκ (fig. 2.2). Let T > 0. Estimate (3.2) is better than (2.1). Then v(t, x, y) verifies P κ,T : the small time expansion of the conjugate heat kernel is Borel summable and its Borel sum is equal to v.

The following corollary deals with the partition function on the torus and is a consequence of the Theorem 3.1.

Corollary 3.3 Let ε > 0 and d ∈ AE * . For each q ∈ ν , let c q be a square matrix acting on d . Assume that c -q = c * q and

q∈ ν e 4π 2 εq 2 |c q | < ∞. Let c(x) := q∈ ν c q e 2iπq.x . Let λ 1 λ 2 • • • be the eigenvalues of the operator H := -∂ 2
x -c(x) acting on d -valued functions defined on the torus (Ê/ ) ν . For each q ∈ ν , there is a function ŵ(q, .) analytic on satisfying

• For every κ > 0, there exist constants C 1 > 0 and C 2 > 0 such that, for every q ∈ ν and τ ∈ S κ ,

| ŵ(q, τ )| C 1 exp C 2 |τ | 1/2 . (3.3) • For every t ∈ + +∞ n=1 e -λnt = (4πt) -ν/2 q∈ ν e -q 2 4t +∞ 0 e -τ t ŵ(q, τ ) dτ t .
(3.4)

Remark 3.4 By the argument of Remark 3.2, Corollary 3.3 implies the following result. Let c as in Corollary 3.3. For each q ∈ ν , there are numbers a 1,q , a 2,q , . . . ∈ and functions R 0,q , R 1,q , . . . ∈ A( + ) such that, for every r 0 and t ∈ + ,

+∞ n=1 e -λnt = (4πt) -ν/2 q∈ ν e -q 2 4t d + a 1,q t + • • • + a r-1,q t r-1 + R r,q (t) , (3.5)
and for each T, κ > 0, there exist K > 0 such that

|R r,q (t)| K r! κ r |t| r ,
for every r 0, q ∈ ν , t ∈ DT .

The index q in (3.5) can be viewed as labeling closed classical trajectories (geodesics) on the torus. Then q 2 denotes the length of such a geodesic. All coefficients of the expansion 3.5 have classical (or geometric) interpretation.

The following theorem deals with the harmonic case (i.e. V (x) = αx 2 -c(x) with α ∈ Ê) and gives a statement about the expansion of the conjugate heat kernel which provides Borel summability. But we do not obtain a statement as precise as in Theorem 3.1 about its Borel transform (the proof is established without working in the Borel plane).

Theorem 3.5 Let ω ∈ Ê ∪ iÊ. Let c be as in Theorem 3.1. Let u be the solution of

   ∂ t u = ∂ 2 x -ω 2 4 x 2 u + c(x)u u| t=0 + = δ x=y ½ (3.6)
• Then there are a number T > 0, functions a 1 , a 2 , . . . ∈ A( 2ν ) and

R 0 , R 1 , . . . ∈ A(D + T × 2ν ) such that u = (4πt) -ν/2 e -(x-y) 2 4t ½ + a 1 (x, y)t + • • • + a r-1 (x, y)t r-1 + R r (t,
x, y) , (3.7) for every r 0, t ∈ D + T and (x, y) ∈ 2ν . • And for each R > 0, there exist K > 0 and κ > 0 such that,

|R r (t, x, y)| K r! κ r |t| r , (3.8
)

for every r 0, t ∈ D + T , (x, y) ∈ 2ν , |x| < R, |y| < R.
Remark 3.6

• If ω ∈ Ê, the heat kernel is uniquely defined as the kernel of an analytic semi-group. In the case ω ∈ iÊ, one can also give a uniqueness statement (see [START_REF] Hargé | Some remarks on the complex heat kernel on ν in the scalar potential case[END_REF]) which allows one to speak about "the" heat kernel.

• By (3.8) and Theorem 2.2, the expansion in (3.7) admits a Borel transform satisfying Q κ,T . In the case ω = 0, the estimate (3.8) is better than that obtained by Remark 3.2 since DT ⊂ D + T (fig. 2.1). In particular we get a uniform estimate when Ret → 0 + and Imt is a non-vanishing constant. Hence, we obtain information about the Schrödinger kernel. In fact, this information is contained in Theorem 3.1: it is no difficult to see that Theorem 3.1 implies the estimate (3.8) for t ∈ D + T .

Proof of the theorems

The proof of our result uses an explicit formula of the heat kernel which has two expressions (compare (5.1) and (4.18)). This formula is based on a matrix (Ω ♮ or Ω) and a path (q ♮ ω or q ω ). There are two cases. In the free case (ω = 0), the matrix Ω is linear in t and the path q ω does not depend on t. Consequently, the proof of our results is simple and working in the Borel plane is natural. In the harmonic case (ω = 0), we must deal with the t-dependence of the matrix and the path. The following subsection is devoted to the study of this matrix (let us emphasize that only Lemma 4.1, in this subsection, is useful for the study of the free case).

The deformation matrix

Let n ∈ AE, ω ∈ , t ∈]0, +∞[ such that |ωt| < π and let (s 1 , . . . , s n ) ∈ [0, t] such that 0 < s 1 < • • • < s n < t. For A ∈ , denote sh A := 1 2 (e A -e -A
). In the following sections, the matrix

Ω ♮ := sh(ωs j∧k ) sh ω(t -s j∨k ) ω sh(ωt) 1 j,k n (4.1)
plays an important role (cf. (4.22) and (5.1)) and some of its properties must be established. However, since we shall also consider complex values of t, we study the following matrix. Assume now that t ∈ . Let (s 1 , . . . ,

s n ) ∈ [0, 1] such that 0 < s 1 < • • • < s n < 1.
Let Ω be defined by Ω := sh(ωts j∧k ) sh ωt(1 -s j∨k ) ω sh(ωt)

1 j,k n . (4.2)
The goal of this section is to study some properties of Ω (Proposition 4.5) in particular when n, the dimension of the matrix, is large. This control, in large dimension, plays an important role in the proof of the Borel summability of the heat kernel expansion. Note that the connection of this matrix with a propagator (cf. (5.7)) seems to be important for the understanding of its properties. The following lemma gives a useful elementary property of Ω := 1 t Ω| ω=0 = s j∧k (1s j∨k ) 1 j,k n and Proposition 4.5, the goal of this subsection, can be viewed as a generalization of this lemma. For (ξ 1 , . . . , ξ n ) ∈ Ê νn , let

Ω • ξ ⊗ n ξ := n j,k=1 s j∧k (1 -s j∨k )ξ j • ξ k . (4.3) Lemma 4.1 For every n 1, (ξ 1 , . . . , ξ n ) ∈ Ê νn and (s 1 , . . . , s n ) ∈ [0, 1] such that 0 < s 1 < • • • < s n < 1 0 Ω • ξ ⊗ n ξ n n j=1 ξ 2 j .
Proof The upper bound of the quantity Ω•ξ⊗ n ξ is elementary and its positivity can be viewed as a consequence of Remark 4.4.

Lemma 4.2 Let T > 0 and M > 0. There exists T > 0 satisfying the following property. Let f be an analytic function f on D T verifying f (0) = 0, f ′ (0) = 1, sup t∈D T |f (t)| M and, for every t ∈ D T ,

Ret = 0 ⇒ Ref (t) = 0. (4.4) Then, for t ∈ D T , Ret > 0 ⇒ Ref (t) > 0. (4.5)
Proof We can choose T > 0, depending only on T and M , such that every analytic function f verifying f (0) = 0, f ′ (0) = 1 and |f (t)| M for t ∈ D T is a one-to-one analytic mapping on D T . For small t > 0, Ref (t) > 0 since f (0) = 0 and f ′ (0) = 1. Then by (4.4) and since f is a one-to-one analytic mapping, one gets (4.5).

Proposition 4.3 Let E be the space of Borel measures µ = n j=1 δ sj ξ j such that n 1, ξ j ∈ Ê ν , s j ∈]0, 1[. Let z ∈ D π . We denote by (., .) z the following bilinear form on

E (µ 1 , µ 2 ) z := 1 0 1 0 sh(zs ∧ s ′ ) sh z(1 -s ∨ s ′ ) z sh z dµ 1 (s) • dµ 2 (s ′ ).
Note that

(µ 1 , µ 2 ) 0 = 1 0 1 0 s ∧ s ′ (1 -s ∨ s ′ )dµ 1 (s) • dµ 2 (s ′ ).
Then, for z ∈ D π ,

∀µ ∈ E, |(µ, µ) z | π 2 π 2 -|z| 2 (µ, µ) 0 . (4.6) Proof For (s, s ′ ) ∈ [0, 1] 2 , let K(s, s ′ ) := sh(zs ∧ s ′ ) sh z(1 -s ∨ s ′ ) z sh z . Then    -d 2 K ds 2 + z 2 K = δ s=s ′ K| s=0 = K| s=1 = 0 . (4.7) Let (ξ 1 , . . . , ξ n ) ∈ Ê νn and (s 1 , . . . , s n ) ∈]0, 1[ n . Let u be the function on [0, 1] defined by u(s) = n j=1 sh(zs ∧ s j ) sh z(1 -s ∨ s j ) z sh z ξ j .
u is continuous and piecewise differentiable on [0, 1]. Let µ := n j=1 δ sj ξ j . By (4.7)

   -d 2 u ds 2 + z 2 u = µ u(0) = u(1) = 0 . (4.8)
For n 1, k, l ∈ {1, . . . , ν} and s ∈ [0, 1], let e n,k,l (s) := √ 2 sin(nπs)δ k=l . (e n,k ) n,k is an orthonormal basis of L 2 ([0, 1], ν ) which diagonalizes the unbounded operator S := -d 2 ds 2 . Let

H 1 0 := f ∈ L 2 ([0, 1], ν ) df ds ∈ L 2 , f (0) = f (1) = 0 = n,k f n,k e n,k n,k |nf n,k | 2 < ∞ , H -1 := n,k f n,k e n,k n,k f n,k n 2 < ∞ . For (f, g) ∈ H -1 × H 1 0 , let f, g := 1 0 f (s) • g(s)ds = n,k fn,k g n,k .
Note that S can be viewed as an isomorphism from H 1 0 to H -1 . The function u belongs to H 1 0 and by (4.8), µ = (S + z 2 )u ∈ H -1 . Moreover, (µ, µ) z = 1 0 u • dµ and π 2 is the lowest eigenvalue of S (every eigenvalue of S are real). Hence, for z ∈ D π , (µ, µ) z = µ, (S + z 2 ) -1 µ ,(µ, µ) 0 = µ, S -1 µ (4.9)

For λ ∈ [π 2 , +∞[ and z ∈ D π , the following inequality holds

+ z 2 ) -1 | π 2 π 2 -|z| 2 × λ -1
(4.10)

Then (4.6) is a consequence of (4.9) and (4.10).

