Gilles Simon

In-Situ 3D Sketching Using a Video Camera as an Interaction and Tracking Device

Keywords: Descriptors (according to ACM CCS): Artificial Intelligence [I, 2, 10]: Vision and Scene Understanding-Modeling and recovery of physical attributes Computer Graphics [I, 3, 6]: Methodology and Techniques-Interaction techniques

In this paper, we present a novel method for in-situ 3D sketching of polyhedral scenes. A video camera is used as both an interaction and a tracking device, which makes the system particularly suitable for handheld devices such as PDAs and mobile phones. The efficiency and accuracy of the method are demonstrated using a miniature scene and a real outdoor scene.

Introduction

Acquiring the 3D geometry of arbitrary scenes has been a primary objective of both the computer vision and graphics communities for many decades. Applications are numerous in various domains such as construction, geographic information systems, robotics and augmented reality (AR). Existing modeling methods usually rely on two separate stages: first some data about the scene (photographs, videos, physical measurements, laser scanners, ...) are acquired onsite. Then these data are treated off-line using some specific manipulations and algorithms (e.g. [HDT * 07]). Unfortunately, this process has two drawbacks: first, some of the involved tasks (data acquisition and/or geometry extraction from these data) can be very tedious. Secondly, it is not guaranteed that the desired structure is fully extractable from the acquired data and additional acquisitions are sometimes required in order to supplement the missing parts. In this paper, we propose to bridge the gap between data acquisition and their exploitation. We describe a purely image-based system for in-situ 3D sketching of polyhedral scenes. The built parts of the scene are immediately shown superimposed on the environment, which allows the user to verify the geometry against the physical world in real-time.

In-situ modeling has been introduced in [START_REF] Piekarski | Tinmith-Metro: New Outdoor Techniques for Creating City Models with an Augmented Reality Wearable Computer[END_REF] under the name of "construction at a distance". The principle is to allow mobile AR users to capture planar shaped objects from the physical world using simple 3D primitives and constructive solid geometry. The user interacts with the computer using a set of pinch gloves and hand tracking and the camera is tracked using an inertial sensor and a GPS; this system thus requires several special equipments and devices and is only suitable for outdoor use. By contrast, a purely image-based interactive model building system has recently been proposed in [START_REF] Bunnun | OutlinAR: an assisted interactive model building system with reduced computational effort[END_REF]. This work has several common aspects with our work, such as the use of a cameramouse and model-based tracking. However, the modeling task is less constrained in our system in that it does not require any initial template to be placed in the scene, nor that the current model partly stays in the camera field of view while new structures are added. Moreover, defining a vertex in [START_REF] Bunnun | OutlinAR: an assisted interactive model building system with reduced computational effort[END_REF] is based on parallax motion which is not well appropriate for modeling objects far from the camera. Finally, our system can be seen as an immersive version of the widely used 3D drawing software Google SketchUp TM (http://sketchup.google.com), whose principles are described e.g. in [START_REF] Stuerzlinger | Comparing SESAME and Sketching on Paper for Conceptual 3D Design[END_REF]. This tool combines some of the features of pencil-and-paper sketching and some of the features of CAD systems to provide a lightweight, gesturebased interface for 3D polyhedral modeling. In the latest releases of SketchUp, the user is able to align the world axes to match a photo perspective. With this done, he can create models using the photo as a direct reference; mouse strokes are converted into 3D-space using inverse ray intersections with the previously defined geometry (or the ground plane by default). All these principles have been taken up in our implementation, but with the important difference that we consider dynamic video images instead of static ones. Moreover, the video camera is used as the interaction device instead of a mouse, which makes our system particularly suitable for mobile devices such as PDAs, wearable computers and mobile phones.

The system alternates between two modes of operations (Fig. 1):

Modeling mode. Pure rotations are applied to the camera, e.g. when using a head-mounted display, this amounts to look around the scene by only turning the head. Using the camera as an interaction device, the user is able both to calibrate the camera and to define the scene geometry (section 2). It must be noticed that this mode may be used alone, providing already interesting contributions to classical single view metrology, as (i) by turning the head the reachable field of view is much larger than the camera's field of view and (ii) mouse manipulations are replaced by camera manipulations, which can be of great interest in a mobile context. Tracking mode. When at least one face of the scene has been described and the camera undergoes a general motion, a 6-DOF (degrees of freedom) camera tracking is performed, based on the available geometry (section 3). This allows the user to get closer to some parts of the scene or make some new faces visible before continuing modeling. In this mode, the geometry previously modeled has to stay (at least partly) visible in the camera field of view, which is not required in modeling mode. A pose recovery procedure based on SIFT features can be called upon user request or each time a tracking failure is detected by the system.

The system starts in modeling mode. Once a 3D geometry has been initialized, switches between modeling and tracking modes are done automatically, depending on the motion applied to the camera (Akaike's motion model selection is used [START_REF] Vigueras | Iterative multi-planar camera calibration: Improving stability using model selection[END_REF]).

Modeling Interactions

A key idea of this work is that user interactions can be done using the video camera itself. Indeed, let us consider that the user wants to draw a line stroke on a video image between two physical points A and B of a real-world scene; there are two possible ways to do this (Fig. 2) : (i) the camera stays fixed and a mouse is used to move a cursor to the two respective endpoints; (ii) the camera is rotated so that the physical points apparently move to a fixed cursor, for instance at the center of the image. The second solution, which is used in our implementation, does not require mouse support. However, the position of point A has to be updated from frame to frame while point B is aimed at. Fortunately, camera rotations only induce homographic deformations of the image, that can be computed easily e.g. using keypoint matches. In both cases a click has to be done when the cursor (or the camera) is correctly positioned. In the second solution, mouse clicks can be replaced by key presses or any other input controls such as buttons, voice commands, etc. In our system, only three keys are used for all operations: one to "click" or "drag and drop" in the image, a second to cancel the current operation or request a pose recovery and a third to scroll the tools menu.

Before being able to model the scene, the user has to calibrate the camera: this is done by indicating two sets of horizontal parallel lines orthogonal to each other (Fig. 3, frame a). In addition, the user can drag and drop the origin of the world frame, providing a ray of possible 3D positions of that point. The depth of the origin is chosen so that the unit up vector has an arbitrary size in the image plane. The user can also change the scale of the scene by moving the extremity of the unit up vector. Once the camera parameters are known, the user can start modeling. New faces are instantiated in contact with existing faces or the ground plane by default. Contacts are guaranteed by snapping the clicked points to existing 3D points which appear in a specific color when close to the cursor: yellow for a face vertex, blue for the middle of an edge and red for the closest point on an edge. Faces under the cursor are also specifically outlined and new 3D vertices can be generated by using inverse ray intersecc The Eurographics Association 2010. tion. In addition, most actions are guided using a line stroke, which itself can be snapped to the projected world axes.

Six different tools have been implemented which are summarized in Tab. 1. Fig. 3 illustrates how a simple house can be modeled using the Add, Extrude, Cut and Move tools: a rectangular face is first created using two line strokes AB and BC: according to the priority rules described at bottom of Tab. 1, AB is instantiated in the ground plane and parallel to the red axis (frame b); the vertex C is dragged in the plane orthogonal to AB (frame c) and finally such that BC is aligned with the blue axis (frame d); the new face is extruded forming a box and the top face of this box is subdivided so that the cut line joins the middles of two opposite edges (frame e); finally, the cut line is moved along the blue axis in order to form the roof (frame f).

Add a rectangular face by clicking three points A, B, C ( * ) . The rectangle is generated in the plane ABC using A and C as diagonally opposite vertices.

Extract texture and SIFT features from the selected face.

Extrude the selected face.

Move the selected vertex, edge or face.

Subdivide the selected face by joining two of its vertices.

Delete the selected face.

( * ) The inverse ray intersection priorities are: A: selected vertex > selected face > ground;

B: selected vertex > projected world axis AB > plane(A); C: selected vertex > projected world axis BC > plane ⊥ [AB]
Table 1: Modeling tools used in our prototype.

Camera Tracking

Camera poses are computed using image content only and the modeled planar polygons [START_REF] Vigueras | Iterative multi-planar camera calibration: Improving stability using model selection[END_REF]: Harris corners are matched from frame to frame using normalized crosscorrelation of patches in a search window; these corners are detected in the whole image for a 3-DOF tracking or inside the projected faces of the model for a 6-DOF tracking; outlier matches are rejected using a RANSAC computation of the planar homographies that best fit the sets of corners between the subsequent views; finally, the camera pose is computed by iteratively minimizing the sum of squared transfer errors (SSTE) of the inlier matches, using the homographies induced by the pose parameters and the equations of the involved planes.

This procedure is fast and robust but has two drawbacks: first, due to the recursive nature of the algorithm, errors induced by noise (e.g. in corner detection) accumulate over time. Secondly, as the size of the search window has to be bounded both to reduce matching ambiguities and to achieve real-time, camera tracking may fail in case of fast motions. To tackle the first problem, we use the texture information extracted by the user during the modeling mode. Like in [START_REF] Reitmayr | Going out: Robust tracking for outdoor augmented reality[END_REF], the current textured model is rendered from the current pose using the OpenGL graphic engine. Harris corners are detected in the rendered view and matched with the Harris corners of the current frame. These matches, which are merged with the set of matches obtained in the previous video frame, tend to reduce the drift. For performance issues, this correction procedure can be called only every k frames (k = 2 in our implementation).

Tracking failures are detected automatically by testing whether the number of inlier matches falls under a threshold. The user is also able to manually require for pose recovery when he visually estimates that it is necessary. Pose recovery is basically based on SIFT feature matching [START_REF] Lowe | Distinctive image features from scaleinvariant keypoints[END_REF]. SIFT features have a higher computational cost than Harris corners but are invariant to image scale and rotation and robust to some extend to change in viewpoint. When the "Extract texture" tool is used, SIFT features are extracted inside the projected faces of the model and stored (in a kd-tree) associated with the current pose. When the recovery procedure is called, the list of textured faces is traversed until one face gets enough matches with the SIFT features of the current frame. If one face is found, it is put at the top of the list (so that it will be evaluated first at the next tracking failure) and a rough estimate of the current pose is obtained by minimizing the SSTE of the inlier matches. This rough pose is then used to render an image of the textured model and the correction procedure is called.

The recovery procedure slightly differs in modeling mode: when the system enters this mode, an initial set of SIFT features is detected in the whole image. These features are transferred from frame to frame using the RANSACestimated homographies. The image is subdivided into a 3x3 grid; when one square of the grid is empty after the SIFT features have been transferred, new SIFT features are detected inside this square and added to the current set of SIFT features (see the outdoor video). When a tracking failure occurs, SIFT features are detected in the current image and matched with the current set of SIFT features.

