Jesús Angulo
email: jesus.angulo@mines-paristech.fr

Statistical Gaussian Model of Image Regions in Stochastic Watershed Segmentation

Stochastic watershed is an image segmentation technique based on mathematical morphology which produces a probability density function of image contours. Estimated probabilities depend mainly on local distances between pixels. This paper introduces a variant of stochastic watershed where the probabilities of contours are computed from a gaussian model of image regions. In this framework, the basic ingredient is the distance between pairs of regions, hence a distance between normal distributions. Hence several alternatives of statistical distances for normal distributions are compared, namely Bhattacharyya distance, Hellinger metric distance and Wasserstein metric distance.

Introduction

Image segmentation is one of the most studied and relevant problems in low level computer vision. Indeed, the state-of-the-art is vast and rich in multiple paradigms. We are interested here on approaches based on statistical modeling of pixels and regions. Examples of methods tting in such a paradigm and having excellent performance are mean shift [START_REF] Comaniciu | Mean Shift Analysis and Applications[END_REF] and statistical region merging [START_REF] Nock | Statistical Region Merging[END_REF]. Hierarchical contour detection is another successful paradigm with approaches based for instance on machine learned edge detection [START_REF] Arbelaez | Contour detection and hierarchical image segmentation[END_REF] or on watershed transform [START_REF] Beucher | The Morphological Approach to Segmentation: The Watershed Transformation[END_REF].

Instead of dealing with a determinist set of contours, the idea of the stochastic watershed [START_REF] Angulo | Stochastic watershed segmentation[END_REF] (SW) is to estimate a probability density function (pdf ) of contours by MonteCarlo simulations. Some variants included multiscale framework [START_REF] Angulo | Multiscale stochastic watershed for unsupervised hyperspectral image segmentation[END_REF],

bagging framework [START_REF] Franchi | Bagging Stochastic Watershed on Natural Color Image Segmentation[END_REF], robust framework [START_REF] Bernander | Improving the stochastic watershed[END_REF], etc. It was shown in [START_REF] Meyer | A stochastic evaluation of the contour strength[END_REF] that the corresponding pdf obtained by SW can be calculated in closed form without simulation by using graph algorithms, for more recent results see also [START_REF] Meyer | Stochastic Watershed Hierarchies[END_REF]. Nevertheless, we focuss here on an approach working on simulations. In particular, our contribution is in the line of [START_REF] López-Mir | Probability Density Function of Object Contours Using Regional Regularized Stochastic Watershed[END_REF], where the estimation of the probability of each contour is based on a regional model of each region, the model being in [START_REF] López-Mir | Probability Density Function of Object Contours Using Regional Regularized Stochastic Watershed[END_REF] the mean color. In the present work, the approach is pushed forward such that each region should be modeled as a multivariate normal distribution. The basic ingredient will be a distance between pairs of regions, hence a distance between normal distributions. In this context, several alternatives of statistical distances for normal distributions are compared.

The rest of the paper is organized as follows. In Section 2, we remind the MonteCarlo framework of stochastic watershed and in particular the simulation of regionalized random germs. Section 3 introduces the contribution of the paper: rst, distances between multivariate normal distributions are reviewed; second, computation of probability density of contours based on a normal region model is formulated. Results are discussed and compared to statistical region merging segmentation [START_REF] Nock | Statistical Region Merging[END_REF].

Remind on Stochastic Watershed

Regionalized Poisson points. We rst consider the notion regionalized random points as well as the algorithm used to simulate a realization of N random germs associated to a spatial density.

A rather natural way to introduce uniform random germs is to generate realizations of a Poisson point process with a constant intensity θ (i.e., average number of points per unit area). It is well known that the random number of points N (D) falling in a domain D, which is considered a bounded Borel set, with area |D|, follows a Poisson distribution with parameter θ|D|, i.e., . Pr{N (D) = n} = e -θ|D| (-θ|D|) n n! . In addition, conditionally to the fact that N (D) = n, the n points are independently and uniformly distributed over D, and the average number of points in D is θ|D| (i.e., the mean and variance of a Poisson distribution is its parameter).

More generally, we can suppose that the density θ is not constant; but considered as measurable function, dened in R d , with positive values. For simplicity, let us write θ(D) = θ(x)dx. It is also known [START_REF] Matheron | Random sets and integral geometry[END_REF] that the number of points falling in a Borel set B according to a regionalized density function θ follows a Poisson distribution of parameter θ(D), i.e., Pr{N (D) = n} = e -θ(D) (-θ(D)) n n! . In such a case, if N (D) = n, the n are independently distributed over D with the probability density function θ(x) = θ(x)/θ(D). In practice, in order to simulate a realization of N independent random germs distributed over the image with the pdf π k (x) we propose to use an inverse transform sampling method. More precisely, the algorithm to generate N random germs in an image m : E → {0, 1} according to density θ(x) is as follows:

1. Initialization: m(xi) = 0 ∀xi ∈ E; P = Card(E) 2. Compute cumulative distribution function: cdf (xi) = k≤i θ(x k ) P k=1 θ(x k ) 3. for j = 1 to N 4. rj ∼ U (1, P ) 5.
Find the value sj such that rj ≤ cdf (xs j ) .

6.

m(xs j ) = 1

Marker-driven watershed transform. Let g(x) and mrk(x) be respectively a (norm of ) gradient image and a marker image. Intuitively, the associated watershed transformation [START_REF] Beucher | The Morphological Approach to Segmentation: The Watershed Transformation[END_REF], W S(g, mrk)(x), produces a binary image with the contours of regions marked by the image mrk according to the strength of contour given by the gradient image g. The classical paradigm of watershed segmentation lays on the appropriate choice of markers, which are the seeds to initiate the ooding procedure.

(a) (b) (c1) (c2) (c3) (d) (e)
Probability density of contours using MonteCarlo simulations of watershed. In the stochastic watershed (SW) approach [START_REF] Angulo | Stochastic watershed segmentation[END_REF], an opposite direction is followed, by spreading random germs for markers on the watershed segmentation. This arbitrary choice will be balanced by the use of a given number M of realizations, in order to lter out non signicant uctuations. Each piece of contour may then be assigned the number of times it appears during the various simulations in order to estimate a probability density function (pdf ) of contours. In the case of uniformly distributed random germs, large regions will be sampled more frequently than smaller regions and will be selected more often.

Using g(x) as density for regionalization of random germs involves sampling high contrasted image areas and it has been proved to be an appropriate choice [START_REF] Angulo | Stochastic watershed segmentation[END_REF].

In this case, the probability of selecting a contour will oer a trade-o between strength of the contours and size of the adjacent regions.

More precisely, given a color image f the associated SW pdf of contours is obtained as follows. Let {mrk n (x)} M n=1 be a series of M realizations of N spatially distributed random markers according to its gradient image g. Each realization of random germs is considered as the marker image for a watershed segmentation of gradient image g in order to obtain the binary image

W S(g, mrk n )(x) = 1 if x ∈ Watershed lines 0 if x / ∈ Watershed lines
Consequently, a set of M realizations of segmentation is computed, i.e., {W S(g, mrk n )} 1≤n≤M . Note that in each realization the number of points determines the number of regions obtained (i.e., essential property of watershed transformation). Then, the probability density function of contours is computed by the kernel density estimation method as follows:

pdf (x) = 1 M M n=1 W S(g, mrk n )(x) * K σ (x). (1)
Typically, the kernel K σ (x) is a spatial Gaussian function of width σ, which determines the smoothing eect.

Then, the image pdf (x) can be segmented by selecting the contours of probability higher than a given contrast [START_REF] Angulo | Stochastic watershed segmentation[END_REF]. Fig. 1 depicts an example of color image segmentation using SW. As we can note from this example, which includes large homogenous areas, well contrasted objects as well as textured zones, the pdf and corresponding segmentation produces relatively satisfactory results. However, large homogeneous areas are oversegmented and textured zones are not always well contoured. Obviously, low contrasted areas (e.g., boundary between dog head and wall) are not properly segmented. All those are well known drawbacks of SW which have been addressed by the robust stochastic watershed (RSW) [START_REF] Bernander | Improving the stochastic watershed[END_REF] or by the regional regularized stochastic watershed [START_REF] López-Mir | Probability Density Function of Object Contours Using Regional Regularized Stochastic Watershed[END_REF]. We adopt here an approach related to the latter one, based on a statistical model of regions.

Let us remind the principle of the RSW [START_REF] Bernander | Improving the stochastic watershed[END_REF] since it will be also used in the results of next section. The fundamental property of watershed is the insensitivity to the placement of seed points, which usually enables the SW segmentation to nd reliably relevant boundaries, but, in the case of false boundaries it works to our disadvantage. The idea of RSW is to introduce a perturbation n (i.e., small amount of noise) into the ooding function g (i.e., gradient) at each realization n, in order to reduce the number of times that a false boundary will appear.

More precisely, the kernel density estimator (1) becomes now

pdf (x) = 1 M M n=1 W S(g + n , mrk n )(x) * K σ (x), (2)
where n (x) is in our experiments a zero-mean Gaussian white noise with an intensity-dependent variance from g(x).