Let v(s)

:= n j=1 s ∧ s j (1 -s ∨ s j )ξ j . Then (µ, µ) 0 = 1 0 v • dµ = 1 0 dv ds 2 since µ = -d 2 v ds 2 .

One gets

Remark 4.4 (., .) 0 is a positive definite symmetric bilinear form on E.

For (ξ 1 , . . . , ξ n ) ∈ Ê νn and 0 < s 1 < • • • < s n < 1, let Ω.ξ ⊗ n ξ := n j,k=1
sh(ωts j∧k ) sh ωt(1 -s j∨k ) ω sh(ωt)

ξ j • ξ k . (4.11) Proposition 4.5 Let ω ∈ Ê ∪ iÊ. There exists T d > 0 such that for ev- ery n 1, (s 1 , . . . , s n ) ∈ [0, 1] n , (ξ 1 , . . . , ξ n ) ∈ Ê νn , t ∈ with the condition 0 < s 1 < • • • < s n < 1, |t| < T d , Ret 0 ⇒ Re (Ω.ξ ⊗ n ξ) 0, (4.12) |Ω.ξ ⊗ n ξ| 2n|t| n j=1 ξ 2 j . (4.13) Proof Let µ := n j=1 δ sj ξ j . Then Ω.ξ ⊗ n ξ = t(µ, µ) ωt and Ω • ξ ⊗ n ξ = (µ, µ) 0 . Let T := π √ 2|ω|
. By (4.6) and Lemma 4.1, (4.13) holds for t ∈ D T . Note that (4.13) can also be obtained directly without using (4.6).

Let us choose arbitrary ξ 1 , . . . , ξ n such that (ξ 1 , . . . , ξ n ) does not vanish. By Remark 4.4, (µ, µ) 0 = 0. Let f be the function defined by b, . . . real and (4.4) holds. Obviously, f (0) = 0 and f ′ (0) = 1. Then we can use Lemma 4.2 and there exists

f (t) = t (µ, µ) ωt (µ, µ) 0 = Ω.ξ ⊗ n ξ (µ, µ) 0 . By (4.6), f is bounded on D T by 2 × π √ 2|ω| . Since f is odd and ω ∈ Ê ∪ iÊ, f (t) = f ( t), for t ∈ D T . Then f (t) = t+at 3 +bt 5 +• • • with a,
T d < T such that Ref (t) > 0 for t ∈ D + T d . Since (µ, µ) 0 ∈]0, +∞[, (4.12) holds for t ∈ D + T d .

The deformation formula

Let (x, y) ∈ 2ν , t ∈ * and ω ∈ Ê ∪ iÊ. Let u ω be defined by

u ω = 4π sh(ωt) ω -ν/2 exp - 1 4 ω sh(ωt) (ch(ωt)(x 2 + y 2 ) -2x • y) . (4.14)
Then, by one variant of Mehler's formula,

∂ t u ω = ∂ 2 x - ω 2 4 x 2 u ω , u ω | t=0 + = δ x=y . Note that u 0 = (4πt) -ν/2 e -(x-y) 2 4t .
We denote by q ♮ ω the extremal path of the action S := t 0 q2 (s) + ω 2 q 2 (s) ds such that q ♮ ω (0) = y and q ♮ ω (t) = x. Then

q ♮ ω (s) = 1 sh(ωt) sh(ωs)x + sh ω(t -s) y . (4.15) Let (ξ 1 , . . . , ξ n ) ∈ Ê νn . Let us define, for s ∈ [0, 1],
q ω (s) := 1 sh(ωt) sh(ωts)x + sh ωt(1 -s) y and, for s = (s 1 , . . . , s n ) ∈ [0, 1] n ,

q n ω (s) • ξ := q ω (s 1 ) • ξ 1 + • • • + q ω (s n ) • ξ n .
Remark 4.6 There exists δ > 0 such that, for every

(t, ω) ∈ 2 , s ∈ [0, 1], (x, y) ∈ 2ν , |q ω (s)| 4 max(|x|, |y|)if |ωt| < δ. (4.16)
Proposition 4.7 Let µ be a measure on Ê ν with values in a complex finite dimensional space of square matrices. Let us assume that for every R > 0

Ê ν exp(R|ξ|)d|µ|(ξ) < ∞.
(4.17)

Let c(x) := Ê ν exp(ix • ξ)dµ(ξ). Let ω ∈ Ê ∩ iÊ. Let v be defined by    v = ½ + n 1 v n , v n (t, x, y) = t n 0<s1<•••<sn<1 Ê νn e -Ω.ξ⊗nξ e iq n ω (s)•ξ d νn µ ⊗ (ξ)d n s.
(4.18)

In (4.18), d n s denotes ds 1 • • • ds n and d νn µ ⊗ (ξ) denotes dµ(ξ n ) • • • dµ(ξ 1 ).
Then there exists

T c > 0 such that v ∈ A(D + Tc × 2ν ) and the function u := u ω v is solution of (3.6). If ω = 0, v ∈ A( + × 2ν ) .
Proof Let T d given by Proposition 4.5 and let δ given by Remark 4.6. We choose

T c = min T d , δ |ω| . (4.19)
Let R > 0 and let

A := exp(4R|ξ|)d|µ|(ξ).
Let (x, y) ∈ 2ν such that |x|, |y| < R. Let t ∈ D + Tc . By (4.12) and (4.16)

| exp(-Ω.ξ ⊗ n ξ) exp(iq n ω (s) • ξ)| exp 4R(|ξ 1 | + • • • + |ξ n |) . (4.20) Hence |v n | (|t|A) n n! ,
so the series in (4.18) converges absolutely. Since R is arbitrary, v is well defined on D + Tc × 2ν . By dominated convergence theorem, one can check that v n and hence v ∈ A(D + Tc × 2ν ). Let us verify that the function u := u ω v, with v given by (4.18), is solution of (3.6). A solution u of (3.6) is given, if we use the relation u = u ω v by a solution v of the conjugate equation

     ∂ t -2 uω ∂ x u ω • ∂ x v = ∂ 2 x v + c(x)v (t = 0) v| t=0 + = ½ . Notice that - 2 u ω ∂ x u ω = ω sh(ωt) ch(ωt)x -y = q♮ ω (t).
Set v 0 := ½. It is then sufficient to verify that v n given by (4.18) satisfies

   ∂ t + q♮ ω (t) • ∂ x v n = ∂ 2 x v n + c(x)v n-1 v n | t=0 + = 0 , (4.21) for (t, x, y) ∈ D + Tc × 2ν , n 1. It suffices to check (4.21) for (t, x, y) ∈]0, T c [× 2ν . By (4.18) and the defi- nition of c v n = 0<s1<•••<sn<t F n d n s,
where

F n := exp n j,k=1 Ω ♮ j,k ∂ zj • ∂ z k c(z n ) • • • c(z 1 ) z1 = q ♮ ω (s1) • • • zn = q ♮ ω (sn) . (4.22)
Here Ω ♮ denotes the n × n matrix defined by (4.1). In the Appendix, we give an heuristic explanation of this result. Here is a rigorous verification. We have

(∂ t + q♮ ω (t) • ∂ x )v n = (boundary) + (interior) where (boundary) := 0<s1<•••<sn-1<t F n | sn=t d n-1 s, (interior) := 0<s1<•••<sn<t ∂ t + q♮ ω (t) • ∂ x F n d n s.
In particular, Ω ♮ j,k | sn=t = 0 for j = n or k = n. Therefore

(boundary) = c(x)v n-1 .
Now we want to prove that (interior) = ∂ 2 x v n . Using, for instance, explicit expressions of q♮ ω (t) and q ♮ ω , we get, for s ∈]0, t[ and m = 1, . . . , ν,

∂ t + q♮ ω (t) • ∂ x q ♮ ω,m (s) = 0. (4.23) Then, if ϕ(z 1 , . . . , z n ) is an arbitrary differentiable function of (z 1 , . . . , z n ) ∈ νn , ∂ t + q♮ ω (t) • ∂ x [ϕ(z 1 , . . . , z n )]| z1 = q ♮ ω (s1) • • • zn = q ♮ ω (sn) = 0. So, since ∂ t Ω ♮ j,k = sh(ωsj ) sh(ωt) sh(ωs k ) sh(ωt) , denoting B j,k := sh(ωsj ) sh(ωt) sh(ωs k ) sh(ωt) , (interior) = 0<s1<•••<sn<t G n d n s,
where

G n := n j,k=1 B j,k ∂ zj • ∂ z k exp n j,k=1 Ω ♮ j,k ∂ zj • ∂ z k c(z n ) • • • c(z 1 ) z1 = q ♮ ω (s1) • • • zn = q ♮ ω (sn)
.