Experiments

All experiments were performed using a Dell Precision M6300 laptop coupled with a simple Logitech webcam. The system runs at video rate in standard mode and 2 to 6Hz when the recovery procedure is called. Two videos are associated with this paper, showing the system in action. A miniature scene is used to assess the accuracy of the method. Several modeling and tracking sequences alternate during the working session (Fig. 3). It can be distinguished in the video between when the system is running in modeling mode and when it is running in tracking mode, as the cross cursor disappears in the second case. Fig. 4 shows the errors obtained on the recovered 3D geometry (scaled so that the recovered distance d 1 is equal to the expected distance). These errors are acceptable for many applications: for instance, to prove that the recovered model is suitable to perform AR, we added a seventh tool to our platform, enabling the user to add virtual objects under the camera cursor. The added objects appear rigidly anchored in the scene, even for a large range of distances from the camera to the scene (Fig. 3 The modeling mode is also illustrated using a real outdoor scene (an athletism stadium with a changing house). The first part of the video shows how the set of SIFT features is expanded. Several abrupt motions are done, requiring the recovery procedure to be called. The camera is then calibrated and some modeling operations are performed before a billboard is added along the track (Fig. 5).

Conclusion

In this paper, we have proved that in-situ modeling is possible using a standard webcam as only device. The system can be improved in several ways. For instance, including edge features to the tracking algorithm [START_REF] Reitmayr | Going out: Robust tracking for outdoor augmented reality[END_REF] would help to handle such scenes like the athletism stadium where little texture information appears. Localization in larger areas would also require to make the recovery procedure more scalable and efficient. Such techniques like the Potentially Visible Sets [AWK * 09] may be useful for that purpose.

Figure 1 :

 1 Figure 1: 3-DOF camera rotations (in modeling mode) alternate with 6-DOF camera motions (in tracking mode). Thick lines represent the modeled faces of the scene.

Figure 2 :

 2 Figure 2: Equivalence between moving cursor / fixed camera (left) and fixed cursor / moving camera (right).

Figure 3 :

 3 Figure 3: Snapshots of the system in use.

Figure 4 :

 4 Figure 4: Recovered geometry with actual distances A (mm), absolute errors ∆ (mm) and relative errors ∆/A (%).

Figure 5 :

 5 Figure 5: One snapshot of the outdoor sequence.

  , frames g,h).

		A(mm)	∆(mm)	∆/A(%)
	d 1 d 2 d 3 d 4 d 5 d 6 d 7 d 8 d 9	61 88 38 92 75 34 139 21 95	Ref -0.3 1.5 4.2 -5.2 -1.3 -8.4 0.8 -4.0	Ref -0.3 3.9 4.6 -6.9 -3.8 -6.0 3.8 -4.2

c The Eurographics Association 2010.


J F Sygnet
email: [sygnet@iap.fr

H Tu

B Fort
email: fort@iap.fr

R Gavazzi
email: gavazzi]@iap.fr

A search for edge-on galaxy lenses in the CFHT Legacy Survey ⋆

Keywords: Gravitational lensing, strong -Surveys -dark matter -Galaxies, spiral -Galaxies, halos

Context. The new generation of wide field optical imaging like the Canada France Hawaii Telescope Legacy Survey (CFHTLS) enables discoveries of all types of gravitational lenses present in the sky. The Strong Lensing Legacy Survey (SL2S) project has started an inventory, respectively for clusters or groups of galaxies lenses, and for Einstein rings around distant massive ellipticals.

Aims. Here we attempt to extend this inventory by finding lensing events produced by massive edge-on disk galaxies which remains a poorly documented class of lenses. Methods. We implement and test an automated search procedure of edge-on galaxy lenses in the CFHTLS Wide fields with magnitude 18 < i < 21, inclination angle lower than 25 • and having a photometric redshift determination. The procedure estimates the lensing convergence of each galaxy from the Tully-Fisher law and selects only the few candidates which exhibit a possible nearby arc configuration at a radius compatible with this convergence (r arc 2r E ). The efficiency of the procedure is tested after a visual examination of the whole initial sample of 30 444 individual edge-on disks. Results. We calculate the surface density of edge-on lenses possibly detected in a survey for a given seeing. We deduce that this theoretical number is about 10 for the CFHTLS Wide, a number in broad agreement with the 2 good candidates detected here. We show that the Tully-Fisher selection method is very efficient at finding valuable candidates, albeit its accuracy depends on the quality of the photometric redshift of the lenses. Eventually, we argue that future surveys will detect at least a hundred of such lens candidates. Conclusions.

Introduction

The Λ-CDM cosmological paradigm has been highly successful at explaining most of the large scale properties of the Universe. Nevertheless it is still to be shown that this model is able to explain its small scale properties. Among these, the formation and dark matter content of disk galaxies, and in particular the rotation curves of disks played a key role at giving evidence for the presence of dark matter on small scales (e.g. Bosma 1978;Rubin et al. 1980;Persic et al. 1996;de Blok et al. 2008;[START_REF] Donato | [END_REF]. Despite tremendous observational and modeling efforts over the last three decades, advances are still plagued by our ability to disentangle both the contributions of baryonic disk and bulges and the contribution of dark matter.

The hypothesis of the maximum bulge or disk (van Albada & Sancisi 1986;Persic et al. 1996;Dutton et al. 2005), is able to fix the mass-to-light ratio of the stellar populations and therefore to put constraints on the dark matter content of galaxies or alternatively on inferences about modifications of the gravitation law ⋆ Based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA, at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institut National des Sciences de l'Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii. This work is based in part on data products produced at TERAPIX and the Canadian Astronomy Data Centre as part of the Canada-France-Hawaii Telescope Legacy Survey, a collaborative project of NRC and CNRS.

(e.g. de Blok & McGaugh 1997;McGaugh 2004;de Blok et al. 2008;[START_REF] Donato | [END_REF]. In order to get further insight onto the relative contribution of different species (stars in disk, stars in bulge and dark matter) one needs to get additional constraints to break the degeneracies (e.g. Kranz et al. 2003;[START_REF] De Jong | Island Universes -Structure and Evolution of Disk Galaxies[END_REF]Ibata et al. 2007).

Another particularly promising method resides in studying the dynamical properties of strong lensing galaxies. Until recently, essentially due to the scarcity of background QSOs, only very few lensing disk galaxies with a low enough redshift, suitable for kinematical observations, were known. For instance, one can mention the early studies of the Einstein Cross Q2237-0305 (Trott & Webster 2002;Trott et al. 2009;van de Ven et al. 2008), B1600+434 [START_REF] Jaunsen | [END_REF][START_REF] Koopmans | [END_REF]Maller et al. 2000), J2004-1349 (Winn et al. 2003), CXOCY J220132.8-320144 (Castander et al. 2006) or Q0045-3337 (Chieregato et al. 2007).

Great advances have become recently possible thanks to the advent of large high-quality imaging and spectroscopic datasets. Automated or visual inspection of high resolution imaging from Hubble Space telescope (HST) (e.g. Marshall et al. 2009;Newton et al. 2009;Faure et al. 2008;[START_REF] Moustakas | [END_REF]Covone et al. 2009) or from the ground (e.g. Cabanac et al. 2007;Kubo & Dell'Antonio 2008, Gavazzi et al. 2010) have just started providing us with many candidate strong galaxy-galaxy lensing events. For spectroscopy, an automated search of superimposed spectra at different redshifts into a given fiber followed-up by high resolution HST imaging turns out to be very efficient on large spectroscopic datasets like the Sloan Digital Sky Survey (SDSS) (Bolton et al. 2004(Bolton et al. , 2006(Bolton et al. , 2008;;Auger et al. 2009;Willis et al. 2005Willis et al. , 2006)). These latter studies are limited to relatively low redshifts (z ∼ 0.2) and most of them are focused on lensing by massive elliptical galaxies with large splitting angle.

Conversely, late-type galaxies, while more numerous (e.g. Bartelmann 2000;[START_REF] Kochanek | Saas-Fee Advanced Course 33: Gravitational Lensing: Strong, Weak and Micro[END_REF]Chae 2010), have a smaller Einstein radius and are thus more difficult to identify. For instance the recently found edge-on disk galaxy OAC-GL J1223-1239 (Covone et al. 2009), with r E ∼ 0. ′′ 43, could only be found with HST imaging. A first automated spectroscopic search for disk galaxies lenses in the SDSS database was done by Féron et al. (2009): they found 8 candidates among 40 000 disk galaxies, two of them being already confirmed as genuine lenses.

Despite these efforts, disk galaxies remain a poorly documented class of lenses, particularly for edge-on lensing galaxies. This situation calls for an improvement. We need more edge-on disk lenses with a low inclination to maximize the success of dynamical studies and simplify the recognition of lensing/lensed structures in imaging data.

In this paper we investigate if the deep multiband subarcsecond imaging data of the Canada France Hawaii Telescope Legacy Survey (CFHTLS) offers a good opportunity to find edge-on disk lenses beyond the redshift range accessible by the SDSS.

The paper is organized as follows. Section 2 presents a calculation of the number of lenses with an i magnitude 18 < i < 21 and an inclination lower than 25 • which should be detected for a given seeing in an imaging survey. Section 3 explains the procedure that we have followed to extract edge-on lens candidates from the CFHTLS Wide survey by estimating an Einstein radius from the Tully-Fisher (TF) relation (Tully & Fisher 1977) and comparing it with the arc radius possibly detected around the galaxy. We also cross-check the validity of this automated procedure by a visual inspection of all the edge-on galaxies present in the survey. In Sect. 4 we discuss our results with an overview on what could be done with future all-sky imaging surveys.

Throughout this paper all magnitudes are expressed in the AB system, and we assume the concordance cosmological model with h = 0.7, Ω m = 0.3, and Ω Λ = 0.7.

Forecasting the frequency of edge-on disk lenses

We attempt here to predict the number of lenses that could be observed within an imaging survey of properties similar to the CFHTLS Wide.

Cross-sections and optical depths

We estimate lensing optical depths by starting from generic assumptions about mass distributions (see e.g. Turner et al. 1984;[START_REF] Schneider | Gravitational Lenses[END_REF]Marshall et al. 2005;[START_REF] Kochanek | Saas-Fee Advanced Course 33: Gravitational Lensing: Strong, Weak and Micro[END_REF]). In particular, we assume that all lenses are Singular Isothermal Spheres (SIS) parameterized by their velocity dispersion σ. They are also assumed to be sparse enough that the probability of multiple lensing is negligible, so that one can sum up individual lensing cross-sections over the probed volume. Arguably, circular symmetry might not be a good description of edge-on disk galaxies. Bartelmann & Loeb (1998) showed that, averaging over all possible inclination angles, the effect of the disk is null for the overall cross-section. It is however found that the efficiency of edge-on lenses is a bit boosted so that a calculation of the total SIS cross-section and the same calculation restricted to inclinations α i ≤ 25 • (corresponding to a third of all possible orientations) consistent with edge-on disks will bracket the right number. So we define a corrective term 1/3 < η < 1 that will multiply the total lensing cross-section of disky galaxies approximated as SIS.