Multivariate Gaussian Model of Regions in SW

As we just discussed, the watershed segmentation of g from N markers (imposed minima) produces a set of thin lines dividing the image domain E into N disjoint

τ = {R r } 1≤r≤N , with i = j ⇒ R i ∩ R j = ∅, such that E = ∪ r R r W S(x) ⇔ W S(x) = E \ ∪ r R r = ∪l i,j .
The watershed lines W S(x) can be decomposed into the curves that separates the regions. More precisely, let us denote by l i,j the curve (or irregular arc segment) dened as the boundary between regions R i and R j , i.e., l i,j = ∂R i ∩ ∂R j .

We obviously have W S = ∪l i,j , but we also note that in the case of three (or more adjacent) regions, their boundary segments intersect at their junctions (or triple points).

The color image values restricted to each region of the partition, P i = f (R i ), can be modeled by dierent statistical distributions. Here we focuss on a multivariate normal model.

Distances for multivariate normal distributions

Let us consider a family of multivariate normal distributions P i of mean µ i and covariance matrix Σ i , i.e., P i ∼ N (µ i , Σ i ). Dierent distances are dened in the space of P i .

Bhattacharyya distance and Hellinger metric distance. The Bhattacharyya distance D B (P 1 , P 2 ) measures the similarity of two discrete or continuous probability distributions P 1 and P 2 . More precisely, it computes the amount of overlap between the two statistical populations, i.e., D B (P 1 , P 2 ) = -log P 1 (x)P 2 (x)dx. is given by

D B (P 1 , P 2 ) = 1 8 (µ 1 -µ 2 ) T Σ -1 (µ 1 -µ 2 ) + 1 2 log det Σ √ det Σ 1 det Σ 2 , (3)
where

Σ = Σ 1 + Σ 2 2 .
Note that the rst term in the Bhattacharyya distance is related to the Mahalanobis distance, both are the same when the covariance of both distributions is the same.

We have 0 ≤ D B ≤ ∞ and it is symmetric D B (P 1 , P 2 ), it is not a metric. But D B does not obey the triangle inequality and therefore it is not a metric.

Nevertheless, it can be metrized by transforming it into to the following Hellinger metric distance D H (P 1 , P 2 ) given by D H (P 1 , P 2 ) = 1 -exp (-D B (P 1 , P 2 )),

such that DH (P1, P2) = 1 - det Σ √ det Σ1 det Σ2 -1/2 e (-1 4 (µ 1 -µ 2 ) T (Σ 1 +Σ 2 ) -1 (µ 1 -µ 2 )) . ( 4
)
Hellinger distance is an α-divergence [START_REF] Amari | Methods of information geometry[END_REF], which corresponds to the case α = 0 and it is the solely being a metric distance. Hellinger distance can be related to measure theory and asymptotic statistics. For more details on Bhattacharyya and Hellinger distances, see for instance [START_REF] Nielsen | The Burbea-Rao and Bhattacharyya centroids[END_REF].

Wasserstein metric distance. The Wasserstein metric is a distance function dened between probability measures on a given metric space based on the notion optimal transport [START_REF] Villani | Optimal transport. Old and new[END_REF]. Namely, the W 2 Wasserstein distance between probability measures µ and

ν on R n is W 2 (µ, ν) = inf E( X -Y 2 ) 1/2
, where the inmum runs over all random vectors (X, Y ) ∈ R n × R n with X ∼ µ and Y ∼ ν. For the case of discrete distributions, it corresponds to the well-known earth mover's distance [START_REF] Rubner | A Metric for Distributions with Applications to Image Databases[END_REF].

In the case of two multivariate normal distributions, the Wasserstein metric distance is obtained in a closed form as [START_REF] Dowson | The Fréchet distance between multivariate normal distributions[END_REF][START_REF] Takatsu | Wasserstein geometry of Gaussian measures[END_REF]:

D W (P 1 , P 2 ) = µ 1 -µ 2 2 + Tr (Σ 1 + Σ 2 -2Σ 1,2 ), (5)
where

Σ 1,2 = Σ 1/2 1 Σ 2 Σ 1/2 1 1/2 .
We note in particular that in the commutative case

Σ 1 Σ 2 = Σ 2 Σ 1 we have D W (P 1 , P 2 ) 2 = µ 1 -µ 2 2 + Σ 1/2 1 -Σ 1/2 2 2 F .

Probability density function estimation

We have now the ingredients to compute by MonteCarlo simulations the probability density function from a color image. The idea is to assign to each piece of contour l i,j between regions R i and R j the statistical distance between the color gaussian distributions P i and P j :

π i,j = D(P i , j ) l k,l ∈W S D(P k , P l ) , (6)
where D(P i , P j ) is any of the distances discussed above. Thus, for any realization n of SW, denoted W S(x, n), one can compute the following image of weighted contours:

P r(x, n) = π i,j if x ∈ l n i,j 0 if x / ∈ l n i,j
where l n i,j is the boundary between regions R i and R j from W S(x, n). Fig. 3 gives an example of three realizations of SW lines W S(x, n) and the corresponding probability maps P r(x, n) using the Bhattacharyya distance.

Finally, integrating across the M realizations, the MonteCarlo estimate of the probability density function of contours is given by

pdf (x) = 1 M M n=1 P r(x, n) * K σ (x). (7)
Obviously, this approach is compatible with the robust stochastic watershed (RSW) variant discussed in previous section, since each realization n of the RSW produces also a tessellation of E since the image of weighted contours P r(x, n) can be computed.

In Fig. 4 is depicted a comparison of image segmentation of Custard using gaussian region model SW and gaussian model RSW. In particular, the pdf (x)

for the three considered distances (Bhattacharyya distance, Hellinger metric distance and Wasserstein metric distance) is provided, as well as the obtained segmentation by taking a probability contrast value which provides a similar degree of segmentation. It is also included the result obtained using only the color mean as the model proposed in [START_REF] López-Mir | Probability Density Function of Object Contours Using Regional Regularized Stochastic Watershed[END_REF]. From this example, and similar ones obtained from more experiments, Bhattacharyya distance produces good results and a better selectivity of contours than Hellinger metric distance. The results obtained by Wasserstein metric distance are also relevant but the inuence of the covariance matrix is less signicant, thus being closer to the results obtained using the distance between only the mean colors. Concerning the comparison between the standard SW and the RSW paradigm, it is visually observed that the gaussian model RSW produces improvements on the obtained segmentation with respect to SW.

We have also included in the gure the results obtained for this example by statistical region merging (SRM) [START_REF] Nock | Statistical Region Merging[END_REF], computed using MATLAB code provided in [START_REF] Boltz | Image segmentation using statistical region merging[END_REF]. Using the scale parameter Q, we have computed nine segmentations, such that the sum of contours from the nine images can be viewed as a contours saliency function. Then, we have selected two values of Q giving segmentation similar to those of the gaussian region model SW. We observe that the detected regions are quite similar, however, the precision of contours in SW-approaches is qualitatively better for such kind of images.

Perspectives

Besides a more quantitative assessment of the performance of the proposed algorithms, other related perspectives can be considered in ongoing research. First, in addition to color information, texture at each pixel x can be described also by its structure tensor T (x) ∈ SP D(2). Thus, texture at each region R i can be described by a zero-mean gaussian distribution N (0, Σ i ), where the covariance matrix is given by Σ i = |R i | -1 x∈Ri T (x). Hence, the approach presented in the paper can be also used to estimate pdf of contours from texture information or from color + texture. Second, the use of some available prior knowledge, typically represented by training images of annotated contours, could be considered in order to have a (semi-)supervised segmentation. In our framework, this goal can be formulated as a problem of distance learning in the space of multivariate normal distributions. The number of realization is M = 50 and the number of random markers at each realization is N = 200 germs. Bottom, image segmentation of Custard using statistical region merging (SRM) [START_REF] Nock | Statistical Region Merging[END_REF]: (e) sum of contours obtained from SRM using nine values of Q (256, 128, 64, 32, 16, 8, 4, 2, 1), (f ) segmentation for Q = 128 and (g) for Q = 32.

Fig. 1 .

 1 Fig. 1. Stochastic watershed segmentation of Custard image: (a) color image f , (b) its color gradient g, (c) top, three realizations of random markers mrkn regionalized from g, bottom, corresponding watershed lines W S(g, mrkn), (d) estimated density of contours pdf , (e) segmentation from d. In (c) and (d) images in negative for better visualization.

Fig. 2 .

 2 Fig. 2. Tessellation τ of E from watershed W S(x).

Fig. 3 .

 3 Fig. 3. Given the Custard color image (a), (b1)-(b3) are, on the top, three realizations of SW lines W S(x, n), at the bottom, the corresponding probability maps P r(x, n) using the Bhattacharyya distance.

Fig. 4 .

 4 Fig. 4. Top, image segmentation of Custard using gaussian region model SW, where the rst row is pdf (x) and second row the segmentation obtained at a given probability contrast; middle, gaussian region model RSW: (a) using only color mean [12], (b) Bhattacharyya distance, (c) Hellinger metric distance, (d) Wasserstein metric distance.