For (α, β) ∈ {1, . . . , ν } 2 , one has ∂ x β q ♮ ω,α = δ α=β × sh(ωs) sh(ωt) . Since q ♮ ω is linear with respect to x,

∂ 2 x [c(z n ) • • • c(z 1 )]| z1 = q ♮ ω (s1) • • • zn = q ♮ ω (sn) = n j,k=1 B j,k ∂ zj •∂ z k c(z n ) • • • c(z 1 ) z1 = q ♮ ω (s1) • • • zn = q ♮ ω (sn)
.

Then ∂ 2 x F n = G n . Hence (interior) = ∂ 2 x v n .
This proves that the function u := u ω v is solution of (3.6).

In the case ω = 0, Ω.ξ ⊗ n ξ = t Ω • ξ ⊗ n ξ and q ω (s) = y + s t (x -y). We can get directly that v ∈ A( + × 2ν ) and that u = u 0 v is solution of (3.6).

Remark 4.8 Weaker assumptions than (4.17) can also be considered. One can assume that, for every m 0,

Ê ν |ξ| m d|µ|(ξ) < ∞. (4.24)
Then, for instance in the free case, v ∈ C ∞ + × Ê 2ν ∩ A + , C ∞ (Ê 2ν ) .

Proof of Theorem 3.1 and Corollary 3.3

In the study of Borel transform of the conjugate heat kernel in the free case, the Borel transform of t -→ t n exp(-Bt), B ∈ , which is related to Bessel functions, plays a specific role. Lemma 4.9 For every z ∈ , let us denote

J(z) := n 0 (-1) n z n (n!) 2 = π 0 cos 2z 1/2 sin(ϕ) dϕ π .
The following estimate holds for every z ∈

|J(z)| exp 2|Imz 1/2 | . (4.25)
These properties are easy to prove and well known since J(z) = J 0 2z 1/2 , J 0 denoting the Bessel function of order 0.

By Lemma 4.1 2|Im(τ 1/2 )| Ω • ξ ⊗ n ξ 2κ 1/2 n(ξ 2 1 + • • • + ξ 2 n ) 2 × 2κn ε 1/2 × ε 2 (ξ 2 1 + • • • + ξ 2 n ) 2κn ε + ε 2 (ξ 2 1 + • • • + ξ 2 n ).
Hence

| Fn | exp R(|ξ 1 | + • • • + |ξ n |) × |τ | n n! exp 2κn ε exp ε 2 (ξ 2 1 +• • •+ξ 2 n ) . (4.32) Let A := exp 2κ ε + ε 2 ξ 2 + R|ξ| d|µ|(ξ).
The integral in (4.30) is absolutely convergent and

|v n | 1 (n!) 2 × (A|τ |) n .
Then the series in (4.31) is absolutely convergent. Since |v| n 0 1

(n!) 2 × (A|τ |) n = J(-A|τ |) and by (4.25) |v| exp 2(A|τ |) 1/2 .
This proves (3.2).

By dominated convergence theorem, one can check that vn and hence v are analytic on S κ × 2ν and hence on 1+2ν since κ is arbitrary.

Let us prove that v is the Laplace transform of v. Let t ∈]0, +∞[. By (4.29) and since t -→ t n exp(-Bt) is the Laplace transform of τ -→ K n (B, τ ),

t n exp(iq n 0 (s) • ξ) exp(-t Ω • ξ ⊗ n ξ) = +∞ 0 Fn (τ )e -τ t dτ t .
Then, by (4.28), v is the Laplace transform of v. Now we may prove Corollary 3.3. Let p t (x, y) be the solution of

   ∂ t p t (x, y) = ∂ 2 x + c(x) p t (x, y) p| t=0 (x, y) = q∈ ν δ x=y+q ½ (4.33)
Since the torus is compact and c is a Hermitian matrix-valued analytic function, the spectrum of H is real, discret and, for t ∈]0, +∞[,

+∞ n=1 e -λnt = [0,1] ν
Tr p t (x, x) dx.

Let µ := q∈ ν c q δ ξ=2πq . Then c satisfies the assumptions of Theorem 3.1. Let v and C be defined as in Theorem 3.1. Then

p t (x, y) = (4πt) -ν/2 q∈ ν exp - (x -y -q) 2 4t +∞ 0
v(τ, x, y + q)e -τ t dτ t .

(4.34) Let κ > 0. By (3.2), the quantity v(τ, x, y + q) is bounded (uniformly in q ∈ ν ) by exp C|τ | 1/2 and C only depends on max |Imx|, |Imy| and κ. Hence the series in (4.34) is convergent for t ∈ + , x ∈ ν and y ∈ ν . Let ŵ(q, τ ) :=

[0,1] ν Tr v(τ, x, x + q) dx, C 1 := sup | Tr M | |M | M = 0 , C 2 := 2 exp 2κ ε + ε 2 ξ 2 d|µ|(ξ) 1/2
.

Here the supremum is taken over non-vanishing d × d complex matrices. Then (3.4) and (3.3) hold.

Proof of Theorem 3.5

The following lemma will be useful for the proof of Theorem 3.5.

Lemma 4.11 Let T > 0, K 1 > 0,and σ 1 > 0. Let f be an analytic function on D + T . Assume that, for every r 0, there exist f r analytic on D T and g r analytic on

D + T such that f = f r + g r , |f r (t)| K 1 σ r 1 r!for every t ∈ D T , |g r (t)| K 1 σ r 1 r!|t| r for every t ∈ D + T . Let σ 2 > σ 1 . Then there exist K 2 > 0, a 0 , a 1 , . . . ∈ , R 0 , R 1 , . . . ∈ A(D + T ) such that    f (t) = a 0 + • • • + a r-1 t r-1 + R r (t) |R r (t)| K 2 σ r 2 r!|t| r , (4.35)
for every r 0 and t ∈ D + T .

Proof Let r 0. Let M:=sup |t|<T |f r (t)| and a 0 := f r (0), . . . , a r-1 := 0). Let w r be the analytic function on D T defined by

1 (r-1)! f (r-1) r (
w r (t) = 1 t r f r (t) -(a 0 + • • • + a r-1 t r-1 ) .
Let q = 0, . . . , r -1 and let t ∈ D T . By Cauchy formula, |a q t q | M . By maximum modulus principle

|w r (t)| M T r (1 + r) K 1 σ r 1 r! T r (1 + r). Let σ 2 > σ 1 . Let us choose K 2 > 0 such that, for r 0, K1σ r 1 T r (1 + r) 1 2 K 2 σ r 2 and K 1 σ r 1 1 2 K 2 σ r 2 .
Let R r (t) := t r w r (t)+g r (t). Then (4.35) holds for t ∈ D + T .

Remark 4.12 In fact, we use a parametric version of Lemma 4.11. The function f may depend on additional parameters. Of course, the constants which appear in the assumptions of Lemma 4.11 must be independent of the parameters. Moreover, we have to assume that functions take their values in a complex finite dimensional space.

Now we prove Theorem 3.5. Instead of proving (3.7) and (3.8) directly, we shall prove the existence of the following factorization

u = u ω v ω (4.36)
where

v ω := ½ + a w 1 (x, y)t + • • • + a ω r-1 (x, y)t r-1 + R ω r (t,
x, y) and R ω r satisfying (3.8). By (4.14), uω u0 is analytic near t = 0. Then (3.7) and (3.8) will hold since Borel summability properties of an expansion do not change if it is multiplied by a convergent expansion near t = 0.

Let v and T c be respectively defined by (4.18) and (4.19). Then v verifies all the properties of Proposition 4.7. We shall use Lemma 4.11 with Remark 4.12. For m 1, w ∈ ,

e w = 1 + w + • • • + w m-1 (m -1)! + w m (m -1)! 1 0 (1 -θ) m-1 e θw dθ.
(4.37)

Let r > 0. Using (4.37) with w = -Ω.ξ ⊗ n ξ in (4.18), one gets, for t ∈ D + Tc , a decomposition v = f r + g r where

f r := ½ + n + m < r n 1, m 0 t n 0<s1<•••<sn<1 Ê νn F n,m d νn µ ⊗ (ξ)d n s, F n,m := 1 m! (-Ω.ξ ⊗ n ξ) m exp iq n ω (s) • ξ) ,
and g r = g

(1)

r + g (2)
r , where In what follows, we shall fix some R > 0 and some ε ∈]0, ε 0 [. We denote A := Ê ν exp εξ 2 + 4R|ξ| d|µ|(ξ)

g (1) r := n + m = r n 1, m 1 t n 0<s1<•••<sn<1 Ê νn 1 0 G n,m dθd νn µ ⊗ (ξ)d n s, G n,m := 1 (m -1)! (-Ω.ξ ⊗ n ξ) m (1 -θ) m-1 exp(-θΩ.ξ ⊗ n ξ) exp(iq n ω (s) • ξ),
and we take arbitrary x, y ∈ ν such that |x| < R and |y| < R.