Ideally, the cross-section X for multiple imaging by a SIS mass distribution at redshift z l of a source at z s is given by the solid angle (in steradians) subtended by the Einstein radius:

X = πr 2 E = π 4π σ c 2 D ls D os 2 .
(1)

However, due to observational limitations, one cannot detect all the lenses of a given r E , the main limitation being seeing and blending with the light possibly coming from the foreground deflector. In addition, for a SIS profile, two images (arcs) are produced at radii θ 1 and θ 2 from the optical axis (with the convention θ 1 > θ 2 > 0) so that the radius of the source is β = (θ 1 -θ 2 )/2 and its Einstein radius is r E = (θ 1 + θ 2 )/2. The detectability of a lens implies that the outermost arc θ 1 ≡ r arc > r cut , with r cut a limiting radius that we set to the seeing FWHM r cut = 0. ′′ 8. The other condition for multiple imaging, β < r E , complements the constraints for the actual lensing cross-section. The full calculation yields

X ′ =            0 if r E < r cut /2 π (2r E -r cut ) r cut if r cut /2 ≤ r E ≤ r cut π r 2 E if r cut < r E (2)
The optical depth τ for strong lensing of a source at z s can be written as the volume integral of the above cross-section integrated over the distribution of the velocity dispersion distribution function of lenses φ(σ) that is assumed to be constant with cosmic time (at least back to redshift ∼ 1 beyond which lensing efficiency quickly falls off).

τ(z s ) = z s 0 dz l f k (χ) 2 dχ dz l ∞ 0 dσ K(z l , σ)X ′ (σ, z l , z s )φ(σ) . (3)
In this equation we have introduced a selection function K(z l , σ) of potential lenses that will depend on observational settings and lens finding strategy. Notice that a purely source oriented approach, typical of radio surveys (Browne et al. 2003;Winn et al. 2003), will have K(z l , σ) = 1. Conversely a lens oriented survey like SLACS (Bolton et al. 2006) will be limited in redshift by the depth of SDSS spectroscopy and will pick the most massive σ ∼ 240 km s -1 galaxies from the start. Since we assume a flat cosmology we can use comoving distances f k (χ) = χ in Eq. ( 3). We use the velocity dispersion of Chae (2010) for SDSS late-type galaxies which reads:

φ(σ) = φ * σ -1 σ σ * α exp - σ σ * β β Γ(α/β) , (4)
with α = 0.69, β = 2.10, φ * = 0.066 h 3 Mpc -3 and σ * = 91.5 km s -1 .

Selection function of lenses

As already anticipated, observational limitations will lead to a non-unity selection function K(z l , σ). Obviously one cannot select directly upon z l or σ, but more realistically the apparent magnitude is used. Therefore we assume that a combination of photometric redshifts (with negligible errors at this stage) and a direct relation between luminosity and velocity dispersion will approximate most of the observational selections effects. More precisely, assuming that lens galaxies are isothermal spheres allows us to relate the velocity dispersion to the rotation velocity σ = V c / √ 2. We then use the latest normalization of the Tully-Fischer relation in the R band from Pizagno et al. (2007) and neglect the intrinsic scatter about the mean relation, giving: log V c = -0.135 M r -5 log 10 h + 20.332 + 2.210

(5)

In addition, we shift apparent i band magnitudes back to the rest frame r band using a redshift dependent k-correction so that M r = i -25 -5 log 10 (D lum /h -1 Mpc) -k i→r . A polynomial fit to the photometric change k i→r yields:

k i→r (z) = -0.36 + 0.65z -0.92z 2 + z 3 + 0.06z 4 (6)
which is obtained by redshifting template of Sbc galaxies from Coleman et al. (1980) using hyperz facility (Bolzonella et al. 2000). Hence a given selection in magnitude i min < i < i max can readily be cast into a redshift-dependent selection in velocity dispersion: K(z l , σ) = Θ(σσ min (z l ))Θ(σ max (z l ) -σ), where Θ is the Heavyside step function.

Population of background sources

Given the lensing optical depth of Eq. ( 3) and a redshift distribution of sources, the number of gravitational lenses can be written as:

N = η A n 0 ∞ 0 dz s p(z s ) τ(z s ) , (7)
where A is the surveyed area and η the term accounting for the boost of the disk introduced in § 2.1. In principle the redshift distribution inferred from photometric redshifts by Coupon et al. (2009) for the CFHTLS Wide survey should be most suitable for our analysis. However, magnification bias, which can create a gain of more than one magnitude in depth, makes better suited the redshift distribution of the HST COSMOS survey down to i < 26, which is closer to the population of faint blue objects we aim at detecting behind foreground edge-on disks. Therefore we use the results of Leauthaud et al. (2007) from COSMOS that can be approximated by the expression:

dn(z s ) dz s = 1 z 0 Γ(a) e -z s /z 0 (z s /z 0 ) a-1 (8)
with z 0 = 0.345 and a = 3.89 (Gavazzi et al. 2007). In addition, at these limiting i < 26 magnitudes, one can achieve a surface density of sources of about n 0 = 300 000 deg -2 .

Results

For the values considered above, we plot in Fig. 1 the predicted number of edge-on disks as a function of the cut radius which is identified to the seeing. For the CFHTLS Wide image quality (seeing ≃ 0. ′′ 8 in g band) and depth we typically get about 0.07 -0.30 detectable lenses per square degree depending on the edgeon disk boost factor η. Therefore for the 124 deg 2 imaged in five u * griz bands, one might discover between 8 and 37 lenses assuming all our hypotheses (SIS density profile, no scatter in the TF relation nor errors with photometric redshifts...) are correct. To assess our calculations we also made predictions for the number of lensing elliptical galaxies assuming a perfect relation between magnitude and velocity dispersion as given by the Faber-Jackson relation whose normalization is taken from Oguri (2006). Down to a limiting AB magnitude i < 22.5 for the deflector, a similar calculation based on the velocity dispersion function for ellipticals also given by Chae (2010) yields the red curve in Fig. 1. It predicts 10 to 100 more lenses with a substantially lower dependence on seeing below 1. ′′ 5. This might be explained by the generally larger Einstein radius (due to greater velocity dispersions) of ellipticals for which the exponential fall-off of the distribution functions drops beyond 1. ′′ 5. Worse seeing will also make the detectability of arcs and counterimages much more challenging beyond 0. ′′ 8. In addition, the light from the deflector, no matter the seeing size, will prevent many small scale detections. For comparison, Marshall et al. (2005) predicted about 50 lenses that could be detected by SNAP (two magnitudes deeper with FWHM∼ 0. ′′ 12). Ten or so systems would remain by degrading depth and image quality to our survey specifications and by restricting ourselves to bright i < 22.5 deflectors. This is consistent with our calculations. This is also in broad agreement with COSMOS observations of strong lensing luminous M r < -20 ellipticals (Faure et al. 2008).

Seeking edge-on disk lenses within the CFHTLS Wide

From the calculations above, we can undertake a comprehensive search of lenses in the CFHTLS Wide survey. The T0005 data release (for a detailed description, see [START_REF] Mellier | The CFHTLS T0005 Release[END_REF] spans 171 fields of one square degree each for which images in at least the three g, r, and i bands are obtained. But, in order to calculate the photometric z l of the possible lenses, we restrict ourselves to the 124 fields for which the five u * , g, r, i, and z bands are present. The typical 80% completeness limit magnitude for point sources is 25.4, 25.4, 24.6, 24.5, and 23.6 respectively. Image quality is also very good with typical seeing FWHM of 0. ′′ 90, 0. ′′ 85, 0. ′′ 75, 0. ′′ 72, and 0. ′′ 71 respectively.

Photometric preselection of bright edge-on galaxies

We start from a catalogue of 28 million extended objects detected with SExtractor (Bertin & Arnouts 1996). We first select 1.02 million objects within the magnitude range 18 < i < 21 as the initial data base. The bright i < 18 low redshift galaxies are excluded because their angular size is too large as compared to their expected Einstein radius. Objects fainter than i = 21 are also discarded because of the low signal-to-noise and the difficulty of future spectroscopic follow-up. The few already known spiral galaxies producing multiply-imaged QSOs happened to be in this magnitude range [START_REF] Koopmans | [END_REF]Maller et al. 2000;Castander et al. 2006;Chieregato et al. 2007). Edge-on disk galaxies are then preselected as highly elongated objects with projected ellipticity 0.7 < e ≡ (a 2 -b 2 )/(a 2 + b 2 ) < 0.92. The lower limit on the ellipticity is chosen because it removes most of the ellipticals (Park et al. 2007) while the upper limit e < 0.92 stands for rejecting most (∼ 90%) of spurious objects like diffraction patterns of bright stars. This ellipticity cut yields 30 444 bright edge-on galaxies.

The Tully-Fischer relation as a proxy for Einstein radius

The multiband (u * griz) photometry of the survey allows the computation of photometric redshifts (Coupon et al. 2009). Along with apparent magnitudes, this additional information enables estimates of absolute magnitudes that can readily be converted into an estimate of their rotation velocity V c using the latest calibrations of the Tully-Fischer relation (Pizagno et al. 2007). Assuming the lensing galaxy can be approximated by a circularly symmetric isothermal mass distribution, the velocity dispersion is σ = V c / √ 2. Using a typical source redshift 1 < z s < 3 and taking into account a possible error ∆M r = -0.37 on the absolute magnitude (corresponding to the 1σ-dispersion of the TF law), we can predict an Einstein radius r E(TF) that is then compared to the seeing size. More specifically, given that the maximum distance of the furthest of the two multiple images of a source is twice its Einstein radius and assuming that this latter image could not be detected if embedded within the seeing disk, we have selected foreground objects by requiring they satisfy the TF cut: 2r E(TF) ≥ 0. ′′ 8.

Applying this last automated cut on the catalogue provides us with 2 064 massive edge-on disk galaxies with the greatest chance of being lenses.

Subsequent visual classification

At this stage, we were unable to avoid visual inspection of color cutout images of the above 2 064 objects. The first reason for that is that the small size of Einstein radii and therefore distance of multiple images to the central deflector is comparable to both the deflector size and seeing disk. Hence SExtractor will fail at deblending arcs and deflector.

Consequently we undertook a systematic inspection of these 2 064 objects and decided to further consider objects that fulfill the following criteria.

(i) Arcs between 0. ′′ 6 and 3. ′′ 5: The inner limit is set by the separability of arc and deflector as said above whereas the outer limit is thought to be just above the innermost radius accessible to the automated arc detectors used by the SL2S collaboration for detecting large separation group and cluster scale lenses (Cabanac et al. 2007). Note that we discard arcs that are definitely too dim for spectroscopic follow-up (estimated to correspond to a magnitude i 24). (ii) Arc and deflector colors are different: In order to reduce the contamination by faint satellite galaxies falling into the main galaxy and producing tidal tails, we require the color of the arc and the deflector to be different. This criterium will not be 100% efficient at removing close pairs but its impact on completeness is very low because the probability that a lens and a source at very different redshifts have the same color is almost negligible. Since precise photometry is difficult on such small scales, color estimation is essentially based on visual inspection of the color images. (iii) Should look like a library of lensed features: Finally, the most stringent criterium that we apply is based on the geometry of the arc and its counterimages, as almost all1 edge-on disk lenses produce very typical image configurations (see e.g. Keeton & Kochanek 1998;Bartelmann & Loeb 1998;Shin & Evans 2007) that we can classify into four families "Bulge arcs", "Fold arcs", "Disk arcs" and "Pairs" as suggested by Shin & Evans (2007) and illustrated in Fig. 2. Naturally, intermediate cases between these categories may exist with about the same probability for an edge-on disk galaxy similar to the Milky Way (Shin & Evans 2007).