Jesús Angulo
email: jesus.angulo@mines-paristech.fr

Morphological Scale-Space Operators for Images Supported on Point Clouds

Keywords: mathematical morphology, point clouds image, metric measure space, idempotent measure, Hamilton-Jacobi semigroup

HAL is

Introduction

With the development of 3D scanning technology it is now easy to generate 3D models from real objects, together with standard images painted on them. These discrete objects can be represented either by surfaces (typically as polygonal meshes) or by scattered point clouds. In this paper we focus exclusively on a point cloud representation. A typical case of such generalized 3D images are the point clouds obtained from range cameras, like the Kinect camera. Indeed, this is the case-study which will be used in the paper to illustrate our methods.

Let us formalize the notion of point cloud and of image on it. A point cloud (X, d) is a nite set of points X = {x i } N i=1 equipped with a distance function d, which corresponds typically to the Euclidean distance in the ambient space where X is endowed. This set of points has been irregularly sampled from a metric space; typically X ⊂ R d (for instance the 3D space) but X can be also a sampled manifold with possible complex topological/geometric structure. We assume unknown inner product and geodesic structure of the underlying manifold. It is also important to note that (X, d) is neither a graph nor a discretized surface with triangulations. Thus there is a lack of natural connectivity (or neighborhood relationship) between the points. A real-valued point cloud (X, d, f ), or image f supported on a point cloud (X, d), is the function f which maps each point of X to a valuation space, typically the extended real line for gray-scale images, i.e., f : Point clouds processing is an active research area which have been approached using geodesic distance methods [START_REF] Mémoli | A theoretical and computational framework for isometry invariant recognition of point cloud data[END_REF], laplace operator [START_REF] Belkin | Constructing laplace operator from point clouds in R d[END_REF] or diusion PDEs [START_REF] Liang | Methods and Algorithms for Scientic Computing Solving Partial Dierential Equations on Point Clouds[END_REF].

{x i } N i=1 -→ R.
Connection between heat method on manifolds and shortest geodesic distance via the Varadhan's formula has been used in [START_REF] Crane | Geodesics in Heat: A New Approach to Computing Distance Based on Heat Flow[END_REF], which integrates the heat ow.

Morphological operators have been also considered for point cloud processing [START_REF] Calderon | Point Morphology[END_REF].

However, here we are interested on processing images painted on point clouds, without modifying the point cloud itself. The problem of smoothing images painted on surfaces using the short-time Beltrami kernel was considered for instance in [START_REF] Spira | A short-time Beltrami kernel for smoothing images and manifolds[END_REF].

In this context, the aim of this paper is to develop the theory and to propose an algorithm for morphological processing of such generalized manifold images.

Mathematical morphology is a nonlinear image processing methodology based on two basic operators, dilation and erosion, which correspond respectively to the convolution in the max-plus algebra and its dual. More precisely, in Euclidean (translation invariant) mathematical morphology the pair of adjoint and dual [START_REF] Serra | Image Analysis and Mathematical Morphology[END_REF][START_REF] Heijmans | Morphological image operators[END_REF]:

operators, dilation (sup-convolution) (f ⊕ b)(x) and erosion (inf-convolution) (f ⊖ b)(x) of an image f : E ⊂ R n → R = R ∪ {-∞, +∞}, are given by
{ δ b (f )(x) = (f ⊕ b)(x) = sup y∈E {f (y) + b(y -x)} , ε b (f )(x) = (f ⊖ b)(x) = inf y∈E {f (y) -b(y + x)} ,
where b : R n → R is the structuring function which determines the eect of the operator. The structuring function plays a similar role to the kernel in classical convolution ltering. The structuring function is typically a parametric family b t (x), where t > 0 is the scale parameter. In particular, the canonic structuring function is the parabolic shape (i.e., square of the Euclidean distance): b t (x) = -∥x∥ 2 2t , such that the corresponding dilation and erosion are equivalent to the viscosity solution of the standard Hamilton-Jacobi PDE [START_REF] Maragos | Dierential morphology and image processing[END_REF]:

u t (t, x) ∓ ∥u x (t, x)∥ 2 = 0, (t, x) ∈ (0, +∞) × E; u(0, x) = f (x), x ∈ E.
Theory of morphological ltering is based on opening and closing operators, obtained respectively by composition product of erosion-dilation and dilation-erosion. Opening (resp. closing) is increasing, idempotent and anti-extensive (resp. extensive).

Evolved lters are obtained by composition of openings and closings [START_REF] Serra | Image Analysis and Mathematical Morphology[END_REF][START_REF] Heijmans | Morphological image operators[END_REF]. Morphological operators are classically dened for images supported on Euclidean spaces. A recent work has introduced mathematical morphology for real valued images whose support space is a Riemannian manifold [START_REF] Angulo | Riemannian Mathematical Morphology[END_REF].

In order to extend morphological operators to process point cloud supported images, one needs to dene dilation and erosion as semigroup operators in framework adapted to point clouds (X, d). That corresponds to a supremal convolution (and inmal convolution) using admissible structuring function on (X, d). These operators are formulated in Section 3. In a more theoretical perspective, we start in Section 2 by introducing the notion of abstract structuring functions formulated on length metric Maslov idempotent measurable spaces, which is the appropriate setting for (X, d). In practice, computation of Maslov structuring function will be approached by a random walk framework to estimate heat kernel on a metric measure space (X, d, µ), followed by the logarithmic trick, as shown in Section 4. Some preliminary examples of morphological processing RGB-valued point cloud images are also given in Section 4.

2 From abstract heat semigroups to Maslov structuring functions Before introducing the morphological framework, let us review the theory of abstract heat kernels on a metric measure space, which is based on recent works by Grigor'yan and co-workers [START_REF] Grigor'yan | Heat Kernel and Analysis on Manifolds[END_REF][START_REF] Grigor'yan | Heat kernels on metric measure spaces[END_REF].

Heat kernel on metric measure spaces

Let (M, d, µ) be a locally compact, separable metric space, endowed with a Radon measure µ. Then a family {p t } t≥0 of measurable functions p t (x, y) on M × M is called a heat kernel is the following conditions are almost surely satised ∀x, y ∈ M and t, s > 0: i) Positivity: p t (x, y) ≥ 0; ii) Total mass inequality:

∫ M p t (x, y)dµ(y) ≤ 1, iii) Symmetry: p t (x, y) = p t (y, x), iv) Semigroup property p s+t (x, y) = ∫ M p s (x, z)p t (z, y)dµ(z), v) Approximation to identity: For any f ∈ L 2 (M, µ) ∫ M p t (x, y)f (y)dµ(y) -→ L 2 f (x) as t → 0 + . In addition, the heat kernel is called stochastically complete (conservative) when ∫ M p t (x, y)dµ(y) = 1.
Given a measurable function f on M , any heat kernel gives rise to the family of operators {P t } t≥0 , called the heat semigroup, where P 0 = id and P t for t > 0 is dened by

P t f (x) = ∫ M p t (x, y)f (y)dµ(y).
(1) such that {P t } t≥0 is continuous, symmetric and a Markovian semigroup P t P s = P s+t .

Heat kernels and heat semigroups arise naturally from Markov processes. Let ({X t } t≥0 ; {P x } x∈M ) be a Markov process on M , that is reversible with respect to measure µ. Assume that it has the transition density p t (x, y), i.e., a function such that ∀x ∈ M, t > 0, and all Borel sets A ⊂ M , one has P

x (X t ∈ A) = ∫ M p t (x, y)dµ(y).
If M is a smooth connected compact Riemannian manifold such that△ µ is the Laplace-Beltrami operator on M , where µ is the Riemannian measure, it is well known that starting from the heat equation ∂u ∂t = △ µ u, we have that, for any y ∈ M , the function (t, x) → p t (x, y) is the smallest positive fundamental solution of the heat equation with a source at y, i.e., if u(x, 0) = u 0 (x), then u(x, t) = ∫ M p t (x, y)u 0 (y)dµ(y). In addition, let {φ k } +∞ k=0 be an orthonor- mal basis of eigenfunctions of Laplace-Beltrami operator -△ µ , with eigenvalues

0 = λ 0 < λ 1 ≤ λ 2 ≤ • • • , then the heat kernel is determined by p t (x, y) = ∑ +∞ k=0 e -λ k t φ k (x)φ k (y).
As mentioned in the introduction, all these dierent viewpoints on heat kernel theory have been considered in the image processing state-of-the-art dealing with meshes and point clouds.

The explicit expression of the heat kernel in the Euclidean space (M, d, µ) = R n is just the Gaussian kernel. In the case of a Riemannian manifold (M, µ), only estimates are available, which typically depends on the geodesic distance in M , see [START_REF] Grigor'yan | Heat Kernel and Analysis on Manifolds[END_REF][START_REF] Grigor'yan | Heat kernels on metric measure spaces[END_REF]. Similarly, in the case of a metric measure space (M, d, µ), the sub-Gaussian estimate has the following form (assuming a walk dimension equals to 2) [START_REF] Grigor'yan | Heat Kernel and Analysis on Manifolds[END_REF][START_REF] Grigor'yan | Heat kernels on metric measure spaces[END_REF]:

p t (x, y) ≍ C V (x, t 1/2 ) exp ( -c d 2 (x, y) t ) (2)
where d(x, y) is a metric, V (x, r) is the volume function of a metric ball on the space and C and c are positive constants.