We begin to check that f r verifies assumptions of Lemma 4.11. Obviously, by definition of Ω.ξ ⊗ n ξ and q n ω (s) • ξ, f r are analytic for t ∈ D π |ω| . By (4.12), (4.13) and ( 4 Then Stirling formula implies the existence of K 1 > 0 and σ 1 > 0 such that for t ∈ D Tc and r 1, |f r | K 1 σ r 1 r!.

(4.39)

We check now that g r verifies the assumptions of Lemma 4.11. Let t ∈ D + Tc . By (4.12), (4.13) and (4.16) Since g r = g

(1)

r + g

(2)

r and by similar arguments which allow us to obtain (4.39), we get the existence of K 1 > 0 and σ 1 > 0 such that for t ∈ D + Tc and r 1,

Appendix

Let u be the solution of (3.6) and v be the function defined by the factorization u = u ω v. The goal of this section is to give a heuristic explanation of the following (deformation) formula

v = ½ + n≥1 v n , (5.1)
where

v n := 0<s1<•••<sn<t F n ds 1 • • • ds n , F n := exp( n j,k=1 Ω ♮ j,k ∂ zj • ∂ z k )c(z n ) • • • c(z 1 ) z1 = q ♮ ω (s1) • • • zn = q ♮ ω (sn)
.

Here t > 0. Ω ♮ and q ♮ ω are defined in section 4 (cf. (4.1) and (4.15)). We prove and use this formula in a rigorous context (cf. (4.22)). This heuristic viewpoint has the advantage to explain the shape of the formula and gives a simple interpretation of matrix Ω ♮ . However, from a rigorous point of view, the proof chosen in Proposition 4.7 seems to be more efficient.

Figure

  Figure 2.2:

  .16), for t ∈ D Tc , •<sn<1 Ê νn H n,m d νn |µ| ⊗ (ξ)d n s where H n,m := exp 4R(|ξ 1 | + • • • |ξ n |) . , T c . Then, for (n, m) ∈ AE 2 such that n + m < r

  <sn<1 Ê νn L n,m d νn |µ| ⊗ (ξ)d n s

  |g r (t)| K 1 σ r 1 r!|t| r . (4.40) By (4.39) and (4.40), we can use Lemma 4.11 with Remark 4.12. Then factorization (4.36) holds and R ω r satisfies (3.8). This proves (3.7) and (3.8). Let us prove the regularity of the functions a 1 , a 2 , . . . and R 0, R 1 , . . . The function V defined by V (x) = -ω 2 4 x 2 + c(x)is analytic on ν . By a property of Minakshisundaram-Pleijel expansion , a 1 , a 2 , . . . ∈ A( 2ν ) and hence R 0 , R 1 , . . . ∈ A(D + Tc × 2ν ).

The minus sign in front of c(x) is chosen for future convenience.

If r = 0, the expansion must be read f (t) = R 0 (t), by the previous convention.

In general, we denote functions defined on the Borel plane by a hat.

Lemma 4.10 Let n 1. For B ∈ , let τ -→ K n (B, τ ) be the Borel transform of the function t -→ t n exp(-Bt). Then, for τ ∈ ,

(1 -θ) n-1 (n -1)! J(θBτ )dθ. (4.26)

Proof Since t n exp(-Bt) = m 0 (-1) m t n+m m! B m , using the definition of the Borel transform, we get the first equality of (4.26). Since

the second equality of (4.26) is a consequence of

) is a consequence of (4.25) and (4.26). Now we prove Theorem 3.1. By Proposition (4.7) with ω = 0, the function u := u 0 v is solution of (3.1) where v :

where q 0 (s) := y + s(x -y). Since q n 0 (s) • ξ does not depend on t, Lemma 4.10 will allow us to obtain a convenient formulation of the Borel transform of v.

Let Fn , vn , v be defined by

Let us check that the integral giving the definition of vn is absolutely convergent and hence that the series giving the definition of v is absolutely convergent. By (4.27)

For proving (5.1), we use Wiener integral representation of u and the following version of Wick's theorem. Let E be a real vector space. Let ., . be a scalar product on E and let A be a symmetric invertible operator. Then

Of course, if E has finite dimension and A > 0, this formula is rigorous and means that the Laplace transform of a Gaussian is a Gaussian.

The justification of the deformation formula starts from the following representation of u u = Texp

where integration is taken over all the paths q : Ê -→ Ê ν such that q(0) = y and q(t) = x. Texp denotes the time-ordered exponential function:

Using the decomposition q(s) = q ♮ ω (s) + w(s) and the fact that q ♮ ω is an extremal path for the action S(q) := t 0 q2 (s) + ω 2 q 2 (s) ds, which is quadratic with respect to q, we get S(q) = S(q ♮ ω ) + S(w). Hence

where integration is taken over all the paths w : Ê -→ Ê ν such that w(0) = 0 and w(t) = 0. By (5.4), with c ≡ 0,

(5.5)

By Taylor formula

.

(5.6) To use (5.2), let us define

(5.7)

As in quantum field theory, the operator A -1 can be viewed as a propagator. Since The deformation formula (5.1) is then a consequence of (5.6) and (5.8).


Florent Lafarge

Mathieu Renaud Keriven

Mathieu Brédif

Combining meshes and geometric primitives

come

Introduction

3D-models of urban scenes are very useful for many applications such as urban planning, virtual reality, disaster recovery or computer games. The reconstruction of such scenes is a well known computer vision problem which has been addressed by various approaches providing integral building representations such as [START_REF] Baillard | Automatic reconstruction of piecewise planar models from multiple views[END_REF][START_REF] Dick | Modelling and interpretation of architecture from several images[END_REF][START_REF] Lafarge | Building reconstruction from a single DEM[END_REF][START_REF] Sinha | Interactive 3D architectural modeling from unordered photo collections[END_REF], but remains an open issue [START_REF] Mayer | Object extraction in photogrammetric computer vision[END_REF][START_REF] Zhu | Special issue on modeling and representations of large-scale 3d scenes[END_REF].

Building modeling

With the new perspectives offered for the aid to navigation by general public softwares such as Street View (Google) or GeoSynth (Microsoft), 3D building modeling is a topic of growing interest. Many works have been recently proposed. Two main families of approaches may be distinguished.

3D-primitive modeling -The first family represents buildings as 3D-object layouts [START_REF] Dick | Modelling and interpretation of architecture from several images[END_REF][START_REF] Han | Bottom-up/top-down image parsing by attribute graph grammar[END_REF][START_REF] Lee | Extraction and integration of window in a 3d building model from ground view images[END_REF][START_REF] Muller | Image-based procedural modeling of facades[END_REF][START_REF] Pauly | Discovering structural regularity in 3D geometry[END_REF][START_REF] Xiao | Image-based façade modeling[END_REF][START_REF] Zebedin | Fusion of feature-and area-based information for urban buildings modeling from aerial imagery[END_REF]. These works efficiently detect and insert various urban objects such as windows or doors in 3D building models. However, these limited parametric descriptions fail to model fine details.

Mesh representation -The reconstruction of buildings with high order details, such as ornament, statues and other irregular shapes, is mainly addressed by mesh generation techniques using Laser scanning [START_REF] Banno | Flying laser range sensor for largescale site-modeling and its applications in bayon digital archival project[END_REF][START_REF] Fruh | An automated method for large-scale, ground-based city model acquisition[END_REF] or multi-view stereo processes [START_REF] Furukawa | Accurate, dense, and robust multi-view stereopsis[END_REF][START_REF] Goesele | Multi-view stereo for community photo collections[END_REF][START_REF] Vu | Towards high-resolution large-scale multi-view[END_REF]. Multi-view stereo techniques have significantly progressed during recent years as underlined in the comparative studies [START_REF] Seitz | A comparison and evaluation of multi-view stereo reconstruction algorithms[END_REF][START_REF] Strecha | On benchmarking camera calibration and multi-view stereo for high resolution imagery[END_REF]. Figure 1 highlights the quality of a mesh, especially for describing high order details. However, buildings are man made objects containing many regular components such as planar or cylindrical shapes. Such a mesh representation gives a large amount of useless information concerning these regular elements which could be more relevantly described by parametric objects (e.g. wall facets by planes or columns by cylinders).

The two families have complementary advantages : semantic knowledge and model compaction for the former, detail modeling and non-restricted use for the latter. A natural idea, but still unexplored, would consist in merging both the families in order to propose a hybrid modeling. Regular elements would be representing by 3D primitives whereas irregular structures would be described by meshes. In this paper, we propose a process for substituting regular mesh patches by 3D-objects. This is of interest for several reasons: (i) the introduction of semantic knowledge in the mesh; (ii) the simplification of the modeling while preserving details; and (iii) the corrections of some errors generated by the multi-view stereo processes.

Figure 1: Modeling by mesh representation output by a multi-view stereo process (data from [START_REF] Strecha | On benchmarking camera calibration and multi-view stereo for high resolution imagery[END_REF], results from [START_REF] Vu | Towards high-resolution large-scale multi-view[END_REF]).

Strategy

Extracting 3D-primitives from meshes without a preliminary segmentation is a difficult problem [START_REF] Dorai | COSMOS -a representation scheme for 3D free-form objects[END_REF]. It has been addressed by [START_REF] Cohen-Steiner | Variational shape approximation[END_REF] for simplifying a mesh into a 3D-plane layout, and then extended by [START_REF] Yan | Quadric surface extraction by variational shape approximation[END_REF] for modeling with quadrics. However, such an approach cannot be efficiently adapted for image-based modeled meshes which contain noise, facet density variations, multi-scale components and errors/approximations resulting of multi-view stereo processes as shown on Figure 1. We adopt a more robust two step strategy consisting in (i) segmenting the mesh, and (ii) fitting 3D-primitives on the obtained partition where it is relevant. Section 2 presents the segmentation process based on a curvature analysis of the mesh. A multi-label energy taking topological smoothness constraints into account is formulated. The optimal labeling is estimated by α-expansion. The 3D-primitive extraction from the obtained partition is then described in Section 3. An error parameter controls the fitting quality and decides whether a mesh cluster has to be substituted by a plane, sphere, cylinder, cone or torus. In addition, a refinement process corrects the eventual errors generated during the segmentation step. Experimental results on real building meshes and also on synthetic data are given in Section 4. Basic conclusions are outlined in Section 5.