At this stage we obtain 11 candidates labeled with a A or B upper script in Table 1 and also shown in the upper panels panels of Fig. 4.

(iv) We then further require that the arc radius is smaller than twice the Einstein radius estimated with the Tully-Fisher method as discussed in § 3.2. 1. Right panel: Photometric redshift distribution for these 16 candidates. This latter filter (hereafter the TF (iv) test) leaves us with only 2 galaxies labeled A in Table 1 which would be our prime candidates for follow-up observations: they occupy the first row of Fig. 4. We notice that, no matter whether we consider class A or A+B candidates, we find numbers that are roughly consistent with our calculation of Sect. 2 although we fully acknowledge that completeness cannot be achieved given the sharp cutoffs we used.

Manual cross-check of the automated procedure

In order to validate the above procedure and to assess the effects of photometric redshift uncertainties, we decided to cross-check it by visually analyzing the complete sample of 30 444 objects of the CFHTLS selected at the end of § 3.1.

The number of objects is large but a trained observer, working in sessions of about two hours a day2 , is able to explore about 1 000 sub-images per hour, corresponding to 4 deg 2 per hour. This procedure would be impossible for an all-sky survey but it translate into about 30 hours for the whole CFHTLS Wide and allows an interesting cross-check of the automated procedure.

We obviously apply the same criteria described above in § 3.3. For information, the first of these steps: (i) An edge-on galaxy must have a faint object nearby, allows us to get rid of more than 99.3% of the potential lenses. It leaves us with 190 interesting galaxies (i.e. 1.5 ± 0.5 per deg 2 ) that deserve closer inspection, using steps (ii) and (iii).

The full visual selection procedure allowed to keep only 16 candidates. Of them, 11 were already selected as class A or B in the more automated procedure above whereas 5 additional systems were found to be worth presenting here although they don't pass the Tully-Fisher tests (neither the first cut 2r E(TF) ≥ 0. ′′ 8 nor the second TF (iv) test r arc ≤ 2r E(TF) ). These class C objects occupy the bottom of Table 1 and Fig. 4, while Fig. 3.4 shows the photometric redshift distribution and the distribution of arc radii for the full 16 candidates.

This time-consuming cross-check therefore validates our use of the TF tests, and shows that errors on photometric redshifts are not, in our case, a major source of concern.

Results & Discussion

Description of the candidates

We have seen that our search of edge-on disk galaxies in 124 deg 2 of the CFHTLS Wide yields 2 A-class good candidates (listed at the top of Table 1 and Fig. 4). We believe they are the only genuine strong lenses detected by our procedure.

-SL2SJ021711-042542 is a spiral galaxy with a photometric redshift z l = 0.57 ± 0.06. It is a bulge arc configuration with an observed arc radius (r arc = 1. ′′ 7) fully compatible with the maximum arc radius allowed by a TF model (2r E(TF) = 1. ′′ 8). -SL2SJ022533-042414 is also a spiral with a photometric redshift z l = 0.24 ± 0.04. It is a bulge arc configuration at r arc ≃ 1. ′′ 4 with a faint red counter image which is compatible with the maximum arc radius of the TF model (2r E(TF) = 1. ′′ 4).

From a pure visual selection (i.e. not applying filter (iv)), we would have selected 9 more candidates (labeled B in Table 1). These uncertain candidates can be roughly classified in two categories.

-85% of them have an arc radius that is too large for their lens magnitude and a small color difference with their lens. They probably correspond to stellar tidal trails produced by galaxy interactions or disrupted satellites. For the most massive satellites falling in the centre of a galaxy potential these structures can be bright enough to be observed, especially when star formation is initiated by gravitational interactions. These thin structures are wrapped around the dark matter halo potential and can mimic arc structures. Some of them are seen straddling the edge-on disk like an arc triplet formed by a naked cusp. We see in Fig. 3.4's left panel a cut-off for r arc > 2. ′′ 5. Its existence shows that this main source of spurious detections are more easily discriminated for large arc radii. Only the high spatial resolution of HST images could discriminate them at smaller radius. -The remaining 15% (namely SL2SJ020933-054012 and SL2SJ140454+544552) correspond probably to Singly Highly Magnified Sources (SHMS) where the arc is formed at a distance greater than 2r E and does not produce multiple images. The optical depth of this strong flexion regime was studied by Keeton et al. (2005) with both isothermal and NFW halo density profiles. He found that the proportion of SHMS among arcs should be less than 20% for distant background sources. For these candidates we shall remember that the baryonic mass within the disk can increase the convergence of the edge-on lens beyond the one considered for the TF test (Bartelmann & Loeb 1998). Therefore if the arc is not too far from the critical line an observer shall not systematically discard them without further considerations.

We also listed 5 class C candidates at the bottom of Table 1 which have been selected only during the cross-check manual procedure ( § 3.4), because their theoretical Tully-Fisher radius r E(TF) is less than half the seeing value. We believe they are interesting systems anyway and are worth presenting here.

Discussion & Prospects

During this detection exercise, several lessons were found, which can be of some use for future surveys.

The Tully-Fisher cut turns out to be an efficient way to strongly decrease the number of line of sight to be visually scrutinized and to get rid of many of the false positive cases. Indeed no bona fide strong lens was found during the manual crosscheck ( § 3.4) out of the 30 444 -2 064 objects discarded by the TF cut ( § 3.2). However we stress that the accuracy of this cut is limited by the quality of photometric redshifts. With a limited number of colors, high redshifts could be mistaken for low ones, specially in the range 0.05 < z l < 0.15 (e.g. Bernstein & Huterer 2009), therefore the absolute magnitude of the lens would be over estimated and thus resulting in too small a value of r E(TF) . We therefore recommend not to implement a too stringent TF cut.

We show in the left panel of Fig. 3.4 the distribution of arc radii. The effect of the seeing cut-off at small radii can be clearly seen: the numerous lenses with arc radius smaller than the seeing disk (0. ′′ 8) are missed. The photometric redshift distribution of the 16 selected candidates lies in the range 0.2 z l 0.6 as shown in the right panel of Fig. 3.4. In this respect, our CFHTLS candidates nicely complement the ones extracted from shallower surveys like SDSS (see e.g. Féron et al. 2009).

We can wonder if the LSST (LSST Collaboration 2009) survey will significantly improve the situation. It will cover 20 000 square degrees of the sky and will reach the depth of the CFHTLS Wide just after the first year of observation and the depth of the CFHTLS Deep after ten years [START_REF] Ivezić | American Institute of Physics Conference Series[END_REF]. The seeing quality at Cerro Pachon in Chile is expected to be 0. ′′ 7 quite comparable to the one at CFHT. Therefore, in theory, we can expect from our calculation (cf. Fig. 1) at least 1 300 massive edge-on lenses in LSST. Its deeper photometry will also allow to get better photometric redshift and hence to improve the chance of keeping good candidates with the TF test. Nevertheless, conservatively assuming the same detection probability as in CFHTLS, one expects to find in the LSST about 1770 candidates instead of the 11 A+B class candidates found here, leading probably to about 320 bona-fide edge-on lenses, which is an excellent perspective.

The situation shall improve even more dramatically with future space borne surveys like JDEM or Euclid (Marshall et al. 2005;Refregier 2009) which will have a better resolution (< 0. ′′ 3). We expect from our calculation that these surveys could detect a few thousand edge-on lenses with Einstein radius substantially smaller than in ground based surveys.

Conclusion

In conclusion, while it is extremely difficult to find edge-on lenses with ground based imaging surveys, the situation is not completely hopeless with sub-arcsecond seeing (< 0. ′′ 8).

A good way to implement a fast search procedure is to preselect a limited subset of galaxies presumably massive enough to produce a detectable arc. The Tully-Fisher law seems quite appropriate for this purpose, since it can predict an Einstein radius for each disk. Galaxies with the largest Einstein radii are then checked for the presence of a faint source nearby which might be a gravitational arc. Extensive visual cross-checking of all disk galaxies present in the survey has shown that this procedure picks most of the interesting candidates, provided reliable photometric redshifts are available.

Whatever the final choice of an observer, the number of candidates that can be choosen for an observational follow up is quite small: here we ended up with only 2 good candidates from 124 square degrees of the CFHTLS Wide. This illustrates how challenging it is to dig out such lenses from any ground based survey. Despite their great number in the sky and as long as wide space surveys are not available, edge-on disk lenses will remain scarce and precious objects for astronomers who try to understand the relative distribution of dark and ordinary matter in spirals.

Fig. 1 .

 1 Fig. 1. Predicted number of gravitational lenses per square degree as a function of image quality. Imaging survey specifications in terms of depth are those of the CFHTLS Wide. The dashed and dot-dashed black lines bracket the number of lensing edge-on disks depending on the edge-on disk boost factor η. The upper solid red curve is for lensing by bright i < 22.5 ellipticals.

Fig. 2 .

 2 Fig. 2. Families of multiple images configurations for lensing by an edge-on disk (yellow/dark) galaxy for different source (blue/light) positions. Red/dark lines (resp. green/light) represent critical lines (resp. caustics) in the image (resp. source) plane. From top left to bottom right: "Bulge arcs", "Fold arcs", "Disk arcs" and "Pairs" (Shin & Evans 2007).

Fig. 3 .

 3 Fig.3. Left panel: Distribution of arc radii for the 190 peculiar lines of sight (solid black curve) of § 3.4, and for the 16 best candidates (dashed red curve) listed in Table1. Right panel: Photometric redshift distribution for these 16 candidates.

The observation of faint counterimages close to the center is very challenging, especially for disk galaxies with substantial dust extinction(Kochanek

2006).

This was found to be a decent limit to preserve human eye accuracy. Repeated visual inspections of several fields were done at different periods to assess the stability of the human decision process.

Acknowledgements. The authors are thankful to the CFHTLS members and the Terapix team for their excellent work in reducing and distributing data to the community. They would also like to thank P. Marshall and J.-P. Kneib for fruitful discussions, and J. Coupon for providing his photometric redshifts calculations. Part of this work was supported by the Agence Nationale de la Recherche (ANR) and the Centre National des Etudes Spatiales (CNES). Part of this project is done under the support of the National Natural Science Foundation of China No.10878003, 10778752, Shanghai Foundation No. 07dz22020, and the Leading Academic Discipline Project of Shanghai Normal University (08DZL805).

Notes. The uncertainty on magnitudes is, in almost every case, of the order of 0.01 and the Einstein radius is the maximum allowed by the TF law. For geometry family, b stands for "bulge arc", f for "fold arc", d for "disk arc" and p for "pair" (see § 3.3). (A) Candidates found with both the automated and the manual cross-check procedures and satisfying the Tully-Fisher (iv) test. (B) Candidates same as (A) , but failing the Tully-Fisher (iv) test. (C) Extra candidates found only with the manual cross-check procedure.