Structuring functions on length metric Maslov measure spaces

We need the counterpart of this abstract theory in the context max-plus mathematics (also known as idempotent analysis [START_REF] Maslov | Méthodes opératorielles[END_REF][START_REF] Litvinov | Idempotent Functional Analysis: An Algebraic Approach[END_REF]). More precisely, the max-plus equivalent of the heat kernel will be here named structuring functions on a length metric Maslov measure space.

Length metric Maslov measure space. Theoretical foundations of Maslov idempotent measure theory [START_REF] Moral | Maslov Optimization Theory: Optimality versus Randomness[END_REF][START_REF] Akian | Densities of idempotent measures and large deviations[END_REF] are based on replacing in the structural axioms of probability theory the role of the classical semiring S (+,×) = (R + , +, ×, 0, 1, ≤)

of positive real numbers by the idempotent semiring:

S (max,+) = ( R, max, +, -∞, 0, ≤). In this context, a change of the measure involves a consistent counterpart to the standard probability theory.

Let (X, d) be a (Hausdor topological) metric space and let m be a Maslov idempotent measure on X, i.e., mapping from X in the max-plus semiring such that for every function f :

X → R, we have A ⊂ X, m f (A) = sup x∈A f (x).
In addition, we should assume that (X, d) is a length space. We remind that a length space is a metric space such that for any pair of points, their distance is the length of the shortest path between the two points. The triple (X, d, m) is called a length metric Maslov measure space. Maslov

m (b t (x, •)) = sup y∈X b t (x, y) ≤ 0 ⇔ b t (x, y) ≤ 0. Completeness (or conservative): m (b t (x, x)) = 0 ⇔ b t (x, x) = 0. Symmetry: b t (x, y) = b t (y, x). Semi-group property: b t+s (x, y) = sup z∈X {b t (x, z) + b s (z, y)} .
Approximation to identity:

sup y∈X {f (y) + b t (x, y)} -→ f (x) as t → 0 + .
Logarithmic trick: from heat kernel to Maslov structuring function.

Our approach is based on the so-called logarithmic connection between linear and morphological system theory [START_REF] Burgeth | An Explanation for the Logarithmic Connection between Linear and Morphological System Theory[END_REF]. The theory of this connection between the usual convolution and the convolution in max-plus mathematics is based on the Cramer transform [START_REF] Akian | Densities of idempotent measures and large deviations[END_REF], which is dened as the Legendre transform of the logarithm of the Laplace transform. In the particular case of a canonic Gaussian kernel of type: 1/

√

2πσ 2 e -x 2 /(2σ 2 ) , it is well known that the Laplace transform is e (σ 2 s 2 )/2 and the nal Cramer transform is -x 2 /(2σ 2 ). In summary the Cramer transform of the Gaussian kernel is equal to quadratic structuring function.

Similarly, we can easily show that, up to a normalizing constant, the logarithm of the heat kernel p t (x, y) in a metric measure space (X, d, µ) is a Maslov structuring function b t (x, y) in the counterpart idempotent metric measure space.

Let us consider for instance the case of the sub-Gaussian estimate of the heat kernel [START_REF] Akian | The Max-Plus Martin boundary[END_REF]. First, a normalization version pt (x, y) is required such that pt (x, x) = 1, i.e., pt (x, y) = V (x, t 1/2 )/Cp t (x, y). Then, the corresponding Maslov structuring function is dened as

b t (x, y) = log pt (x, y) ≍ -c d 2 (x, y) t , (3)
and, without loss of generality, we x c = 1/2. The nonpositivity, total mass inequality, completeness, symmetric and approximation to the identity are obvious. In order to prove the semigroup property, we need to assume that d(x, y) is a metric distance in a length space such that [START_REF] Lott | Hamilton-Jacobi semigroup on length spaces and applications[END_REF] b t+s (x, y) = sup

z∈X {b t (x, z) + b s (z, y)} = sup z∈X { - d 2 (x, z) 2t - d 2 (z, y) 2s } } = -inf z∈X { d 2 (x, z) 2t + d 2 (z, y) 2s } } = - d 2 (x,
E t f (x) = inf y∈X {f (y) -b t (y, x)} .
D t f (x) = sup y∈X { f (y) -d 2 (x,y) 2t } and E t f (x) = inf y∈X { f (y) + d 2 (x,y) 2t }
, are equal to the so-called Hamilton-Jacobi semigroup on length spaces, introduced by Lott and Villani in [START_REF] Lott | Hamilton-Jacobi semigroup on length spaces and applications[END_REF] and extendedly considered more recently by Ambrosio et al. in [START_REF] Ambrosio | Calculus and heat ow in metric measure spaces and applications to spaces with Ricci bounds from below[END_REF] for the study of Ricci curvature bounds in metric spaces.

Properties of semigroups δ bt and ε bt

The following properties can be easily proved using the properties of a Maslov structuring function.

1. (Adjunction) For any two real-valued functions f and g on (X, d, m), the pair (ε bt , δ bt ) forms an adjunction, i.e., ∀x ∈ X

δ bt (f )(x) ≤ g(x) ⇔ f (x) ≤ ε bt (g)(x).
2. (Duality by involution) For any function f (x) and ∀x ∈ X, one has δ bt (f

)(x) = -ε bt (-f )(x). 3. (Increaseness) If f (x) ≤ g(x), ∀x ∈ X, then δ bt (f )(x) ≤ δ bt (g)(x) and ε bt (f )(x) ≤ ε bt (g)(x)
, ∀x ∈ X and ∀t > 0.

(Extensivity and anti

-extensivity) δ bt (f )(x) ≥ f (x) and ε bt (f )(x) ≤ f (x),
∀x ∈ X and ∀t > 0.

(Ordering property)

If 0 < s < t then ε bt (f )(x) ≤ ε bs (f )(x) ≤ f (x) δ bs (f )(x) ≤ δ bt (f )(x)
, ∀x ∈ X. 6. (Scale-space) For any function f and ∀x ∈ X, and for all pair of scales s > 0 and t > 0, we have the Maslovian semigroup:

δ bt (δ bs (f )) (x) = δ bs+t (f )(x), ε bt (ε bs (f )) (x) = ε bs+t (f )(x).
In addition, the semigroups operators δ bt (f ) and ε bt (f ) are continuous in t.

We note that these properties are a natural generalization of some well known properties of morphological scale-spaces [START_REF] Jackway | Scale-Space Properties of the Multiscale Morphological Dilation-Erosion[END_REF][START_REF] Maragos | Dierential morphology and image processing[END_REF].

Multi-scale openings and closings and other derived operators

From these two basic well formalized operators, all the morphological ltering theory is generalized to images on (X, d, m). In particular, as a consequence of the classic theory of morphological operators [START_REF] Serra | Image Analysis and Mathematical Morphology[END_REF][START_REF] Heijmans | Morphological image operators[END_REF], the composition products of the adjoint operators (ε bt , δ bt ) lead to the multi-scale opening and closing:

γ bt (f )(x) = δ bt • ε bt (f )(x) = sup z∈X inf y∈X {f (y) -b t (y, z) + b t (z, x)} , φ bt (f )(x) = ε bt • δ bt (f )(x) = inf z∈X sup y∈X {f (y) + b t (z, y) -b t (x, z)} .
Having the opening and closing, all the other morphological lters dened by composition of them are easily obtained, such as the alternate sequential lters.

It is also possible to extend the geodesic dilation [START_REF] Heijmans | Morphological image operators[END_REF], denoted by δ geo bt (m; f ), of a marker function m constrained by the reference function f , dened as

δ geo bt (m; f )(x) = δ bt (m)(x) ∧ f (x)
and t being a small scale. The iteration of this operator until converge is called geodesic reconstruction [START_REF] Heijmans | Morphological image operators[END_REF]. This is an extremely useful operator which, based for instance on an opening m = γ bt (f )

as marker, removes the bright objects and leaving intact the contours of dark structures.

(a) According to the theory discussed above, to compute a Maslov structuring function on a point cloud is equivalent to compute the heat kernel followed by the logarithmic trick. In the case of point clouds, like those of the Kinect camera, the initial ambient space embedding (X, d Euclid ) is not straightforwardly useful for image processing since the intrinsic geometry of the 3D objects is not taken into account by d Euclid (x, y). Local geometry of the underlaying sampled manifold could be approximated by means of manifold learning techniques (e.g., ISOMAP). That is basically computation of geodesic distances which are then plugged into the heat kernel. Another possibility involves to transform the point cloud into a triangular mesh surface. Heat kernel can be obtained by numerical computation of eigenfunctions of Laplace-Beltrami operator.

(b) (c) (d) (e)
We consider here an alternative approach based on random walk paradigm, which does not need a mesh nor an explicit graph. Hence, heat diusion can be modeled as a large collection of hot particles taking random walks on space X starting at x i ∈ X: any particle that reaches a distance point y after a small time has had little time to deviate from the shortest possible path. Probabilities of random walks of length p can be computed by looking at the power p of the transition probability matrix of the stationary Markov chain. A similar approach was used in [START_REF] Angulo | Stochastic Morphological Filtering and Bellman-Maslov Chains[END_REF], to formulate the notion of stochastic morphological ltering on Euclidean images.