Mesh segmentation

Let us consider a three dimensional boundary mesh M defined as a tuple {V, E, F} of vertices V , edges E and triangular faces F. We aim to segment the vertices of the mesh M into subsets corresponding to regions of interest.

Geometric attributes based on curvature analysis

Many kinds of local geometric attributes have been proposed in the literature for segmenting synthetic meshes such as multi-scale blowing bubbles, 3D feature descriptors or skeleton knowledge. The comparative studies proposed in [START_REF] Attene | Mesh segmentation -a comparative study[END_REF][START_REF] Shamir | A survey on mesh segmentation techniques[END_REF] present the most efficient techniques for extracting information from synthetic meshes. Most of these techniques cannot be adapted to meshes generated by multi-view stereo processes due to the problems mentioned in Section 1. Local differential geometry estimates are known to be robust for analyzing the mesh topology. The principal curvatures k min and k max and their associated direction vectors w min and w max measure how the surface bends by different amounts in different directions (see Figure 2). In order to distinguish the various types of shapes, this curvature information is used to label the mesh according to four labels of interest: planar (k max = k min = 0), developable convex (k min = 0 < k max ), developable concave (k min < k max = 0) and non developable surfaces (k min k max = 0). Let us consider L = {1, 2, 3, 4}, the label set corresponding to the classes mentioned above respectively. Let l = (l 1 , ..., l N ) be a label configuration in L N , associated with the N vertices of the mesh M. By denoting G σ (k) = exp(-k 2 /2σ 2 ) the non normalized centered Gaussian distributions with a standard deviation σ , we can express the probability of each label at the vertex i as a combination of the curvature distributions:

Pr(l i |k (i) min , k (i) max ) =          G σ (k (i) min )G σ (k (i) max ) if l i = 1 G σ (k (i) min )(1 -G σ (k (i) max )) if l i = 2 (1 -G σ (k (i) min ))G σ (k (i) max ) if l i = 3 (1 -G σ (k (i) min ))(1 -G σ (k (i) max )) if l i = 4 (1)
Figure 2 presents the behavior of this probability in function of the couple (k min , k max ). The label configuration maximizing

∏ i∈V Pr(l i |k (i) min , k (i)
max ), denoted by l P , is simple to compute and provides an interesting estimator in the case of synthetic meshes as we can see with the Fandisk and Cup models presented on Figure 3. However, the results obtained from non synthetic meshes are clearly more limited. Additional information has to be taken into account to improve results.

A multi-label energy model

The energy of the configuration l is formulated using both a consistency term and topological smoothness constraints, balanced by the parameter β > 0:

U(l) = ∑ i∈V D i (l i ) + β ∑ {i, j}∈E V i j (l i , l j ) (2)
Consistency The consistency D i (l i ) which measures the coherence of the label l i at the vertex i is computed using the probability Pr(l i |k

(i) min , k (i)
max ) (see Eq.1) such as:

D i (l i ) = 1 -Pr(l i |k (i) min , k (i) max ) (3)
The sensitivity of the consistency term is controlled by the standard deviation σ of the principal curvature distributions (See Figure 2). Taking a low σ value makes the consistency term more selective with planar and developable labels and favors the non developable one.

On the contrary, a high value has to be chosen for dealing with noise corrupted meshes.

Topological smoothness constraints

The term V i j represents a pairwise interaction potential between adjacent vertices i and j. It expresses prior knowledge about the optimal labeling.

V i j (l i , l j ) = 1 if l i = l j min(1, a||W i -W j || 2 ) otherwise ( 4
)
where a is a scale factor fixed proportionately to the mean edge length e of M, and W i and W j are 6 × 1 vectors combining the principal direction vectors and their curvatures:

W = k min .w min k max .w max (5)
This term introduces spatial smoothness constraints which take into account the mesh topology. Two principles define the behavior of V i j :

• Smoothness on regular surfaces -In order to favor the label homogeneity in a neighborhood, adjacent vertices are penalized if their labels are different. This principle acts like the Potts model (See Figure 3-2 nd and 4 th rows).

• Edge preservation -The boundaries are preserved by taking into account the principal direction vector variations of adjacent vertices with similar labels. The mesh is then partitioned according to changes of local differential geometry. For example, it allows the separation of two connected planes with different normals (See Figure 3-Corner model).

Optimization by α-expansions

Finding the label configuration that minimizes the energy U requires advanced optimization techniques since U is a non convex function defined in a multi-label space. We use the α-expansion algorithm [START_REF] Boykov | Fast approximate energy minimization via graph cuts[END_REF] based on the Graph-cuts theory. One can easily check that our energy fits the requirements for this method. This algorithm allows us to quickly reach an approximate solution close to the global one. To accelerate the convergence, l P is chosen as initialization. Note that faster algorithms such as Logcut [START_REF] Lempitsky | Logcut -efficient graph cut optimization for markov random fields[END_REF] could be used. However, the time savings would be minor since we have a small number of labels. Figure 3 shows results of the segmentation stage. The proposed multi-label energy significantly improves the results compared to the l P estimator for the non synthetic meshes. The various parts are correctly identified: walls, roofs or stairs are associated with the planar label -columns, corners or vaultings with developable convex or developable concave labels -and ornaments or statues with the non developable one. The edges are accurately localized due to the detection of principal direction vector variations (See 3-3 rd row). It allows us to extract these components easily by a region growing process (see Figure 4). The next stage consists in fitting 3D-primitives to the obtained partition.

Geometric shape extraction

In the sequel, we call cluster a connected region of same label extracted by the previous process. Each cluster of the segmented mesh is then compared to a set of 3D-primitives composed of planes, spheres, cylinders, cones and tori. They represent the most common regular shapes which can be found on buildings.

Extraction strategy-In order to avoid an exhaustive comparison between a cluster and all the types of 3D-primitives, the labeling information obtained in the previous stage is used to drive the shape extraction. A cluster labeled as a planar component is then compared to a plane, developable convex and developable concave clusters to cylinders and cones, and non developable clusters to spheres and tori. An error parameter ξ controls the fitting quality. If the quadratic error between the optimal primitive and the cluster is lower than ξ , the cluster is substituted by the detected primitive. Otherwise, the rejected cluster is compared to the other types of 3D-primitives. This second fitting test prevents wrong labelings generated by scale ambiguities. For example, the large vaultings on Figure 3 are mistakenly labeled as 'planar' clusters due to the low values of their principal curvatures. This additional test correctly fits these vaultings to cylinders (See Figure 5). Finally, if the cluster is still rejected during this second test, it keeps its triangular mesh representation.

Object fitting -Severals works such as [START_REF] Schnabel | Efficient ransac for point-cloud shape detection[END_REF][START_REF] Verma | 3D building detection and modeling from aerial LIDAR data[END_REF] have been proposed to detect shapes in point clouds containing outliers. Contrary to point clouds, meshes have generally less outliers and exhibit useful topological information. Outlier rejection based techniques such as the RANSAC algorithm are not required for our problem due to our preliminary segmentation. Plane fitting can be easily performed using a Principal Component Analysis (PCA). However, fitting spheres, cylinders, cones or tori has no closed-form solution when the dataset only represents an unknown portion of the whole shape. Thus, it requires an iterative non-linear minimization, typically using a Levenberg-Marquardt optimization. We base our fitting on [START_REF] Marshall | Robust segmentation of primitives from range data in the presence of geometric degeneracy[END_REF], that proposes a parametrization and a first order Euclidean distance approximation to spheres, cylinders, cones and tori, that behaves well as curvatures vanish. This allows numerically stable fittings of more complex shapes on a dataset close to a simpler shape (sphere, cone, cylinder or torus fitting of an almost planar patch, cone fitting of an almost cylindrical patch, torus fitting of a spherical or a conical patch...). A multi-initialization strategy using local differential geometry estimates -Relying on a non-linear optimization, the quality of the fitting process depends on its initialization. [START_REF] Marshall | Robust segmentation of primitives from range data in the presence of geometric degeneracy[END_REF] estimates an initialization from a global criterion. Here, for more robustness, we propose to let multiple initializations based on various local estimates compete, and keep the overall best fit (see Figure 4-left). Differential geometry estimates have already been computed for each vertex to drive the segmentation. Considering a small set of seed vertices covering the whole patch, we initialize a non-linear optimization for each seed vertex position using its differential geometry estimates. The parameterizations in [START_REF] Marshall | Robust segmentation of primitives from range data in the presence of geometric degeneracy[END_REF] use an arbitrary point on the shape to parametrize the whole shape using its local differential geometry (normal vector, principal curvatures and directions). Spheres and cylinders are completely parameterized using the local estimates of a seed vertex. Cones, which are generalized cylinders with a center at infinity, are initialized using the locally estimated cylinder. Turning to tori, they contain an inner and an outer circle, where the normals are orthogonal to the axis of revolution. Supposing a seed vertex is on such a circle then yields two possible torus initializations which are optimized independently.