Lorenzo Brasco

Giuseppe Buttazzo

Filippo Santambrogio

A Benamou-Brenier approach to branched transport

Keywords: AMS Subject Classification (2000): 49J45, 49Q20, 49Q10, 90B18, 60K30 Optimal transport, branched transport, continuity equation, functionals on spaces of measures, optimal networks

The problem of branched transportation aims to describe the movement of masses when, due to concavity effects, they have the interest to travel together as much as possible, because the cost for a path of length ℓ covered by a mass m is proportional to m α ℓ with 0 < α < 1. The optimization of this criterion let branched structures appear and is suitable to applications like road systems, blood vessels, river networks. . . Several models have been employed in the literature to present this transport problem, and the present paper looks at a dynamical one, similar to the celebrated Benamou-Brenier formulation of Kantorovitch optimal transport. The movement is represented by a path ρ t of probabilities, connecting an initial state µ 0 to a final state µ 1 , satisfying the continuity equation ∂ t ρ + div x q = 0 together with a velocity field v (with q = ρv being the momentum). The transportation cost to be minimized is non-convex and finite on atomic measures:

Introduction

The optimal mass transportation theory consists in the study of transporting a given mass distribution µ 0 on Ω (that we assume to be a compact and convex subset of R d ) into a final configuration µ 1 , by minimizing the total transportation cost, the latter being suitably defined: clearly, µ 0 and µ 1 are required to satisfy the mass balance condition Ω dµ 0 = Ω dµ 1 . From now on, we will assume that they are normalized to be probability measures. The cost for moving a unit mass from a position x to a position y is taken equal to c(x, y), a function a priori given, which determines the nature of the problem and provides the total minimal cost C(µ 0 , µ 1 ) = min Ω×Ω c(x, y) dγ(x, y) : γ ∈ Γ(µ 0 , µ 1 )

(1.1)

where Γ(µ 0 , µ 1 ) is the class of admissible transport plans, i.e. probabilities on the product space Ω×Ω having first and second marginals given by µ 0 and µ 1 respectively. The cases c(x, y) = |x-y| p 1 Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5 -56127 Pisa, ITALY brasco@mail.dm.unipi.it, buttazzo@dm.unipi.it 2 CEREMADE, UMR CNRS 7534, Université Paris-Dauphine, Pl. de Lattre de Tassigny -75775 Paris Cedex 16, FRANCE filippo@ceremade.dauphine.fr 1 with p ≥ 1 have been particularly studied, and the cost C(µ 0 , µ 1 ) in (1.1) provides, through the relation W p (µ 0 , µ 1 ) = C(µ 0 , µ 1 ) 1/p , the so-called Wasserstein distance W p which metrizes the weak* convergence on the space of probabilities P(Ω). A very wide literature on the subject is available; we simply mention the books [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measure[END_REF][START_REF] Villani | Topics in optimal transportation[END_REF][START_REF] Villani | Optimal transport, old and new[END_REF] where one can find a complete list of references. Thanks to the fact that the space W p (Ω) of probability measures endowed with these distances turns out to be a geodesic space, dynamical models for optimal transportation are of particular interest. Being a geodesic space means that the distance between two points is always equal to the infimum of the lengths of the curves connecting these points, and that this infimum is actually a minimum:

W p (µ 0 , µ 1 ) = min 1 0 |ρ ′ t | Wp dt : ρ ∈ Lip([0, 1]; W p (Ω)), ρ 0 = µ 0 , ρ 1 = µ 1 ,
where |ρ ′ | Wp is the metric derivative of the measure-valued Lipschitz curve ρ, defined as (we refer the reader to [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measure[END_REF], for more details)

|ρ ′ t | Wp = lim h→0 W p (ρ t+h , ρ t ) h .
Since the curves connecting two points of this space are actually curves of measures, they can be described through the so-called continuity equation: it is well-known (see [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measure[END_REF], Theorem 8.3.1) that for every Lipschitz or absolutely continuous curve ρ t in the space W p (Ω) (p > 1 for simplicity) there exists a map q from [0, 1] into the space of vector valued measures, such that q t ≪ ρ t (hence q t = v t • ρ t , v being the velocity vector) which represents the flux q = ρv and satisfies

∂ t ρ + div x q = 0 and v t L p (ρt) = |ρ ′ t | Wp , (1.2)
(the degenerate case p = 1 being a little bit more involved, since q t ≪ ρ t is no more guaranteed and the L 1 -norm has to be replaced by the mass of the measure q t , see [START_REF] Ambrosio | Lecture notes on optimal transport problems[END_REF]).

On the other hand, every time that we have a pair (ρ, q) satisfying ∂ t ρ + div x q = 0 with q ≪ ρ, so that

q t = v t • ρ t , we can infer that |ρ ′ t | Wp ≤ v t L p (ρt)
. This means that one can minimize the functional

A p (ρ, q) := 1 0 R d |v t | p dρ t dt if q ≪ ρ and q t = v t • ρ t , +∞ otherwise, (1.3)
which is nothing but the integral in time of the kinetic energy when p = 2, and the cost in (1.1) can be recovered through the equality

C(µ 0 , µ 1 ) = min 1 0 R d dq t dρ t (x) p dρ t (x) dt : ∂ t ρ + div x q = 0, ρ 0 = µ 0 , ρ 1 = µ 1 .
The problem above is the one which was proposed by Benamou and Brenier in [START_REF] Benamou | A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem[END_REF] as a dynamical version of optimal transportation. It has the advantage that it is the minimization of a convex functional of ρ and q, under linear constraints. Other variants of mass transportation problems have been studied and can be expressed in this way by considering in (1.3) other convex functions of the pair (ρ, q). Recently, Dolbeault, Nazaret and Savaré introduced in [START_REF] Dolbeault | A new class of transport distances between measures[END_REF] new classes of distances over P(R d ) based on the minimization of the functional (where λ is a given reference measure on R d and ρ and q are identified with their densities w.r.t. λ)

1 0 R d Φ(ρ, q) dλ dt, where Φ(ρ, q) = |q| p h(ρ) p-1 = |q| h(ρ) p h(ρ), p ≥ 1,
which are connected to the non-linear mobility continuity equation ∂ t ρ + div x h(ρ)v = 0 (a treatment of the limiting case h(ρ) ≡ 1, corresponding to consider Φ(ρ, q) = |q| p , can also be found in [START_REF] Brenier | Extended Monge-Kantorovich theory[END_REF]). If the function h is concave (for example h(ρ) = ρ β , with β ∈ [0, 1]), this problem turns out to be convex as well. The main interest that motivated Dolbeault et al. to the study of these distances lies in the possible applications to diffusion equations of the type of the non-linear mobility continuity equation ∂ t ρ + div x h(ρ)v = 0 above, where the vector field v depends on ρ in a way such that the equation can be interpreted as a gradient flow of a given functional with respect to these new family of dynamical distances. Moreover, the equations of the geodesics (which are similar to a mean-field game system, see [START_REF] Lasry | Mean field games[END_REF]) and the conditions for these distances to be finite are being studied in [START_REF] Nazaret | Non-linear mobility distances on probability measures: geodesics via PDE[END_REF].

In connection with congestion effects and crowd motion, other models include penalizations on high densities: in [START_REF] Buttazzo | An optimization problem for mass transportation with congested dynamics[END_REF] the case

Φ(ρ, q) = |q| p ρ p-1 + cρ 2 p ≥ 1, c ≥ 0
has been considered as a model for crowd motion in a congested situation (for instance in case of panic). This problem as well is convex.

A completely different situation occurs in the case opposite to congestion, when concentration effects are present and the mass has the interest to travel together as much as possible, in order to save part of the cost. This happens very often in many applications, as discovered by Gilbert who in [START_REF] Gilbert | Minimum cost communication networks[END_REF] formulated a mathematical model for the transportation of signals along telephone cables. More recently the Gilbert's model has been refined and considered in the framework of mass transportation, under the name of branched transport, to emphasize the fact that transport rays may bifurcate. All these models have in common the fact that the cost for a mass m moving on a path of length ℓ is proportional to m α ℓ (0 < α < 1, so that (m 1 + m 2 ) α < m α 1 + m α 2 ). In [START_REF] Bernot | Traffic plans[END_REF][START_REF] Bernot | The structure of branched transportation networks[END_REF][START_REF] Bernot | Optimal transportation networks. Models and theory[END_REF][START_REF] Bernot | Figalli: Synchronized traffic plans and stability of optima[END_REF][START_REF] Maddalena | A variational model of irrigation patterns[END_REF] for every 0 < α < 1 a transportation cost from ρ 0 to ρ 1 is considered through a suitable use of probabilities defined on spaces of curves in Ω, with [START_REF] Maddalena | A variational model of irrigation patterns[END_REF] (the so called irrigation patterns model) dealing with the case of a single source ρ 0 = δ x 0 . See Section 4 to have a glance at the details of these models and their formulations. On the other hand, the model of [START_REF] Xia | Optimal paths related to transport problems[END_REF] can be seen as the natural extension of the original Gilbert's model and uses vector measures having prescribed divergence ρ 0 -ρ 1 : these vector measures are the continuous generalization of the finite weighted and oriented graphs that were present in Gilbert's original formulation.

A first attempt to obtain a dynamical formulation of branched transportation through curves of measures was made in [START_REF] Brancolini | Path functionals over Wasserstein spaces[END_REF], and later refined in [START_REF] Brasco | Curves of minimal action over metric spaces[END_REF][START_REF] Brasco | An equivalent path functional formulation of branched transportation problems[END_REF]: in these papers the starting point is the geodesic formulation of the Wasserstein distance, where the length functional is modified considering an energy of the type

1 0 g(ρ t ) |ρ ′ t | Wp dt.
The weight function g is a local term of the moving mass, forcing the mass to concentrate and thus giving raise to branching phenomena. These models are not satisfactory yet, because they are in general not equivalent to those by Gilbert, Xia or Bernot-Caselles-Morel. A tentative to perform some modifications in the functionals 3 defined on curves of measures so as to obtain equivalence with the other models has been made in [START_REF] Brasco | Curves of minimal action over metric spaces[END_REF], where on the other hand some quite involved distinction between moving mass and still mass has been done. In all of these models, the branched transportation is studied avoiding the Benamou-Brenier approach consisting in the minimization of a suitable cost F(ρ, q) under the constraint of the continuity equation ∂ t ρ + div x q = 0, that we believe is the most natural for this kind of problems. The only approach to dynamical branched transportation using the continuity equation is, as far as we know, the one of [9]. Yet, to prove semicontinuity and hence existence, even in this model, the problem is reduced to the minimization of a functional of the form

θ α dH 1 (x, t)
(which is the energy of Xia in [START_REF] Xia | Optimal paths related to transport problems[END_REF]) and the dynamical features are not completely exploited.