In practice, the algorithm is based on the following steps.

1. Euclidean distance matrix. Given a point cloud (X, d Euclid ), compute the matrix of all Euclidean distances between pairs of points of X: D(i, j) = d Euclid (x i , x j ), ∀x i , x j ∈ X. This matrix can be made sparse by hard thresholding at ϵ (this is equivalent to so-called ϵ-graph).

2. From Euclidean distance matrix to stochastic matrix. Transform distances of each point i to the others (column i of matrix D) into transition probabilities using Boltzmann-Gibbs distribution depending on temperature T : P T (i, j) = Z -1 i e -D(i,j) 2 /T , Z i = ∑ j e -D(i,j) 2 /T . 3. Probability of going from point i to j after p steps: Compute power to p of stochastic matrix P T . Each i column of the corresponding matrix can be interpreted as the approximation to the heat kernel at point i, k Heat T (x i , x j ), on the underlying manifold when p → +∞.

Examples of morphological processing

Fig. 3 depicts some examples of morphological processing of a RGB-valued point cloud f (x i ), using the multi-scale structuring functions b t (x i , x j ) obtained by the random walk paradigm. We rst note that each color component is processed separately; but using the same structuring function for the three components.

The eects of morphological operators are as expected: erosion ε bt (f )(x i ) in Fig. 3(b) to (f ), (resp. dilation δ bt (f )(x i ) in (g)) enlarges dark structures and reduces bright ones (resp. enlarges bright zones and reduces dark areas). Opening γ bt (f )(x i ), in (h) removes bright structures at the corresponding scale t, without modifying the dark ones. The dierence between the original image f (x i ) and the opening, given in (i), shows the removed bright zones.

Considering for instance the erosion ε bt (f )(x i ) at a given scale t = 1, it is compared, on the one hand, the eect of the temperature parameter T . As usual in such probabilistic framework, for very low temperatures, e.g. T = 0.05 as in (b), the estimated heat kernel is too tightly adjusted to the local shape of the point cloud. On the contrary, with high temperatures, e.g. T = 0.4 in (c), the (a) Our estimation of the heat kernel, based on random walks, is not very precise.

(b) (c) (d) (e) (f) (g) (h) (i)
A better approximation of underlying geodesic distances can be obtained by a local tangent space approximation and local mesh construction, by introducing the measure of Voronoi volumes, etc. However for the purpose of computing the Maslov structuring function on the point cloud, the random walk estimation can be considered as satisfactory.

In metric measure spaces, the heat kernel can be also approximated using Laplace-Beltrami eigenfunctions. For the need of morphological operators, the equivalent paradigm involves harmonic functions in max-plus algebra and eigenfunctions of Hopf-Lax-Oleinik semigroups. The theory is known in the Euclidean case (also some results in graph optimization) [START_REF] Sobolevskii | Aubry-Mather theory and idempotent eigenfunctions of Bellman operator[END_REF][START_REF] Akian | The Max-Plus Martin boundary[END_REF]. This point will be explored in ongoing research.

Fig. 1 .

 1 Fig. 1. Example of RGB-valued point cloud from a Kinect image: (a) f color (x), (b)f depth (x), (c) f (xi), (d) zoom-in of (c).

Fig. 1

 1 depicts an example of RGB-valued point cloud obtained from a Kinect image, based on the 3D reconstruction of the scene according to the f @AFJD (x) image.

  the context of mathematical morphology, correspond respectively to the multi-scale dilation δ bt (f )(x) = D t f (x) and erosion ε bt (f )(x) = E t f (x) of function f by structuring function b t . By the way, we note that by the symmetry, one has b t (x, y) = b t (y, x). In max-plus mathematics on Hilbert spaces, dilation and erosion are known as the Hopf-Lax-Oleinik semigroups. By considering for instance the sub-gaussian estimate as Maslov structuring function (3), the obtained semigroups in (X, d, m):

Fig. 2 .

 2 Fig. 2. Heat kernel at a given point: (a) RGB-valued point cloud f (xi) of N points; (b) matrix of all-pairs of Euclidean distances d Euclid (xi, xj), 1 ≤ i ≤ N , 1 ≤ j ≤ N ; (c) matrix of all-pairs of heat distances d Heat T (xi, xj), 1 ≤ i ≤ N , 1 ≤ j ≤ N ; (d) in red, a given point x k , (e) heat kernel pt(x k , xi) at this point x k .

4 .

 4 Fig.2illustrates the approach for a RGB-valued point cloud image f (x i ) of N points. Including the shape of the heat kernel at a given point. Structure of the matrix of Euclidean distances with the respect to that of the heat distances,obtained as d Heat T (x i , x j ) = √ -2T log k Heat T (x i , x j ), are compared too. We note that this distance can be considered as an approximation based on the wellknown Varadhan's formula which link the short time heat kernel to the geodesic distance, i.e.,

Fig. 3 .Fig. 4 .

 34 Fig. 3. Morphological processing a RGB-valued point cloud: (a) original image f (xi). Erosion ε b t (f )(xi) with (b) t = 1, T = 0.05, (c) t = 1, T = 0.4, (d) t = 0.5, T = 0.1, (e) t = 1, T = 0.1, (f) t = 2, T = 0.1. (g) Dilation δ b t (f )(xi) with t = 1, T = 0.1. (h) opening γ bt (f )(xi) and (i) top-hat transform f (xi) -γ bt (f )(xi) with t = 1, T = 0.1.

  structuring function. A family {b t } t>0 of Maslov idempotent measurable functions b t : X × X → R is called for us a Maslov structuring function in (X, d, m) if the following conditions are satised ∀x, y ∈ X and all t, s > 0

	Nonpositivity and total mass inequality:


Eloïse Libourel
email: eloise.libourel@enpc.frnomenclature

The Mediterranean railway corridor: territorialising multiple visions of a macro-regional axis

Keywords: Corridor, Mediterranean railway corridor, Spain, Territorialisation, Trans-European Networks of Transport, Transport geography

The Mediterranean railway corridor is a project of a macro-regional trans-European axis of transport. In its Spanish part, it goes through Mediterranean regions which have historical, demographical, economic and even political common structures. But they are also marked by important gaps and a double lack of economic and political stakeholders' cohesion in a competitive context and of railway infrastructure. In this context, the MRC appears as a common goal and a catalyst: it creates new forms of cooperation between the actors and promotes a new image of the Mediterranean corridor as a possible coherent territory, shared by all the coastal regions. The transport corridor project, even if it is not yet realised as an infrastructure, already works as a strong factor of territorialisation along the Spanish Mediterranean axis. A special form of longitudinal and polymorphic territory, fruit of a complex system of particular and diverging interests seems to be emerging in these regions as a specific "corridor territory".

Résumé

Le corridor ferroviaire méditerranéen (MRC) est un projet d'axe de transport transeuropéen d'échelle macrorégionale. Dans sa partie espagnole, il traverse des régions méditerranéennes dotées de structures historiques, démographiques, économiques voire politiques communes, mais aussi marquées par d'importantes fractures et par un double manque de cohésion entre les acteurs politiques et économiques dans un contexte de concurrence et d'infrastructures ferroviaires. Dans ce contexte, le MRC apparaît comme un objectif commun et comme un catalyseur : il aboutit à la création de nouvelles coopérations entre acteurs et promeut une image de l'axe méditerranéen, possible territoire cohérent partagé par toutes les régions littorales. Ce projet de corridor de transport, si l'infrastructure n'est pas encore réalisée, fonctionne déjà comme un puissant facteur de territorialisation de l'axe méditerranéen espagnol. Un territoire singulier, longitudinal et polymorphe, fruit d'intérêts particuliers divergents, semble émerger dans ces régions comme « territoire de corridor » spécifique.

Mots-clés :

Corridor ; Corridor ferroviaire méditerranéen ; Espagne ; Géographie des transports ; Réseau transeuropéen de transport ; Territorialisation

Introduction

The Mediterranean railway corridor is one of the Trans-European Network of Transport projects held by the European Commission to improve interconnection and interoperability throughout the continent for freight and passengers. It is presented as a priority project and as a part of the core network since the 1990's in official documents (Ministerio de Fomento, 1993;European Commission, 1996;European Commission, 2011;Ministerio de Fomento, 2012). The MRC is a great axis from Algeciras to Ukraine in the very last version up to the EU documents. In Spain, it is mainly both a freight and a high-speed passenger axis linking the main ports and cities to France and the rest of Europe through a new, interoperable, international gauge set of tracks. The discussion about the length and modalities of its implementation is very pregnant: the cost of brand new tracks is very high, but the improvements of the actual tracks can't be a long-term option because of their low level of service. The first part should be implemented from the French border to Valencia before the end of 2015. This Spanish part of the trans-European corridor is set in a particular political and economic context. The project of linking the main Mediterranean cities and regions with a railway axis seems to be a response to the lack of powerful infrastructure in these regions and an opportunity to connect with the rest of Europe. But even if the project is present in all the infrastructure master plans of the government since the 1990's, it stands against the historical politics of radial axis centred on Madrid [START_REF] Comín Comín | años de historia de los ferrocarriles españoles[END_REF][START_REF] Bel | España, capital París. Origen y apoteosis del Estado radial: del Madrid sede cortesana a la « capital total[END_REF]. Today, it also appears as an expensive project in an economic crisis context particularly strong in the Mediterranean regions [START_REF] Baron-Yellès | L'Espagne aujourd'hui : de la prospérité à la crise[END_REF][START_REF] Chastagnaret | Banques, caisses d'épargne, immobilier et politiques publiques Une lecture historienne de la crise financière espagnole : l'Espagne toujours diferente ?[END_REF].