Experiments

Our approach is tested on real meshes generated by the multi-view stereo technique proposed in [START_REF] Vu | Towards high-resolution large-scale multi-view[END_REF]. Figure 4 shows the potential of the method on some details whereas Figure 5 presents results on various larger scenes. There is, to our knowledge, no other method proposing hybrid representations. However, we evaluate our results qualitatively and quantitatively with a visual evaluation, a compression rate study and an accuracy improvement experiment.

Visual evaluation -The obtained hybrid representations are promising and provide interesting simplified modelings of the original meshes while preserving details. The overall rough components of buildings are reconstructed by 3D-primitive layouts with an accuracy controlled by ξ . Such object layouts are very useful since they allow the introduction of semantic information in the modeling. Structural components such as walls, roofs, windows or dormer windows can be easily identified from the obtained primitives by a subsequent basic analysis 1 as one can see on Figure 5-4 th row. The results reveal the reconstruction of interesting fine details such as thin pipes located at the vaultings on Figure 5-2 nd row or small statue heads on Figure 5-last row.

Compression rate-The compression rate, defined as the ratio between the original mesh and the hybrid representation, is function of the error parameter ξ . Table 1 shows that it indeed also depends on the scene: a scene containing many regular components (e.g. Church) has a better factor than one composed of many irregular shapes (e.g. Fountain-P11). The experiments presented Figure 5 are conducted with ξ = e and give both a good compression ratio and visually acceptable results. Figure 6 compares our method with a state-of-the-art decimation method (the one in CGAL library2 ). For a given compression rate, our representation gives a better description than the decimated mesh which is uniformly degraded with no semantic awareness. Indeed, taking the geometric regularity of the scene into account is relevant for buildings: on this detail, planes and cylinders are clearly identified.

Accuracy improvement-In order to quantify in what extent our representation is better than the decimated one, we evaluate the error with respect to the (range scanned) ground truth. In fact, our method even allows the corrections of some errors already contained in the nondecimated model! For instance, the noisy balls and waveform ornament are regularized by half a sphere and a set of toroidal patches respectively on Figure 4. The error occupancy histogram, measured with respect to the standard deviation Σ of the ground truth accuracy (see [START_REF] Strecha | On benchmarking camera calibration and multi-view stereo for high resolution imagery[END_REF]), quantifies this improvement. The number of low-error vertices if higher for the hybrid representation than for the original stereo mesh, mainly transferring from the 2Σ bin to the Σ one (Figure 6 bottom). This is indeed a first step toward a more extensive evaluation, since this improvement seems to concern subset of the mesh the has to be identified and closely analyzed. Yet, the benchmark website of [START_REF] Strecha | On benchmarking camera calibration and multi-view stereo for high resolution imagery[END_REF] only outputs global statistics and does not easily allow this investigation.

Limitations-Our approach cannot extract piecewise 3D-primitives merged in a single cluster. For example, the Ω shape ornaments above the doors of the Herz-Jesu mesh (see Figure 5-2 nd row) are not reconstructed because they are composed of cylinders and tori. The compression rate could be improved by proposing a process for fitting several objects per cluster. Moreover, the process is not well adapted to smoothness variations over the mesh. To solve this problem, β could be locally adjusted according to some estimated local quality.

Figure 6: From left to right, top : details of ground truth [START_REF] Strecha | On benchmarking camera calibration and multi-view stereo for high resolution imagery[END_REF], original mesh [START_REF] Vu | Towards high-resolution large-scale multi-view[END_REF], state-ofthe-art decimated mesh (compression rate=3.6), and our representation (same compression rate). Bottom row : error of the original mesh with respect to ground truth (white=low, black=high), error of our representation, histogram of the errors [START_REF] Strecha | On benchmarking camera calibration and multi-view stereo for high resolution imagery[END_REF].

Conclusion

We propose an hybrid representation of noisy 3D models such as buildings obtained by multi-view stereo. This representation merges meshes and 3D-primitives. It provides high compression rates while keeping details, introduces semantic knowledge despites noise cor-ruption, and even improves accuracy of the original reconstruction. Both the proposed multilabel energy formulation for mesh segmentation and the contributions for 3D-primitive fitting could be used for others meshing applications. In the future, we will study the simultaneous generation of meshes and 3D-primitives during the multi-view stereo process. This would allow us to take interactions between meshes and primitives into account, but would require more complex models and advanced 3D-primitive samplers.

Figure 2 :

 2 Figure 2: Principal curvaturesleft: representation of (k min , k max ), right: map of the label dominance in function of k min and k max (blue sector indicate that the highest probability is obtained for the 'planar' label, red for 'developable convex', green for'developable concave' and yellow for'non developable').

Figure 3 :

 3 Figure 3: Mesh segmentationfrom top to down: original mesh, l P estimator (blue='planar', red='developable convex', green='developable concave' and yellow='non developable'), edge term of the regularizing part of the energy (||W i -W j || 2 ) (red=high values, black=low values), our labeling result after energy minimization.

Figure 4 :

 4 Figure 4: 3D-primitive extractionleft: multi-initialization using local differential geometry estimates (highlighted here as small purple patches), right: examples of fitted primitives with segmented meshes (top) and hybrid representations (down).

Figure 5 :

 5 Figure 5: Large scenesfrom left to right: original mesh, segmented mesh, 3D-primitives (purple=plane, pink=cylinder, blue=cone, yellow=sphere, green=torus), hybrid representation. From top to bottom: Church, Herz-Jesu-P25, Fountain-P11, Castle-P30, Calvary.

Table 1 :

 1 Compression rates in function the error ξ . ξ = 0.1 e ξ = 0.5 e ξ = e ξ = 5 e ξ = 10 e

	Church	1.27	3.55	4.59	10.43	10.43
	Herz-Jesu-P25	1.12	3.57	5.93	10.78	11.34
	Fountain-P11	1.09	2.33	3.6	6.34	11.51
	Castle-P30	1.1	2.19	3.96	8.87	11.21

For example, one can detect walls by just selecting tall surface planar primitives which are vertical.

http://www.cgal.org/

Acknowledgments

The authors are grateful to the EADS foundation for partial financial support.


Philippe Signoret
email: philippe.signoret@univ-fcomte.fr

Territories and governance, a research-action actor's point of view

Keywords: territory, governance, observation, complex system

Once repositioned four different experiences in relation to issues of governance, in a first step, we shall come back very briefly on our initial and very traditional approach of the governance and of the relationships between actors. Then, drawing on the four axes of analysis suggested by A. Dauphine, presenting different situations encountered, we will propose an other way we used to better understand governance by and for observation a complex system: the territory.

I. Introduction

The individual looks for its resources in a background, mobilizes them in multiple project plans before acting and modifying it. The background is not neutral, inert. As a result of physical, chemical or human phenomena, it affects certain action skills. The Territory is home to multiple interactions between different hardware and intangible components. The actor is then placed at the heart of the complex system to observe [START_REF] Moine | Le territoire : comment observer un système complexe[END_REF][START_REF] Signoret | Participative systematization and conceptual maps, methodological proposal to enhance competences, better understand and monitor local issues and strengthen decision making, in 7[END_REF]. Among the elements of the territory system, individuals and groups interact in "a process of coordinating actors, social groups, institutions, to achieve specific goals discussed and defined collectively in fragmented environments and uncertain" (Bagnasco and Le Gales, 1997) that can be called System of social action, or more generally "System of actors" [START_REF] Moine | Le territoire : comment observer un système complexe[END_REF]. It then returns to a more or less formal organization of the actors in a governance system that we define as the seat of the confrontation of needs, projects and power. The governance system appears as a subsystem of the territory and as an avatar of the territorialization [START_REF] Signoret | Participative systematization and conceptual maps, methodological proposal to enhance competences, better understand and monitor local issues and strengthen decision making, in 7[END_REF](Signoret, , 2009)). In fact, this "the sum [combination] of different ways in which individuals and institutions, public and private, manage their common affairs. [...] It includes official institutions and governments with executory powers as well as informal arrangements on which people and institutions have agreed or that they perceive as their interests" (Smouts, 1998 p. 88). So, governance and territory are inseparable concepts. When one tries to define the governance, it often leads to distinguish a first institutional level and a second applied to the business world. We consider these two levels are closely intertwined as to make them nearly inseparable; this is especially the case when we think about knowledge production and therefore entrepreneurship to the territorial intelligence support. For managers, the study of entrepreneurial phenomenon (or the fact ...) refers to three levels of analysis (cognitive, structural and praxeological) within a system of actors and an organization [START_REF] Verstraete | Proposition d'un cadre théorique pour la recherche en entrepreneuriat, PhE = f [(CxSxP) ⊂ (ExO)[END_REF]. For geographers, the territorial development approaches are very similar to those implemented by managers but with the introduction of three complementary components: place, space and time. A combination of these two approaches would be closer to the Territory as we understand it. Thus, A. [START_REF] Dauphine | Les théories de la complexité chez les géographes[END_REF] suggests us to understand the complexity of the territory by several angles of approach. The first focuses on the structure and the interrelationships between different elements; it is the cognitive base of the phenomenon analysed. The second is to take into account the different spatial and temporal scales and in particular to define the proper level of relevance. The third focuses on understanding the organization of elements in space but also in terms of hierarchy. Finally, the fourth tends to better understand the functioning of the system by introducing into the analysis patterns the causalities and the feedbacks. We find all this in the foundations of the structural geography which, according to G. Desmarais [START_REF] Desmarais | La géographie structurale[END_REF], is "to describe and explain the appearance, organization, transformation and evolution of forms, both natural and cultural present on the surface of the Earth".