In the present paper we follow a more direct approach: for all pairs (ρ, q) verifying the continuity equation, with ρ 0 = µ 0 and ρ 1 = µ 1 , we define a functional F(ρ, q) and we show that this functional is both lower semicontinuous and coercive with respect to a suitable convergence on (ρ, q), and this provides directly the existence of an optimal dynamical path. The paper is organized as follows:

• in Section 2 we give the precise setting and state the main results;

• Section 3 is devoted to the proofs giving the existence of an optimal path ρ t ;

• in Section 4 we show that our model is equivalent to the other models of branched transportation available in the literature, comparing it to the traffic plan model of [START_REF] Bernot | Traffic plans[END_REF], which is one of the most flexible (and anyway equivalent to the others, as shown in [START_REF] Bernot | Optimal transportation networks. Models and theory[END_REF], Chapter 9);

• in the Appendix we deal with some inequalities involving Wasserstein distances and branched distances, that is distances over the space of probabilities given by the minima of some branched transportation problems. These inequalities have already been studied in [START_REF] Morel | Comparison of distances between measures[END_REF] and [START_REF] Devillanova | Elementary properties of optimal irrigation patterns[END_REF], but some very precise issues concerning d α and W 1/α are very close to the topics of this paper and deserve being treated here. New and simpler proofs are provided.

Problem setting and main results

In this section we fix the notation and state the main results of the paper. In the following Ω will denote a given subset of R d , where all the mass dynamics will take place; for the sake of simplicity we assume that Ω is convex and compact. The space P(Ω) of all Borel probabilities on Ω can then be endowed with the weak* convergence, which is metrized by the Wasserstein distances (see the Introduction). In the following, we will also use the notation M(Ω; R d ) to indicate the space of R d -valued Radon measures over Ω, while L k will indicate the k-dimensional Lebesgue measure.

The main objects to be considered will be pairs (ρ, q) with

ρ ∈ C [0, 1]; P(Ω) , q ∈ L 1 [0, 1]; M(Ω; R d ) (2.1)
satisfying the continuity equation formally written as (here ν stands for the outer normal versor to

∂Ω) ∂ t ρ + div x q = 0, in [0, 1] × Ω q • ν = 0, on [0, 1] × ∂Ω, (2.2)
whose precise meaning is given in the sense of distributions, that is

1 0 Ω ∂ t φ(t, x) dρ t (x) + Ω D x φ(x, t) • dq t (x) dt = 0 (2.3)
for every smooth function φ with φ(0, x) = φ(1, x) = 0.

Definition 2.1. We denote by D the set of all pairs (ρ, q) satisfying (2.1) and (2.3). Moreover, given µ 0 , µ 1 ∈ P(Ω), we define the set D(µ 0 , µ 1 ) of admissible configurations connecting µ 0 to µ 1 as D(µ 0 , µ 1 ) = (ρ, q) ∈ D :

ρ 0 = µ 0 , ρ 1 = µ 1 .
The velocity vector v can be defined as the Radon-Nikodym derivative of the vector measure q with respect to ρ: v = dq dρ .

Among all pairs (ρ, q) ∈ D satisfying the continuity equation above, we consider a cost function F(ρ, q) of the form

F(ρ, q) = 1 0 F (ρ t , q t ) dt, (2.4)
where F is defined through

F (ρ, q) := G α (|v| 1/α • ρ) if q = v • ρ, +∞ if q is not absolutely continuous w.r.t. ρ and G α (0 < α < 1
) is a functional defined on measures, of the kind studied by Bouchitté and Buttazzo in [START_REF] Bouchitté | New lower semicontinuity results for nonconvex functionals defined on measures[END_REF]: G α (λ) = +∞ if λ is not purely atomic, while (# stands for the counting measure)

G α (λ) = Ω |λ({x})| α d#(x) = i∈N |λ i | α , if λ = i∈N λ i δ x i .
In this way our functional F becomes

F(ρ, q) = 1 0 Ω |v t (x)|ρ t ({x}) α d#(x) dt = 1 0 i∈N |v t,i |ρ α t,i dt, (ρ, q) ∈ D,
and the dynamical model for branched transport we consider is

B α (µ 0 , µ 1 ) := min (ρ,q)∈D(µ 0 ,µ 1 ) F(ρ, q). (2.5)
Our main goal is to show that the minimization problem (2.5) above admits a solution. This will be obtained through the direct methods of the calculus of variations, consisting in proving lower semicontinuity and coercivity of the problem under consideration, with respect to a suitable convergence.

Remark 2.2. We point out that the weak* convergence of the pairs (ρ, q) does not directly imply the lower semicontinuity in (2.5), since the functional is not jointly convex. On the other hand, if (ρ n , q n ) ∈ D and we assume (ρ n t , q n t ) ⇀ (ρ t , q t ), for L 1 -a.e. t ∈ [0, 1], then a simple application of Fatou's Lemma would lead to the desired semicontinuity property of F (because one could prove that F is a lower semicontinuous functional on measures, as a consequence of the semicontinuity of G α and of the convexity of (x, y) → |x| p /y p-1 ).

In order to prove in the easiest possible way a semicontinuity result, we will introduce a convergence which is stronger than the weak convergence of measures on [0, 1] × Ω, but weaker than weak convergence for every fixed time t. This convergence will be compatible with the compactness we can infer from our variational problem. Definition 2.3. We say that a sequence (ρ n , q n ) τ -converges to (ρ, q) if (ρ n , q n ) ⇀ (ρ, q) in the sense of measures and sup

n∈N, t∈[0,1]
F (ρ n t , q n t ) < +∞.

Theorem 2.4. Let (ρ n , q n ) be a sequence such that F(ρ n , q n ) ≤ C, then up to a time reparametrization, (ρ n , q n ) is τ -compact.

Theorem 2.5. Let (ρ n , q n ) ∈ D be a sequence τ -converging to (ρ, q). Then

F(ρ, q) ≤ lim inf n→∞ F(ρ n , q n ).
As a consequence we obtain the following existence result.

Theorem 2.6. For every µ 0 , µ 1 ∈ P(Ω), the minimization problem (2.5) admits a solution.

Remark 2.7. We remark that, for some choices of the data µ 0 , µ 1 and of the exponent α, the statement of Theorem 2.6 could be empty, because the functional F could be constantly +∞ on every admissible path (ρ, q) joining µ 0 to µ 1 . This issue will be solved in Section 4, where the equivalence with other variational models for branched transportation will be proven. Since for these models finiteness of the minima has been widely investigated, we can infer for instance that if α > 1 -1/d then every pair µ 0 and µ 1 can be joined by a path of finite energy. On the other hand, if α ≤ 1 -1/d, µ 0 = δ x 0 and µ 1 is absolutely continuous w.r.t. L d , then there are no finite energy paths connecting them.

Proofs

A preliminary inequality to all the proofs is the following: if q ≪ ρ, then q t = v t • ρ t and

F (ρ t , q t ) = i ρ t ({x i }) α |v t (x i )| = i ρ t ({x i })|v t (x i )| 1/α α ≥ i ρ t ({x i })|v t (x i )| 1/α α = v t L 1/α (ρt) , (3.1)
due to the sub-additivity of the function x → x α . This inequality and its consequences will be discussed in the Appendix as well. In particular it also follows

F (ρ t , q t ) ≥ v t L 1 (ρt) = |q t |(Ω). (3.2)

Proof of Theorem 2.4

Due to the fact that the functional F is 1-homogeneous in the velocity, it is clear that reparametrizations in time do not change the values of F. By reparametrization, we mean replacing a pair (ρ, q) with a new pair (ρ, q) of the form ρt = ρ ϕ(t) , qt = ϕ ′ (t)q ϕ(t) (which equivalently means that q is the image measure of q through the inverse of the map (t, x) → (ϕ(t), x)). Thanks to this invariance, if (ρ n , q n ) is such that F(ρ n , q n ) ≤ C, then one can define a new pair (ρ n , qn ), with

F (ρ n t , qn t ) = F(ρ n , qn ) = F(ρ n , q n ) ≤ C, for every t.
After that, we only need to prove compactness for the weak convergence of measures on [0, 1] × Ω, a fact which only requires bounds on the mass of ρn and qn . The bound on ρn is straightforward, since for every t the measure ρn t is a probability, while for qn , which is absolutely continuous w.r.t. ρn , it is enough to use (3.2) in order to bound the mass of q by C.

This allows to extract a subsequence (ρ n k t , qn k t ) which converges weakly to a pair (ρ, q). The only nontrivial point is that we a priori restricted our attention to pairs (ρ, q) where ρ ∈ C [0, 1]; P(Ω) and q ∈ L 1 [0, 1]; M(Ω; R d ) , so that we need to prove that ρ is continuous and that q is of the form q t dt. Yet, the inequality (3.1) applied to the pairs (ρ n , qn ), proves a uniform bound on the L 1/α norm of the velocities, which implies that the curves ρn are uniformly Lipschitz continuous according to the distance W 1/α , and this property is inherited by the limit measure ρ.

For the decomposition of q, just use the inequality (3.1), thus obtaining a uniform bound on v n t L 1/α (ρ n t ) , which a fortiori gives a uniform bound on the Benamou-Brenier functional

A 1/α (ρ n , q n ) = 1 0 v n t 1/α L 1/α (ρ n t ) dt.
This functional being lower semicontinuous, we can deduce the same bound at the limit: this in particular implies that q is absolutely continuous w.r.t. ρ, with an L p density. Since ρ is a measure on [0, 1] × Ω which is of the form ρ t dt, the same disintegration will be true for q.

This means that we have actually found an admissible pair (ρ, q) which is the τ -limit of (ρ n k t , qn k t ) and the proof is complete.

Proof of Theorem 2.5

We consider here a sequence (ρ n , q n ), where q n = v n • ρ n (otherwise the functional F would not be finite-valued), satisfying the continuity equation and such that (ρ n , q n ) τ -converges to (ρ, q). First of all we define a sequence of measures m n on [0, 1] × Ω through

m n = i ρ n t ({x i,t }) α |v n t (x i,t )|δ x i,t dt,
where the points x i,t are the atoms of q n t (i.e. the atoms of ρ n t where the velocity v n t does not vanish). We notice that F(ρ n , q n ) = m n ([0, 1] × Ω).

In order to prove lower semicontinuity of F we can assume F(ρ n , q n ) to be bounded. This bound implies the convergence m n ⇀ m, up to the extraction of a subsequence (not relabeled). It is clear that, on this subsequence, we have

lim n→∞ F(ρ n , q n ) = lim n→∞ m n ([0, 1] × Ω) = m([0, 1] × Ω),
then in order to prove the desired semicontinuity property, it is enough to get some proper lower bounds on m.

Notice that, since we have m n ([a, b] × Ω) = b a F (ρ n t , q n t ) dt and F (ρ n t , q n t ) ≤ C (by definition of τ -convergence), we know that the marginal of m n on the time variable is a measure with an L ∞ density bounded by the constant C on [0, 1]. This bound is uniform and will be then satisfied by the limit measure m too: in particular, this implies that we can write m = m t dt.