In Spain, the notion of Mediterranean corridor has a double signification. On the one hand, it obviously refers to the macro-regional European railway infrastructure project. On the other hand, it also refers to a complex and poorly defined Mediterranean space spread along the Mediterranean Sea, from the French border to an undefined point between Valencia and Algeciras, frequently Alicante or Murcia. This littoral axis seems to have some symbolical coherence in the minds, but it is difficult to call it a "territory" because of its lack of unity or common identity. Territory indeed refers to a clearly defined spatial structure with a coherent identity or opportunity of identification, possibly a political coherence, and a collective appropriation.

In this context, one question has to be asked: how a new macro-regional railway infrastructure as the MRC project can lead to the creation or the renewal of a relevant territory specific to this corridor. The aim of this paper is to show how the concept of corridor, as it is defined in several fields including economy and transport, can be useful in geography to describe a process of territorialisation of a determined spatial field. It therefore proposes the concept of "corridor territory" as the consequence of the acts of the different stakeholders taking part to the progressive implementation or the MRC macro-project.

The qualitative methodology used in this research to establish the different actors' point of views will be first presented. A literature review of the concept of "corridor" and of the link between railway infrastructures and territory will then be set to highlight the existing gap between the multiplication of the use of "corridor" in planning and its very few connection to territorial aspects. The actors' multiple visions of the MRC and of its territorial impacts will then be presented as a result of a series of interviews. This will lead to the main findings of the paper as far as the territorialisation of this macro-regional axis is concerned.

Methodology: a qualitative approach of the MRC

In order to demonstrate how the MRC project leads to particular processes of territorialisation through the visions and acts of different stakeholders, the paper will be mainly based on a qualitative methodology.

A literature review on the topics of the "corridors" and of the link between railway and territory will first be step. It will allow questioning two main aspects. First, the concept of corridor was used in relation with territory in very few cases and it is now used mainly in a conceptual way paradoxically in planning documents. Second, the link between railway and territory is present in a large literature and it should help us making a relation between the concept of corridor and its plural forms of spatial consequences in the MRC case study.

In order to approach the specific territorialisation process along the MRC project, this research is based on the methodology of interviews with relevant actors. A first step is the identification of the stakeholders who take part in the debate and the actions around the Mediterranean corridor: the European Commission is supposed to choose the priority projects up to the proposals of each member state of the EU; in Spain, the great infrastructures are a national competence and are implemented by ADIF, the infrastructure manager; the Autonomous Communities have a set of competences; the economic sector and its lobbies also claim for the MRC; academic personalities also take part into the debate. The identification of the actors was made by crossing two sources: the institutional competences are a good way to identify part of them; the observation of the events and actuations around the MRC through the media and the networking allow to complete the list. The main point of the methodology is to collect their opinion through interviews. Since 2011, 43 interviews were realised, mainly in the late 2013, in several Spanish cities, in Brussels and in Paris. During the interviews, a series of questions about the MRC were asked, among which the personal definition given by each actor, his position regarding it and the concrete actuations planned or implemented in relation to the MRC. The question of the meaning of "corridor" was also asked. The public declarations of the stakeholders in the media or in institutional documents were also analysed as a complementary source. Thanks to the answers, we have been able to make a typology of these actors according to the analysis of their different visions, confronted with the concrete actions initiated on the Mediterranean axis.

From corridor to territory: a missing link in the state-of-the art

The Mediterranean railway corridor refers to the notion of "corridor" as it is used in transportation field. The term is very often part of the institutional discourse to define an important and large scale transport line (European Commission, 2011), having or not a territorial reference. But literature shows that the great railway infrastructure may have important effects on territory and therefore can't be considered independently.

"Corridor": a complex multidimensional concept

The concept of "corridor" emerged in the 1960's in the United States in the field of spatial analysis and network analysis (Haggett & Chorley, 1969;[START_REF] Kansky | Structure of Transportation Networks: Relationship between Network Geometry and Regional Characteristics[END_REF]. At this time, it was quite open to the notions of space, territorial organisation and geography in a rather descriptive way. It was particularly used to describe urban processes linked to the intensification of transit along structuring transportation axis, as C.F.J. [START_REF] Whebell | Corridors: a Theory of Urban Systems[END_REF] and B.E. [START_REF] Prentice | Transport Gateways and Trade Corridors[END_REF] could show in North-American case studies. In its first acceptation, the notion of corridor was directly linked to the transport infrastructures or at least to the transportation axis perceived through flows of transit and mobility. But in this sense, it was not necessarily connected to the idea of a territory as a coherent and continuous space or to the idea of territorial organisation and planning at any scale. Nonetheless, the idea of a corridor as a continuum of space with particular characteristics emerged in the 1970's and is still used nowadays in ecology. In the 1990's, the notion of "corridor", in the transport approach, moved to the field of economy. It was used in multi-criteria economic analysis and supply and demand models.

European case studies mainly appear in the 2000's in institutional reports of the European Commission and in national scientific productions. The Spanish case is original because of an older use of the term "corridor" to define great transport axis, particularly for the Mediterranean regions [START_REF] Boira | Euram 2010: La via europea[END_REF]Centre for Innovation in Transport, 2012). In the European institutional documents as the TEN-T (European Commission, 2011), the term "corridor" refers to the great trans-national transport axis projects. It seems to be used in a prospective and political vision, as a language instrument to give a reality to these projects. Thus, European "corridors" are not infrastructuresor they are a set of discontinuous pieces of different infrastructures to be linked together.

Neither are they territories because they don't take into account the identity or possible unity of the spaces they go through. This technical use of the notion of corridor leads to an abstraction of the concept.

Indivisible concepts: railway infrastructure and territory

The idea of a strong link between transport and territory is historically established by scientific literature as reminded by F. [START_REF] Plassard | Les effets des infrastructures de transport, modèles et paradigmes[END_REF]. Railway especially appears as an instrument for planning and territorial unification since the beginnings in the 19 th Century. In Europe the train was used as a territorial coherence tool [START_REF] Troin | Rail et aménagement du territoire[END_REF]Offner & Pumain, 1996) and some authors even talk about railway as "architecture of the territory" [START_REF] Grillet-Aubert | Recherche : état des lieux et perspectives[END_REF]. The same assessment is made at a regional level [START_REF] Button | Transport and Industrial and Commercial Location[END_REF][START_REF] Banister | Quantification of the non transport benefits resulting from rail investment[END_REF]). The regional or local studies of the impact of railway infrastructure on the territory usually give a prominent role to the actors and the analysis of their position and actuations [START_REF] Wolfram | Planning the Integration of the High-speed Train[END_REF]. In the whole literature, the link between railway and territory is established, even if [START_REF] Offner | Les effets structurants du transport : mythe politique, mystification scientifique[END_REF] demonstrates that there is no "structural effect" of the infrastructure, which leads to the idea that its relationship to the territory is quite complex.

A missing link with territory

The term "corridor" as used by the European institutions and countries in their infrastructure programs shows that territory is very rarely present in the debate at this macro-regional scale. This is a paradox when all the transport policies are most often oriented towards a better cohesion or equity of the (national) territory. It is as if the step to European level made it difficult for the states to consider a possibly coherent territory which would not fit with their borders and would oblige them to consider new forms of territoriality.

The notion of "corridor", as it appears in literature, seems to have slipped from spatial to economic analysis [START_REF] Hall | Cities, Regions and Flows[END_REF]. Nonetheless, the reactivation of the concept of transportation corridor in the context of the TEN-T projects could paradoxically bring it back to geography. De facto, the trans-European networks need to take root in the specific territories in which they are implemented. Therefore, the implementation phase of European projects, which can be quite long, can appear as a privileged phase of territorialisation.

Actors, territory and the MRC: from a plural space to new territorial cooperations

The trans-European projects, as they plan long term and long distance infrastructures, go through different territorial entities, as states, regions or cities, but also more fuzzy forms of territoriality. The Spanish Mediterranean regions have strong identities and traditions at political, economic, social, demographic and institutional levels. It makes it a plural space marked by common Mediterranean affiliations, but also strong divisions in which the different actors are moving and changing their relationships; as it is noticeable through the interviews.

The Mediterranean as a plural space and a coherent axis?