The response scales overlap and combine. The actors in land planning and development are more and more interconnect. But, territorial policies are too often sectored and there is a lack of coordination of public action. New patterns of work to be developed; it is the demand of both administered and local authorities in a context of modernizing public practices. In addition, the culture of sharing can help in the fight against the emergence of practices at different speeds (Ormaux & Signoret, 2005). Finally, it appears that the development of monitoring tools is an approach that must rely heavily on the future users [START_REF] Moine | How the local governance system is influenced by the creation of an observatory : the OSER 70 experiment[END_REF]. We are then confronted with different situations in which the traditional political model is denounced by a variety of actors involved in governance characterized by greater interdependence (interaction) between the authority and collective action. This new situation is reminiscent of the first concept involved in the definition of any system.

In our research on the territorialisation, the actor's game, as "ongoing process of cooperation and accommodation between different and conflicting interests" (Smouts, 1998, p. 88), remained central to our concerns, until to take the risk of proposing an approach of the Territory sometimes too anthropocentric. So, our interest in governance issues is not new and we have been, alternatively and sometimes simultaneously, placed in positions of agent, actor, observer and/or decision maker: the basic functions of an individual acting as an element of the governance system. We therefore propose to draw lessons from our involvement in various systems of governance that we were allowed to cross over the research.

To do this, we will rely on four research projects, conducted at different scales, taken as an illustration of the evolution of the approach and methods used by local actors, but also of our own representation of an organization and its functioning. Two of them were conducted in the framework of the preparation of development and land planning projects. The two others are part of the development of observation systems (tools and methods for territorial intelligence).

Once repositioned these experiences in relation to governance issues, as a first step, we will return briefly to our original and very classic approach of the governance and relationships between actors. Then we will enter the heart of the matter, drawing on the four axes of analysis suggested by A. Dauphine, presenting different situations encountered. Our thinking is rational and empirical, theoretical and pragmatic and we could summarize our motivations in a few words: observing to better understand, communicating to contribute to the practices modernization, clustering needs and decisions to foster social innovation (Bouchard, 1999). We will talk a lot of complex system and modelling, in support to the construction of representations, discourses and illustrations.

II. Four research actions in the heart of the system of governance

According to Claude Lacour (2003, p. 325) "It is fruitful to focus more on functional and institutional logics of organizations that deal with space, social, urban and less on obvious to undefined". During our research, we were faced with the field reality. What seemed clear and obvious is now marked with the seal of the complexity. In this communication, it is with great caution that we will return to four achievements.

First project: The Territorial Coherence Plan (SCOT) is a planning document that sets basic guidelines for organizing the territory and the evolution of urban areas, established by Act No. 2000-1208 of 13th December 2000 on solidarity and urban renewal. The first action (the White Paper SCOT) [START_REF] Dubos-Paillard E | SCOT Du Grand Besançon -Livre blanc[END_REF] was located far upstream from the formalization of the land planning pattern that is expressed through such document. It was indeed responding to a request of the association of communes:

-Identify and articulate policy, planning and programming documents; -Cross the points of view and reveal the thinks kept dark or the inconsistencies; -Focus on important planning purpose to provide opportunities for development.

Second project: Project of the Great Besançon agglomeration (CAGB). It was not a command to the organizing authority, but an opportunity that was offered to us to take part, as an observer, in a participatory process in the context of a territorial development project. Citizens and elected officials were required to share their points of views about a diagnosis and a project taking into account people's expectations; speakers proposed new areas of thought, technicians fed the reflections through cases and situations analysis.

Third project: OSER 70. Its objective was to develop an observation tool (platform for storing and processing data shared via the Internet), for the department of Haute-Saône, in anticipation with the regional level complementarity. Our work was at the intersection of the expression of the needs of future users and ITC development: translating the needs and explaining the technological tool as part of a collaborative implementation. The supply of the tool, the "path of the data", has also been the subject of special attentions in order to remove obstacles and organize the flow of data [START_REF] Moine | How the local governance system is influenced by the creation of an observatory : the OSER 70 experiment[END_REF] Fourth project: this research includes the issues of technological development of the previous project but for the city of Nîmes. However, taking into account the experience in the field, we suggested not to provide toolkit, but to take the silent partner in a process favouring the use of the tool, but also involving a larger community concerned in local issues related mainly to employment (Signoret, 2009b).

As regards the geography of the places concerned, we are here in the presence of very contrasted situations (See table 1) and that if we consider the effect of dispersion dependant on the number of communities concerned, the weight of the population and of his representation implied in the measure. So, our objective is not to compare similar projects even different approaches in comparable places. Indeed, we could observe that the local context necessarily did not intervene in the choice of the method planned to contribute to the territorial development. In addition, the instrumentation of management and processing of geographical data become a major issue for the participative process in land planning (De Sède Marceau, 2002). The successive projects led us to re-examine certain approaches and that less in terms of methods and technics of the geography than in the manner of apprehending the relation researcher-local operator and thus our own relation of actor (agent) to a local context. Project management must be considered with lot of attention.

The comparison of the four projects on governance criteria can be very delicate and that more especially as they do not fit in same temporalities. However, the grid of analysis (See table 2), built starting from the references borrowed from sociology (P. Lascoumes, M. Koebel, P. Le Gales) and from political science (W. Genieys, B. Jobert, G. Pinson), enable to raise part of the difficulties of a compared analysis. It shows the complementarities of the new modes of interpellation established between the actors as well as the new forms of relations between, on one hand, citizens (or stakeholders to be more general) and, on the other hand, public authorities. The territorial intelligence provide evidences of the need to support the multi-disciplinary exchanges but also, in a social innovation goal, to better consider non-institutionalized and anonymous expert testimonies which then alternatively act as proposers and applicants of the public intervention.

III. Governance and complexity

In 2004, we were considering governance as a component of a triptych (combining policy, functional and operational levels), in which also appear disparate considerations (Collective interest, economy of means, culture of decision, coordination, pedagogy, territorial cohesion). Our second vision of governance could be described as a stack of levels of response from citizens to the European institutions through the various levels of the state organization, the whole being obviously governed by a certain idea of the hierarchisation of these various layers. All that was corresponding to a linear and vertical representation of the public intervention. Our approach of this subject is now based on complexity.

The spatial dimension

Paul Villeneuve establishes relationships between geographical scales and social cohesion (See table 3). If we proceed to the same exercise with the instruments available for the implementation of the public policies of land planning and territorial development, we will arrive at a similar result:

-Global (synoptic) scale: land planning guidelines of the State and the broad guidelines laid down by the European Union. -Macroscale: SRADT regional land planning and sustainable development of the territory plan (SRADDT), departments plans, regional nature park, etc ; the SCOT is more often just between macro and mesoscale. -Mesoscale: the scale of intervention would be more about the commune (e.g.: land use plan) and of the intercommunality which received, since the last laws of decentralization, by transfer of competences, new missions (e.g.: urban displacement plan). -Microscale: they are generally land operations (ZAC, ZAD,…), but also of documents of orientation and/or programming (POIH, ZEP, ZFU,…) which can be decline at the level of the quarter or smaller.

Through this "interest from now on carried to the emergence of new relationship between the places (or the regions) and the world, and to the establishment of new inter-territorial or intergovernmental relations (various forms of regionalization at the continental scale, international exchanges of the regions, networks of cities, governments of urban area…)" and which is presented like "a certain territorialization of the systems of action" (Palard, 2003, p. 317), one can perceive a superposition of scales of intervention which introduces the need for a consistency of the approaches and work methods. Indeed, for example, since it is a question of setting up a SCOT, this one coming to print the territory of a development strategy as regards urbanization and economic development, it becomes inconceivable that the actualization of the Land use plans does not take into account of the options decided at the other scales.

Territories and governance, a research-action actor's point of view -Philippe Signoret -UFC UMR 6049 ThéMA -IT-GO 2010 P 5/8

An archipelagic definition of the Territories (Signoret, 2009a) forces to go beyond the one that propose a representation based on a defined and limited space. Then it can help us to better understand the different forms of power and conflicts. Here, the stretch imports then less than the infrastructures of communication or transport and than the interconnectivity of the places and the actors. For aught, by the subjacent powers that it conceals, the scale is very important. We have to take into account fitments of spaces corresponding, for example, with administrative and electoral cuttings or with spaces of projects that partition the operations whereas the new economic and social contexts invite us to transgress these perimeters because and take into account what occurs to their margins or, in addition, to admit that the territory does not necessarily correspond to a continuous space. Organisational levels, their interrelationships, and their overlaps have also to be taken into account. As with the nesting of spaces, temporalities are different and consequently it is more difficult to put consistency in the large variety of planning or programming instruments so why the necessary sustainability of the tools dedicated to the territorial intelligence (observatories). Finally, the question of the scale also relates to the level of precision of information and thus returns to the definition of concepts and the role of the relations of causality in the comprehension of the mechanisms which influence the dynamics of the territory. [START_REF] Villeneuve | Effets d'échelle et cohésion sociale[END_REF]

A structural approach

We recalled in the introduction the definition given by [START_REF] Desmarais | La géographie structurale[END_REF] to the structural geography. According to the National Center of Textual and Rexical Resources (CNTRL) the structure indicates the "ordering, between them, of the components of a built unit, which makes this unit a coherent whole and provides its specific aspect". Thus, we should first understand the space structure: the fitting of the geographical objects and the nature of the relations between objects which nourish the representations of space. But, within the framework of this exercise we will avoid this approach to concentrate us on the structure of the system of actor: better understanding of what will affect the multiple representations, the actors network, and their capacity to act, arrange, use and manage space, all this contributing in return to better understand the organization/structure of the anthropized space [START_REF] Moine | Le territoire : comment observer un système complexe[END_REF].