Let us fix a closed set Q, as well as a time interval [a, b], and take the function

χ M (x) := (1 -M dist(x, Q)) + , x ∈ Ω,
where ( • ) + stands for the positive part: observe that χ M is positive, takes the value 1 on Q, is M -Lipschitz and vanishes outside a 1/M -neighborhood of Q. Indicating by 1 E the characteristic function of a generic set E, i.e. the function which takes the value 1 on E and 0 elsewhere, we consider φ(t, x) = χ M (x) α 1 [a,b] (t), which is upper semicontinuous on [0, 1] × Ω. Then we have

φ(t, x) dm(t, x) ≥ lim sup n→∞ φ(t, x) dm n (t, x) = lim sup n→∞ b a i ρ n t ({x i }) α |v n t (x i )|χ M (x i ) α dt,
where the points x i are, as before, the atoms of q n (and we omitted the dependence on n and t). We then decompose the product

ρ n t ({x i }) α χ M (x i ) α as ρ n t ({x i })χ M (x i ) α-1 • ρ n t ({x i })χ M (x i ) (where ρ n t χ M > 0). Notice that ρ n t ({x i })χ M (x i ) ≤ χ M dρ n t .
Then we can estimate the right-hand side in the previous inequality as b a i

ρ n t ({x i }) α |v n t (x i )|χ M (x i ) α dt ≥ b a χ M dρ n t α-1 i ρ n t ({x i })|v n t (x i )|χ M (x i ) dt = b a χ M dρ n t α-1 χ M d|q n t | dt.
We go on by estimating from above χ M dρ n t : we have

Ω χ M (x) dρ n t (x) ≤ Ω χ M (x) dρ n s (x) + M W 1 (ρ n t , ρ n s ),
which is a consequence of the definition of W 1 by duality with 1-Lipschitz functions (see [START_REF] Villani | Topics in optimal transportation[END_REF], Theorem 1.14). To estimate the W 1 distance we use W 1 ≤ W 1/α and the following fact

W 1/α (ρ n t , ρ n s ) ≤ t s |(ρ n z ) ′ | w 1/α dz ≤ t s v n z L 1/α (ρz) dz,
then applying inequality (3.1) we have in the end

Ω χ M (x) dρ n t (x) ≤ Ω χ M (x) dρ n a (x) + CM (b -a), for every t ∈ [a, b].
In this way we have

b a χ M dρ n t α-1 χ M d|q n t | dt dt ≥ χ M dρ n a +CM (b -a) α-1 b a χ M d|q n t | dt = χ M dρ n a + CM (b -a) α-1 φ 1/α d|q n |.
Hence, we may go on with

φ dm ≥ lim sup n→∞ χ M dρ n a + CM (b -a) α-1 φ 1/α d|q n | ≥ χ M dρ a +CM (b -a) α-1 φ 1/α d|q|.
In the last inequality the second factor has been dealt with in the following way: suppose |q n | ⇀ σ, then we have σ ≥ |q|; moreover φ ≥ φ where φ(t, x) := χ M (x)1 (a,b) (t), and this last function is l.s.c. and positive, so that

lim inf n→∞ φ 1/α d|q n | ≥ lim inf n→∞ φ1/α d|q n | ≥ φ1/α dσ ≥ φ1/α d|q| = φ 1/α d|q|,
since the boundaries t = a and t = b are negligible for |q|.

After that, we can divide by (b-a) (keeping for a while M fixed) and pass to the limit as b → a. This gives, for L 1 -a.e. a ∈ [0, 1],

χ M (x) α dm a (x) ≥ χ M (x) dρ a (x) α-1 χ M (x) d|q a |(x).
We let now M → ∞, so that χ M monotonically converges to the characteristic function of the set Q, and we have, by dominated convergence w.r.t. m a , ρ a and |q a |,

m a (Q) ≥ ρ a (Q) α-1 |q a |(Q). (3.3)
In the last term the convention 0

•∞ = 0 is used (if |q a |(Q) = 0
). This inequality is proven for closed sets, but by regularity of the measures it is not difficult to prove it for arbitrary sets. Actually, if S ⊂ Ω is an arbitrary Borel set, we can write

m a (S) ≥ m a (Q) ≥ ρ a (Q) α-1 |q a |(Q) ≥ ρ a (S) α-1 |q a |(Q),
for every Q ⊂ S closed, and take a sequence of closed sets

Q k such that |q a |(Q k ) → |q a |(S), since |q a | is, for L 1 -a.e. a ∈ [0, 1]
, a finite (and hence regular) measure on the compact set Ω. We want now to prove that

• q ≪ ρ, • q a = v a • ρ a is atomic for L 1 -a.e. a ∈ [0, 1] (i.e. ρ a is atomic on {v a = 0}) • m a (Ω) ≥ F (ρ a , q a ), for L 1 -a.e. a ∈ [0, 1].
This would conclude the proof. The first statement follows from the inequality (3.1), and the behavior of the Benamou-Brenier functional A 1/α (ρ n , q n ) as in the proof of Theorem 2.4, which guarantees an L 1/α density of q w.r.t. ρ. As a consequence, since ρ is a measure on [0, 1] × Ω which disintegrates w.r.t. the Lebesgue measure on [0, 1], the same will be true for q and we can write q t = v t • ρ t .

For the second statement, take the inequality m a (S) ≥ ρ a (S) α-1 |q a |(S) which is valid for any Borel set S, and apply it to sets which are contained in the Borel set

V ε := {x ∈ Ω : |v a (x)| > ε}.
For those sets, we have easily m a (S) ≥ ερ a (S) α . This means that the measure λ := ε 1/α ρ a V ε satisfies the inequality λ(S) α ≤ m(S) for every Borel set S ⊂ V ε . Since m is a finite measure, this implies that λ is atomic (see Lemma 3.1 below). If the same is performed for every ε = 1/k, this proves that ρ a is purely atomic on the set {x : |v a (x)| = 0}, that is q a = v a • ρ a is purely atomic.

Once we know that q a is atomic we infer that F (ρ a , q a ) = i ρ a ({x i }) α-1 |q a |({x i }) and we only need to consider Q = {x i } in (3.3) and add up:

m a (Ω) ≥ i m a ({x i }) ≥ F (ρ a , q a ),
which finally concludes the proof. Lemma 3.1. Take two finite positive measures λ and µ on a domain Ω, and α ∈ (0, 1). Suppose that the inequality λ(S) α ≤ µ(S) is satisfied for every Borel set S ⊂ Ω. Then λ is purely atomic.

Proof. Consider a regular grid on Ω of step 1/k, for k ∈ N, and build a measure λ k by putting, in every cell of the grid, all the mass of λ in a single point of the cell. This measure λ k is atomic and we have

G α (λ k ) = i λ(S i ) α ≤ i µ(S i ) = µ(Ω) < +∞,
where the S i are the cells of the grid. If we let k goes to ∞, the step of the grid goes to zero and we obviously have λ k ⇀ λ. On the other hand, the functional G α is lower semicontinuous (see [START_REF] Bouchitté | New lower semicontinuity results for nonconvex functionals defined on measures[END_REF]) and this implies

G α (λ) ≤ lim inf k→∞ G α (λ k ) ≤ µ(Ω) < +∞.
In particular, λ is atomic, thus proving the assertion.

Proof of Theorem 2.6

In order to prove existence, one only needs to take a minimizing sequence and apply Theorem 2.4 to get a new minimizing sequence which is τ -converging: this new sequence is obtained through reparametrization (which does not change the value of F) and by extracting a subsequence. Since the constraints in the problem are linear, i.e. ρ i = µ i for i = 0, 1 and the continuity equation, the limit (ρ, q) will satisfy the same constraints as well. The semicontinuity proven in Theorem 2.5 allows to obtain the existence of a solution.

Equivalence with previous models

In this section we prove the equivalence of problem (2.5) with the other previous formulations of branched transport problems, existing in literature. In particular, as a reference model we will take the one presented in [START_REF] Bernot | Optimal transportation networks. Models and theory[END_REF], in which the energy is defined as

E α (Q) = C 1 0 |σ(t)| α-1 Q |σ ′ (t)| dt dQ(σ),
where C = C([0, 1]; Ω), Q is probability measure over C and concentrated on the set Lip([0, 1]; Ω) (traffic plan) and for every x ∈ Ω, the quantity |x| Q is the multiplicity of x with respect to Q, defined by

|x| Q = Q ({σ ∈ C : x ∈ σ([0, 1])}) .
Given µ 0 , µ 1 ∈ P(Ω), the corresponding minimum problem is then given by

d α (µ 0 , µ 1 ) = min Q∈T P (µ 0 ,µ 1 ) E α (Q),
where T P (µ 0 , µ 1 ) is the set of traffic plans with prescribed time marginals at t = 0, 1, that is

T P (µ 0 , µ 1 ) = {Q ∈ C : Q concentrated on Lip([0, 1]; Ω), (e i ) ♯ Q = µ i , i = 0, 1},
and e t : C → Ω is the evaluation map at time t, given by e t (σ) = σ(t) for every σ ∈ C.

Remark 4.1. We point out that this model is completely equivalent to the one developed by Xia (see [START_REF] Xia | Optimal paths related to transport problems[END_REF] for the presentation of the model and [START_REF] Bernot | Optimal transportation networks. Models and theory[END_REF], Chapter 9, for the equivalence), which is based on a relaxation procedure, starting from an energy defined on finitely atomic probability measures µ 0 and µ 1 . In particular, thanks to this relaxed formulation, we get that for every µ 0 and µ 1 , there exist two sequences µ n 0 and µ n 1 of finitely atomic probability measures, weakly converging to µ 0 and µ 1 respectively, and such that

d α (µ n 0 , µ n 1 ) → d α (µ 0 , µ 1 ). ( 4.1)
We also need to consider a slight modification of the functional E α above, introduced in [START_REF] Bernot | Figalli: Synchronized traffic plans and stability of optima[END_REF],

C α (Q) = C 1 0 |(σ(t), t)| α-1 Q |σ ′ (t)| dt dQ(σ),
where now the synchronized multiplicity

|(x, t)| Q is |(x, t)| Q = Q({σ ∈ C : σ(t) = x}).
This second multiplicity accounts for the quantity of curves passing at the same time through the same point, while the one used in the definition of E α considered all the curves passing eventually through the point: in this sense, the model corresponding to the energy C α is more dynamical in spirit. As a straightforward consequence of the definition of the two multiplicities, we get

|σ(t)| Q ≥ |(σ(t), t)| Q , (4.2)
so that E α (Q) ≤ C α (Q). Concerning the comparison between the minimization of E α and C α , we recall the following result (see [START_REF] Bernot | Figalli: Synchronized traffic plans and stability of optima[END_REF], Theorem 5.1).