During the interviews, the historical dimension of the MRC, first claimed by orange producers in the 1920's, and of the Mediterranean regions was very often evoked as follows. Historically, the Mediterranean Sea has a common heritage and shares a common culture. Nonetheless, this Mediterranean cohesion was slightly modified with time. The constitution of national States connected politically the Mediterranean regions with the other components of each state and therefore disconnected them from each other [START_REF] Béthemont | Géographie de la Méditerranée : du mythe unitaire à l'espace fragment[END_REF]. Recently the Mediterranean heritage was rediscovered through regional cooperation in a European frame (Institut de la Méditerranée, 2004) -Interreg, Union for the Mediterranean, etc. In Spain, demographic and economic indicators seem to confirm the coherence of the Mediterranean space from a geographical point of view. Apart the region of Madrid, the major growth rates of demography and of economic indicators in the last 15 years are concentrated along the Mediterranean coast (up to the national institute of statistics). This dynamism is reinforced by the presence in the Spanish Mediterranean regions of the main ports (Valencia, Barcelona and Algeciras) and economic sectors, such as automotive industry, ceramics and textile. These exporting companies need an improvement of the railway axis to Europe, but the economic dynamism goes hand in hand with a strong competition between ports (Valencia is very proud to overtake Barcelona in TEUs) and industries (Ford in Valencia and Seat in Barcelona on the European market). From a political point of view, the situation seems to be very propitious to a Mediterranean good understanding, at least for Catalonia and Valencia Community, both governed by a conservative majority. But historically, a strong rivalry has been opposing both regions and it is still very active. One of the elements building a reputation of unity for the Mediterranean axis is the dominant political philosophy in Spain. It is indeed set on the idea of a cultural and institutional domination of Castile and the region of Madrid [START_REF] Loyer | Géopolitique de l'Espagne[END_REF][START_REF] Romero | España inacabada[END_REF] based on ideas from literary and philosophical works.

In the field of transports, the construction of the route and rail networks are quite good indicators. As Germà Bel shows in his book España, capital París (2010) transportation networks were always a way to create and reinforce the nodal position of Madrid and at the same time to build an artificial unity of the national territory. As underlined by all the regional scale actors, this political choice lead to the construction of a radial highway and railway network, centred on Madrid. Its consequence is a lack of coastal infrastructure in the Mediterranean regions which were the poor relation of the infrastructure politics. Nonetheless, if we look at the dominant flows, it is noticeable that the Mediterranean corridor is the most important of the country [START_REF] Monzón | Evaluación de los Impactos Territoriales y Económicos de Planes de Infraestructuras de Transporte[END_REF]. In this context, the MRC seems to make sense, and we can assume that it is coherent with an existing territory, at least from transportation flows' point of view. Nonetheless, even though the MRC project is supported as a European priority, the Spanish central government doesn't make it a real priority as far as funding and construction are concerned but one project among others.

Nonetheless, as José Ortega y Gasset underlines in his book España invertebrada (1921), the whole country lacks of great "vertebrating" axis. It is especially true when considering the Mediterranean axis from railway point of view. Two high-speed lines, Madrid-Levante (with a branch to Valencia and further to Castellón and a branch to Alicante) and Madrid-Barcelona don't take into account the 200km missing between Castellón and Tarragona to complete the network, which by now has medium speed lines and 40km of one-track line. This absence of continuity along the Mediterranean axis is in a way based on very deeply anchored in the psychology and ideas of institutional actors -as shown by the interviews-but also of everyone.

Almost all the actors agree that the Mediterranean axis is fragmented and appears as a quite rejected territory at a national level. But a corridor exists de facto, underlined by the importance of freight traffic, passenger demand and tourism. This gap between the political philosophy and the common ideas about Mediterranean space as a non-coherent and non-appropriated region and the existence of real transport demand and needs appears as a paradox. The existing functional disjunctions seem to disappear in transport field.

Multiple and diverging stakeholders' visions for a single objective: the MRC as a territorial catalyst

If the different interviewed actors seem to agree on the MRC as a necessity for the coastal regions as well as for the national economy, this consensus is based on rather divergent visions and conceptions of the corridor. It was often very difficult for the interviewed actors to express in a direct way an opinion about the question of the territorialisation, because they are not familiar with the concept and don't take this particular issue into account in their actuation. Nonetheless, academic actors (some of them being geographers) and some of the lobbyists are very concerned about territory [START_REF] Boira | L'Eix Mediterrani: entre les dinàmiques locals i la perspectiva megaregional[END_REF]. Political actors also refer very often to the idea of "territory", meaning an institutional perimeter in which they are elected or have a competence in terms of planning. Therefore, their vision of the territory mainly depends on the political scale they represent.

Even if the question of the territorialisation of the MRC is not directly present in the content of interviews, the answers of the stakeholders can be analysed and their vision of the political and economic relationships along the Mediterranean axis and of the new complementarity or cooperation to be created allow us to have an idea of their vision of what could be the territory of the Mediterranean corridor.

A typology can be set on the criteria of the definition of the MRC given by the actors, their more or less spatially integrated vision and their concrete territorialised proposals. Table 1 (below), by comparing stakeholders' priorities and visions of the MRC, leads to several findings. First, the impact of the main objectives of each actor on his vision of a corridor territory is obvious. National or regional objectives as those of the Central Government or of the Autonomous Governments lead to an addition of institutional perimeters, whereas a European integration perspective or the local development objectives mainly lead to non-institutional perimeters. The latter are mainly based on functional considerations. Second, the question of the extension of the MRC, in its Spanish part, is variable depending on a more or less extensive vision of it. European documents and [START_REF] Ferrmed | Ferrmed Global Study. Ferrmed Great Axis Rail Freight Network and its area of influence, Scandinavia-Rhein-Rhone-Western Mediterranean. Feasibility, conclusions and recommendations[END_REF] and EU Core Net Cities informs promote a trans-continental axis which would be the link between Africa and Northern Europe: in this case, all the Spanish Mediterranean regions (Andalusia, Murcia, Valencia and Catalonia) would be part of it. But the economic field has a more limited vision, taking into account Catalonia, Valencia and most of the time Murcia, and sometimes a little part of Andalusia till Málaga. But regional political actors even exclude Andalusia. This region has a topographical and a political gap which makes it more likely to be connected to Seville-Madrid axis than to the Mediterranean corridor. Third, there is a huge domination of a freight railway axis in the claiming of the different stakeholders, whereas a passenger axis is mainly a demand of the cities. A freight axis would be economically profitable and gathers all economical actors, whereas a passenger high-speed rail axis would cost much more than its social profitability would make it acceptable [START_REF] Bel | España, capital París. Origen y apoteosis del Estado radial: del Madrid sede cortesana a la « capital total[END_REF]. But the idea of a high-speed passenger corridor is present in the discourse of the stakeholders as an attractiveness element and as an argument to convince the inhabitants of the opportunity of the corridor. The consecutive territorial vision is ambiguous: one argument in favour of HSR is to improve the accessibility of the whole coast for tourism, giving it a homogeneous sense. However a HSR service would only stop in the main cities and maybe improve their metropolitan status, but it would reinforce the fragmentation of the Mediterranean space by increasing the gap between the main cities and the secondary ones.

Even if it's possible to make a difference between economic, political and academic stakeholders according to their priorities and to their visions of what could be a Mediterranean corridor territory, and despite their divergent objectives, they all are involved in the MRC project and cooperate into some debate and proposition task forces, such as IIVEE-Euram, an institute which promotes a Mediterranean Euro-region [START_REF] Boira | Euram 2010: La via europea[END_REF]. Despite their different objectives and positions about the modalities of the project, all these stakeholders are part of a great pro-MRC cluster lobby. This scheme shows how actors coming from diverse origins with quite unlike cultures and knowledge participate to a common effort in favour of the Mediterranean corridor. It seems impossible to draw one coherent vision of the MRC up to the results of the analysis of stakeholders' priorities, but there is still an active effect on the Mediterranean macro-region as if all stakeholders were moving towards a common objective through different ways. This movement towards the MRC implementation allows to define the notion of "corridor territory".

Main findings: towards the creation of a "corridor territory"

The Spanish Mediterranean regions have some homogeneity according to indicators such as demography, political representation or economy at a small scale. But if we look at a more precise scale, it is fragmented and it seems difficult today to talk about a territory as an appropriated and identified coherent portion or space organised with common interests and cooperation. Nonetheless, the actions of the different stakeholders in the MRC project renew the idea of a relevant corridor territory through political, economic and social (circumstance) cooperation, and through a qualitative jump from regional straight to European scale. The analysis of the main findings lead to the justification of the concept of "corridor territory".

Cooperation and the regional jump towards European scale

The MRC project as a part of the TEN-T projects introduces a major change in the way stakeholders act. Indeed, the Spanish Autonomous Communities are directly represented in the EU institutions through the Committee of the Regions. The principle of subsidiarity, in use in European institutions, implies that the decisions should be taken at the closest practical level for the citizen [START_REF] Kersbergen | The politics of subsidiarity in the European Union[END_REF]. Therefore the regions have increasing weight and credibility even though they are theoretically not competent for infrastructures in Spain.