The structure precedes the organization. It is a kind of compost in which appear intrinsic potentialities, endogenous and exogenic influences. Understanding the territorial structure requires to have the means of reconsidering the regulations and the optimization of the territorial system who organizes itself. Indeed, it seems fruitless to want to turn over an unfavourable situation of a given perimeter, for example in terms of employment, simply while exploiting an increase in the attractivity of place realizing zones of activities having for vocation to accommodate activities with strong technological value. Indeed, the profile of the populations of the ailing districts, where unemployment rate exceeds the 25% of the working population, only very seldom matches to the potential of qualified job creation. The ignorance of the territorial structure cannot be an excuse in such an obvious situation. Thus, the authorities single to ignore some of the elements of the system and is at the origin of the thinks kept dark.

"The competition between the municipalities have for result a scattering of the areas of activities, a degradation of the landscape, even a weakening of economic efficiency." (Chevailler & Signoret, 2007, p.159). In answer to the superposition of the frame space, to the scattering of competences and to the local development issues, it is necessary to rationalize the territorial organization taking into account the difficulties related to the "critical size" allowing, in particular in rural environment, "to carry out the budgetary effectiveness necessary for the development and the implementation of projects". It also runs up against obsolete logics of tax competition which could be still exerted by the means of the 'taxe professionnelle', a clashing cocktail being able to lead the State authorities to Territories and governance, a research-action actor's point of view -Philippe Signoret -UFC UMR 6049 ThéMA -IT-GO 2010 P 6/8 impose regroupings (Chevailler & Signoret, 2007).

Causes or effects? The forces involved are often differentiated by specific languages (vocabularies), partitioned areas for intervention, labor practices that make the economy of collaboration and participation, and, as we mentioned earlier, different levels of intervention and dilution of responsibilities. If all of those causes are put in our triptych (decisional -operational -functional), then we can easily explain a part of why local development has failed. We are here on the diagnosis and empirical approach, which may refers to sociology and political science, is nevertheless an important research base for geographers, especially in modelling and observation of territorial systems.

Organizational entry

The organization and the development of territories of projects (SCOT, CAGB), go with shared interests or the confrontation of demands (it is one of the objectives of the participative workshops). The will to deal with a common destiny, to organize a collective territory, then imposes "prioritization of the levels of intervention, of the urban functions and of the services, while taking into account and distinguishing what concerns the local services and what must concern the higher services which have to be organized at the level of a department, of a region or the level of the network" (Chevailler & Signoret, 2008, p. 17). Moreover, the complexity of the distribution of the competences between the communities can, under the effect of a bad reading, nourish the feeling of an incoherent stratification where everyone does every thing. The observation of the various levels of intervention (competences, complementarities, redundancies, inconsistencies), or of the circuits of information and the practices then makes possible to better understand the territorialisation and the organization of the system complexes Territoiry.

Concerning observatories, the corollary is to well understand and organize all the line production of geographical information and knowledge of the local phenomena. Thus, an observatory cannot be proposed like a simple software but like an information system, applied to a geographical space, appealing, for its optimization and the production of added-value, to a system of actor in which each one plays a determining role: software development, data flow management, exploitation or the valorization of information, etc. Obviously, all this have to be replaced within a framework of a larger local governorship. During the meetings which marked out the deployment of the observatories, we have heared that the change generated was the cause of new demands. The new organization which is set up then forces to re-examine the economic model of the observatory: to specify the needs for the maintenance and the development of the process. Such an exercise then reveals that the only reorganization of the existing resources, diluted between the actors of the preceding organization, makes possible to ensure the main means but also, in many cases, to carry out substantial resources savings.

Impact of logics and behaviours of actors on the construction of the territory

Here, we have to consider the behavioural psychology (e.g. Law of effect of Edward Lee Thorndike) and the sociology of the organizations (e.g. Theory of the strategic actor of Michel Crozier and Erhard Friedberg, 1977). The individuals use spaces of freedom and the organization of the territory is the product of the actor acts such as the political competitions between elected officials (SCOT), the mobilization of the populations (and the elected officials) around a question which concern themselves (PCAGB), agreement of the changes in the practices (Oser70) or the personalization of the exchanges and the will to share a common language in a sociocognitive process (OMEN). The participation of the actors then often depends on the satisfaction of an individual or collective interest what makes possible to grant an additional legitimacy to the process or, at the opposite, to reinforce antagonisms aiming to ruin it. In all the cases, the issue reinforces the actor in the role and the posture which it defends, Then the institutionalization of the actor can be considered as the ultimate stage of the maturity of the play.

If we stick to the systemic approach [START_REF] Crozier | L'acteur et le système : Les contraintes de l'action collective[END_REF], the empirical observation of the actor can offer to better understand what will energize the structure and will govern the organization of the territorial system. Within our framework on "participative Systematization", we met representatives of P 7/8 about thirty organizations, joined together during the meetings of groupware. We thus could observe, without necessarily seeking to analyze them in a scientific way, various attitudes. Some of them concern what we will call "Lobby": to take the benefit from the meet to flaunt one's a corporative logic, to handle by the speech, to occupy the speaking time, etc. Some others can be associated with institutional logics which, when the reference marks are disturbed, cause a certain distrust with respect to the others or contradiction in the scales of intervention; the quality of the institutional actor is then a crucial factor. Some behaviours do not appear directly related to the organization represented but raise more of individual logics. That initially goes with the adhesion and the appropriation of the process suggested. But that also implicate the ability of the participant to act in the process with the authority of its organization, its competences, its knowledge of the other, its will to cooperate, its culture of the responsibility, its detachment compared to its trade, its political opinions and convictions, its perception of the change, its relationship to the power. All that influences the quality of the individual actor, its reactivity during the exchanges, its will to take part, but also on its construction of the representation of the context and thus on its spatialized vision. Lastly, the project supervisor and the authority also have an important role: choice of a work method, complexity of the subjects of study, means implemented to achieve the goals, awaited and proclaimed repercussions, avoidance or forgets of certain actors, governorship within the work group (competence of the steering committee and of the technical one, collective validation mode, etc).

Our matter focuses more on the defect than on the virtue of the stakeholders. But, the construction of the individual and collective representations of the territory is subjected to the influence of the individuals and of the organizations. Behind the construction of a shared information system, inevitably hide convergent or contradictory strategies which will be the keys of success of the device of territorial intelligence and of the efficiency of the means which will be put to the service of the information and the knowledge. du partage de l'information et de la connaissance.

IV. Conclusion

First of all, it appeared difficult to us to take the diagrams of causality (fourth shutter of the approach of A. Dauphiné) like a specific study axis of governorship of the territories (of and by the observation). Nevertheless, we can find them throughout the speech and we use them in our method of analysis of the structure and organization of the territory or in the play of the actors. Its also a means to study the feedbacks between space, system of actor, representations, intervention and individual or collective mediation what is not done in an explicit way. According to Jean-Pierre [START_REF] Gaudin | Pourquoi la gouvernance ?[END_REF], "the governance, it would be thus simply public action in networks, a relational practice of nonpreset and always to reinvent co-operations, remotely linked to hierarchical reinforcements of the past and routinized procedures.". The various situations described, show that space, time but also the object of the public action and the nature of the stakeholders mobilized around a shared project have a strong impact on the organization of the system of governance adapted to the situation and that all have to evolve in time. So, if the actions are based on collaborative or participative process, the observatories contribute to a better organization of the competences and thus increase the ability of the actors to face the challenges. The models of organization then find their relevance and contribute to the determination (differentiation) of the territories.

V. Bibliography

  BAGNASCO A. et LE GALES P., 1997, Les villes européennes comme société et comme acteur, in Villes en Europe, Ed. La Découverte, pp. 38 BOUCHARD C. (Dir.), 1999, Recherche en sciences humaines et sociales et innovations sociales. Contribution à une politique de l'immatériel, Conseil québécois de la recherche sociale, Les publications du Québec, Québec, p. 7. CANET R., 2004, Qu'est-ce que la gouvernance ?, Conférence prononcée dans le cadre du Séminaire Les nouveaux modes de gouvernance et la place de la société civile, organisé par le Service aux collectivités de l'UQAM, -Montréal, Écomusée du fier CHEVAILLER JC. SIGNORET P., 2007, Logiques d'acteurs et intercommunalité en Franche-Comté, in La Franche-Comté ou la réalité d'un territoire en interface, Revue géographique de l'Est, Tome XLVII, numéro 3, AGE, Nancy, pp. 151-160

Table 1 -

 1 Characteristics of the four sites of project

		SCOT	CAGB	OSER70	OMEN
	Number of municipalities	133	59	545	27
	Number of intercommunal organizations	6	1	30	1
	Population (in thousands)	203	180	236	240
	Number of elected concerned	56	140+85	Many thousands	103

Table 2 -

 2 Summaries of research activities conducted within the framework of territorial projects

Table 3 :

 3 Effects of scale and social cohesion

	Echelle	Level of aggregation	Dominant value	Political orientation	Relation of property
	Micro	Individual	Liberty	Liberalism	Private
	Meso	Community	Solidarity	Communitarianism	Commune
	Macro	Society	Egality	Socialisme	State

Territories and governance, a research-action actor's point of view -Philippe Signoret -UFC UMR 6049 ThéMA -IT-GO 2010 P 3/8