As the MRC doesn't reach a full consensus in Spain, and principally in Madrid which would be put aside, the Mediterranean regions had an important mobilisation at the European Commission. That was the starting point of a dialog between Catalonia and Valencia, two neighbour regions which were competing with each other and had no project in common. This cooperation started in the 1990's when the MRC was first included in the TEN-T but was particularly reinforced in the last years. It is interesting to see that the Secretaries of Transport of both regions are now present together at the same meetings, as at the meeting organised by IIVEE in February 2013 in Peñíscola, at mid-distance from Barcelona and Valencia. The IIVEE is represented in both regions.

The creation in June 2013 of the EU Core Net Cities new lobby of the cities of the European core network was also an interesting event. It showed the increasing political cooperation along the Mediterranean axis, but also the remaining gaps. The initiative was due to Ferrmed. A large number of Spanish Mediterranean cities were represented (17 out of 27 European cities) including small ones, which confirms the existence of an increasing continuity and a common interest of the Mediterranean actors for MRC. But the city of Valenciathird Spanish and second Mediterranean metropoliswas not represented. The political cooperation of the Mediterranean corridor is still incomplete, but it is under construction and is reaching a very interesting level: institutions at all scales and of all sizes feel concerned by a single project.

But the political cooperation seems to have come for institutional reasons -that is to say to support the MRC project at the European Commission, whereas the economic actors began their coming together earlier and on much more rational basis. The opportunity offered by the MRC for all the economic sectors and for the Mediterranean harbours is very important (Spain has an Iberian gauge railway network which cannot connect directly with European gauge tracks in France and the rest of the continent and which creates a loss of time and additional costs). The Chambers of Commerce and Industry of the different Autonomous Communities began to cooperate around the MRC project. They were influenced by a set of ideas from economists such as Germà [START_REF] Bel | España, capital París. Origen y apoteosis del Estado radial: del Madrid sede cortesana a la « capital total[END_REF] and by academic and economic institutes such as IIVEE.

In 2004, Ferrmed was officially created and installed as a lobby at the European Commission. It emerged from the first attempts of economic cooperation between the Mediterranean regions and from the conviction that a strengthened cooperation was necessary to support the MRC as a part of a wider network. Ferrmed proposes a "Great Axis" from Casablanca to Saint-Petersburg. Nonetheless, the Spanish part of this axis is in the centre of the lobby's action: it was created and is managed by Joan Amorós, a Catalan, and a huge proportion of its members is Spanish. Ferrmed, as well as IIVEE and some other structures (Business Eumed, EU Core Net Cities, etc.) are now having a role of interface between the different stakeholders. They are the structures in which economic and political actors can meet as well as civil society members. Therefore, they are the sign that a real territorial cooperation is emerging in the Spanish Mediterranean corridor.

A corridor territory made of a set of actuations and symbols

All the stakeholders which strongly support the MRC have their own interests and therefore project in the Mediterranean corridor their representations. It contributes to the constitution of a territory as a symbolically appropriated space.

A set of images and representations of the Mediterranean corridor exist. They are mainly maps from the lobbies, the Government and the research institutes, but also pieces of advertising from the municipalities (Gandía created the image of an AVE train arriving on the beach) or the private actors (like Pro-AVE business association) and caricatures published in the media or on the internet. These representations are very diverse, but they all promote the idea of a coherent Mediterranean axis through images that people can keep in mind. As an example, Ferrmed map of the Greater axis [START_REF] Ferrmed | Ferrmed Global Study. Ferrmed Great Axis Rail Freight Network and its area of influence, Scandinavia-Rhein-Rhone-Western Mediterranean. Feasibility, conclusions and recommendations[END_REF] represents a project as it was officially proposed to the European Commission together with impact and feasibility studies: it actually is an executive prospective map. Nevertheless, it is also an image of a structuring European corridor based on intermediate cities, which are the majority of the lobby. Thus, this map is both a proposal for European networks and the symptom of a synthesis of different stakeholders' interests. It is very well-known and present in mind of all the stakeholders. Business Eumed, a lobby of commerce and industry companies, presents a map of the "Mediterranean orbital train" which, according to an interview with the director [START_REF] Libourel | L'insertion de Valence dans le corridor ferroviaire méditerranéen. Stratégies multiscalaires et aménagement urbain[END_REF], is mainly a representation to consider and keep in mind, and not a real executive project. The idea is to create a vision of a fuzzy territoriality around the Mediterranean to strengthen the cooperation between the Mediterranean companies for commerce and industry. The profusion of these representations (a large set of symbols could be added) makes the Mediterranean corridor exist as a coherent territory despite its multiple definitions and perceptions. A real "corridor territory" structured by the infrastructure is progressively emerging from the MRC project.

At the same time, the events organised by some of the actors (especially the lobbies) joining together a large set of stakeholders at different levels and from different territories contribute to the unification of the Mediterranean corridor as a relevant frame for political and economic action. Sectorial groups are emerging as representatives of the interests of the Mediterranean corridor as a coherent territory, such as Cierval, the business association of the industrials of all the Mediterranean regions from Catalonia to Murcia and some provinces of Andalusia. The implementation of a provisional "third rail" to adapt the existing tracks to European gauge is also a factor of unification of the interests and of cohesion of the territory because it allows the whole corridor to access Europe.

At a local level, numerous projects of urban infrastructure (high-speed rail stations, urban renewal around the stations, logistic parks…) claim their affiliation to the MRC and use it as a justification. There is indeed a huge expectation based on the implementation of the railway corridor: it is supposed to increase the dynamism of all the Mediterranean axis. Even if the projects presented at a local scale are not directly in relation with the MRC, the trans-European project appears as a way to justify investment at a local scale and to create a global vision of the Mediterranean coast as a dynamic territory in Europe.

Thus, in a moment of economic crisis which is particularly strong in the Mediterranean regions and in which the Catalan exception is particularly strong also, the MRC project acts as a catalyst. It renews a Mediterranean coherent perception and makes the idea of a relevant "corridor territory", appropriated by the stakeholders and by the people, possible. The MRC is used as an instrument for diverse aims, but it still is a territorial entity more and more identified and perceived as a coherent bloc in Spain and Europe.

Conclusion

The Mediterranean railway corridor is an infrastructure project at a macro-regional scale, and even at a transcontinental level. In European terms, it is a technical or even a technocratic vision of a more or less ideal link between several European regions. But in Spain, where the Mediterranean regions lack of powerful railway infrastructures but have a strong freight and passenger demand, it takes a particular dimension.

An analysis of the geographical skills of the Mediterranean axis showed certain coherence in the demographic, economic and political indicators at a macro-regional level, even if an analysis of the situation at a more detailed scale shows some important gaps. But according to the series of interviews made with the different stakeholders of the MRC and to their actuations, these gaps are being passed over by new political, economic, social cooperation which make the Mediterranean corridor exist in the facts and discourses, even if it still doesn't exist as an infrastructure.

The political revival of the concept of corridor leads to the positive appreciation of a latent territoriality through its symbolical appropriation. The Mediterranean regions are not yet a complete "corridor territory" as it is not fully appropriated and organised in a coherent way. But we can assume it is positively under construction thanks to the mobilisation and conjunction of interests produced by the Mediterranean railway corridor project. This territorialisation process is visible through the discourses of the different stakeholders, but also through positive acts of economic and political cooperation. Of course, this does not mean that competition is now over, but it means that the stakeholders are aware of the need to give body to a territorial structure in order to make the MRC essential.

This Spanish Mediterranean corridor case study leads geography to integrate the notion of territory to the notion of corridor when it takes the form of a territorial phenomenon.

Table 1 .

 1 Classification of stakeholders according to their vision of the Mediterranean corridor.

	Type of territorial	Actor	Main objectives	Concrete projects
	vision of the MRC
	Integrated / macro-	European	Promoting European networks	-One end of a great TEN-T
	regional vision	Commission	interconnection and interoperation	-Railway connexion to North
				Africa
		Spanish government	-Completing HSR network	-From French border to Algeciras
		(Grupo Fomento :	-Improving the connexion to	as a maximalist hypothesis
		ADIF / RENFE)	Europe	-to Murcia a mid-term
			-Improving competitiveness
		Mediterranean	-Euro-regional integration,	Integrating Catalonia, Valencia and
		regions (Catalonia	-Connexion of main Spanish	Murcia regions
		and Valencia	harbours,
		mainly)	-South-north axis avoiding Madrid
		Academics	-Promoting a Mediterranean great	Catalonia, Valencia and Murcia in a
			region, with particular cohesion	Euro-regional perspective
			and identity,
			-Reinforce the economical weight
			of the Mediterranean
		Lobbies (Ferrmed,	-Connecting industries of the	-Piece of a project from Casablanca
		Business Eumed,	Mediterranean regions,	to Saint-Petersburg,
		AVE, IIVEE)	-Opening Spanish coast to Europe	-Along the Mediterranean
	Sectorial / cluster	CCI	Improving the connexion of	-Connect the main cities to Europe
	vision		Spanish industries to Europe	-International clusters
		Port authorities	Become a gateway for Europe	Connect ports to European market
	Isolated / localist	Cities	-Reach a metropolitan status in	-AVE stations, urban renewal
	vision		Europe	-Events and communication about
				the MRC

ADIF Administrador de Infraestructuras Ferroviarias (Spanish nfrastructure manager) AVE Alta Velocidad Española (Spanish high-speed)