Theorem 4.2. Let µ 0 , µ 1 ∈ P(Ω), with µ 0 a finite sum of Dirac masses. Then for every α ∈ [0, 1] we get min Q∈T P (µ 0 ,µ 1 )

E α (Q) = min Q∈T P (µ 0 ,µ 1 ) C α (Q).
We are now in a position to state and prove a result giving the equivalence between our model and the one relative to the energy E α . Theorem 4.3. For every α ∈ (0, 1) and µ 0 , µ 1 ∈ P(Ω) we get

B α (µ 0 , µ 1 ) = min Q∈T P (µ 0 ,µ 1 ) E α (Q) = d α (µ 0 , µ 1 ). (4.3)
Proof. We first prove the inequality B α (µ 0 , µ 1 ) ≥ d α (µ 0 , µ 1 ). Clearly, if B α (µ 0 , µ 1 ) = +∞ there is nothing to prove; otherwise, take (ρ, q) optimal, which implies, by the way, that q = v • ρ and that q is atomic. Thanks to the superposition principle (see [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measure[END_REF], Theorem 8.2.1) we can construct a probability measure Q ∈ C such that ρ t = (e t ) ♯ Q and Q is concentrated on absolutely continuous integral curves of v, in the sense that

C σ(t) -σ(0) - t 0 v s (σ(s)) ds dQ(σ) = 0, for every t ∈ [0, 1].
Using this information, together with the fact that E α ≤ C α and exchanging the order of integration, we get

E α (Q) ≤ C α (Q) = C 1 0 |(σ(t), t)| α-1 Q |σ ′ (t)| dt dQ(σ) = 1 0 C |(σ(t), t)| α-1 Q |σ ′ (t)| dQ(σ) dt = 1 0 C |(σ(t), t)| α-1 Q |v t (σ(t))| dQ(σ) dt = 1 0 Ω |(x, t)| α-1 Q |v t (x)| dρ t (x) dt.
Then we observe that, by virtue of the fact that ρ t = (e t ) ♯ Q, there holds

|(x, t)| Q = Q({σ ∈ C : σ(t) = x}) = ρ t ({x}),
so that we can rewrite the last integral as

1 0 Ω ρ t ({x}) α-1 |v t (x)| dρ t (x) dt = 1 0 ρ t ({x}) α-1 d|q t |(x) dt = 1 0 i∈N |v t,i |ρ α t,i dt, which then gives d α (µ 0 , µ 1 ) = min Q∈T P (µ 0 ,µ 1 ) E α (Q) ≤ F(ρ, q) = B α (µ 0 , µ 1 ).
In order to prove the reverse inequality, we first prove that

B α (µ 0 , µ 1 ) ≤ min Q∈T P (µ 0 ,µ 1 ) C α (Q). (4.4)
Take Q ∈ T P (µ 0 , µ 1 ) optimal for C α , then we know that there exists a pair (ρ, q) which is a solution of the continuity equation, with ρ t = (e t ) ♯ Q and q t = v t • ρ t . The velocity v may be chosen as

v t (x) = σ ′ (t) dQ t,x (σ),
where Q t,x is the disintegration of Q with respect to the evaluation function e t (see [START_REF] Lisini | Characterization of absolutely continuous curves in Wasserstein spaces[END_REF] for this representation formula of the velocity field v). This means that each Q t,x is a probability measure concentrated on the set {σ ∈ C : σ(t) = x} and Q = Q t,x dρ t (x). Therefore, arguing as before

C α (Q) = 1 0 C |(σ(t), t)| α-1 Q |σ ′ (t)| dQ(σ) dt = 1 0 Ω |(x, t)| α-1 Q |σ ′ (t)| dQ t,x (σ) dρ t (x) dt ≥ 1 0 Ω |(x, t)| α-1 Q |v t (x)| dρ t (x) dt = 1 0 Ω ρ t ({x}) α-1 Q |v t (x)| dρ t (x) dt,
that gives the desired inequality (4.4) since, even if we do not know that q t or ρ t are atomic we can restrict the last integral to the set of atoms of ρ.

Summarizing, up to now we have shown

d α (µ 0 , µ 1 ) ≤ B α (µ 0 , µ 1 ) ≤ min Q∈T P (µ 0 ,µ 1 ) C α (Q),
and equality holds whenever µ 0 is a finite sum of Dirac masses, thanks to Theorem 4.2. In order to conclude, it is enough to notice that thanks to Remark 4.1, we may take two sequences µ n 0 and µ n 1 of finitely atomic probability measures such that µ n 0 ⇀ µ 0 , µ n 1 ⇀ µ 1 and

d α (µ n 0 , µ n 1 ) → d α (µ 0 , µ 1 ), thus getting d α (µ 0 , µ 1 ) ≤ B α (µ 0 , µ 1 ) ≤ lim inf n→∞ B α (µ n 0 , µ n 1 ) ≤ lim n→∞ d α (µ n 0 , µ n 1 ) = d α (µ 0 , µ 1 ),
hence concluding the proof.

Remark 4.4. Observe that in the previous Theorem, we did not only prove the equality of the minima, but we also provided a natural way to pass from a minimizer of our formulation à la Benamou-Brenier to a minimizer of the traffic plans model and back. The two problems are thus equivalent in the sense that they describe the same kind of energy and the same optimal structures of branched transport: the simple equality of the minima (4.3) is just a consequence of this more important fact.

Appendix: the distances d α and W 1/α This last section is devoted to estimates between the distance d α induced by the branched transport and the Wasserstein distances W p . In particular, in [START_REF] Morel | Comparison of distances between measures[END_REF] the following estimates are proven for α > 1 -1/d and p ≥ 1:

d α ≤ C W d(α-1)+1 p .
As far as lower bounds on d α are concerned, the most trivial one is d α ≥ W 1 but [START_REF] Devillanova | Elementary properties of optimal irrigation patterns[END_REF], Theorem 8.1, also proves d α ≥ W 1/α , which is slightly better. Moreover, for scaling reasons (w.r.t. the mass) it is not possible to go beyond p = 1/α in this last inequality.

In this paper we already needed to estimate some branched transport cost in terms of W 1/α distances and metric derivatives. In this section we prove the inequalities

W 1/α ≤ d α ≤ C W d(α-1)+1 1/α ∀α ∈ (1 -1/d, 1].
These inequalities are just a particular case of those that are already known. We restrict our attention to p = 1/α in the first one and p ≥ 1/α in the second one (as we said, they are proven in [START_REF] Devillanova | Elementary properties of optimal irrigation patterns[END_REF][START_REF] Morel | Comparison of distances between measures[END_REF]), but the proof we will provide is different and somehow simpler.

The first inequality will be approached through the formulation of branched transport we gave in this paper, but the main tool (i.e. inequality (3.1)) is essentially in common with [START_REF] Devillanova | Elementary properties of optimal irrigation patterns[END_REF] and [START_REF] Maddalena | Transport distances and irrigation models[END_REF]. What is different is the way to extend this idea to generic measures, i.e. non-atomic ones. Theorem 4.5. For every µ 0 , µ 1 ∈ P(Ω) we get

W 1/α (µ 0 , µ 1 ) ≤ d α (µ 0 , µ 1 ). ( 4

.5)

Proof. Let us consider an optimal transport γ j between a j (µ 0 ) and a j (µ 1 ), for the cost c(x, y) = |x -y| 1/α , that is γ j ∈ P(Ω × Ω) and it is of the form

γ j = 2 jd i,k=1 M j (i, k)δ x i j ⊗ δ x k j ,
with the 2 jd × 2 jd matrix {M j (i, k)} i,k belonging to the convex set M given by

M =    {a i,k } i,k : a i,k ≥ 0, 2 jd i=1 a i,k = µ 1 (Q k j ), 2 jd k=1 a i,k = µ 0 (Q i j )    .
We know by optimality that {M j (i, k)} i,k can be taken to belong to Ext (M), the set of extremal points of M, which consists of the so-called acyclic matrices (see [START_REF] Denny | The support of discrete extremal measures with given marginals[END_REF]). They are those matrices belonging to M such that the following property holds:

s r=1
a irkr a irk r+1 = 0, for every 2 ≤ s ≤ 2 jd and every set of indices i 1 < • • • < i s ∈ {1, . . . , 2 jd }, k 1 < • • • < k j ∈ {1, . . . , 2 jd } (the convention i 2 jd +1 = i 1 and k 2 jd +1 = k 1 is used). This implies in particular that #{(i, k) :

M j (i, k) = 0} ≤ 2 • 2 jd , (4.8)
that is {M j (i, k)} i,k has at most 2 • 2 jd non-zero entries: in other terms, this optimal transport plan γ j does not move more than 2 • 2 jd atoms. Setting |x i j -x k j | = ℓ i,k , we then get Proof. It is enough to show the validity of (4.9) for p = 1/α, then the general case will be just a consequence of the monotonicity property of the Wasserstein distances, i.e.

W 1/α ≤ W p , for every p ≥ 1/α.

Using the triangular inequality, (4.6) and (4.7), we get for every j ∈ N d α (µ 0 , µ 1 ) ≤ d α (µ 0 , a j (µ 0 )) + d α (a j (µ 0 ), a j (µ 1 )) + d α (a j (µ 1 ), µ 1 )

≤ C 2 (d(1-α)-1)j + d α (a j (µ 0 ), a j (µ 1 ))

≤ C 2 (d(1-α)-1)j + C W 1/α (a j (µ 0 ), a j (µ 1 )) 2 jd(1-α) , and W 1/α (a j (µ 0 ), a j (µ 1 )) ≤ W 1/α (a j (µ 0 ), µ 0 ) + W 1/α (µ 0 , µ 1 ) + W 1/α (a j (µ 0 ), µ 1 )

≤ C 2 -j + W 1/α (µ 0 , µ 1 ), which finally gives d α (µ 0 , µ 1 ) ≤ C 2 (d(1-α)-1)j + C W 1/α (µ 0 , µ 1 ) 2 jd(1-α)

= C 2 (d(1-α)-1)j 1 + W 1/α (µ 0 , µ 1 ) 2 j .

It is now sufficient to choose the index j in such a way that diam(Ω) 2 j ≤ W 1/α (µ 0 , µ 1 ) ≤ diam(Ω) 2 j-1 , which in turn yields 2 (d(1-α)-1)j (1 + W 1/α (µ 0 , µ 1 )2 j ) ≤ C W 1/α (µ 0 , µ 1 ) d(α-1)+1 , thus giving the thesis.

Remark 4.9. As we briefly mentioned, observe that the distances d α and W 1/α have exactly the same scaling with respect to the mass.

Remark 4.10. We point out that the very same µ 0 and µ 1 of Example 6.19 in [START_REF] Bernot | Optimal transportation networks. Models and theory[END_REF] show that the exponent d(α -1) + 1 in inequality (4.9) cannot be improved.

W 1 / 8 )Theorem 4 . 8 .

 1848 α (a j (µ 0 ), a j (µ 1 )) = and Jensen's inequalityd α (a j (µ 0 ), a j (µ 1 )) ≤ 2 jd i,k=1 M j (i, k) α ℓ i,k = (i, k) : M j (i, k) = 0}) 1-α ≤ C W 1/α (a j (µ 0 ), a j (µ 1 )) 2 jd(1-α) , concluding the proof. Let α ∈ (1 -1/d, 1], then for every p ≥ 1/α, we get d α (µ 0 , µ 1 ) ≤ C W p (µ 0 , µ 1 ) d(α-1)+1 ,(4.9)with a constant C depending only on d, α and on the diameter of Ω.

Acknowledgements. The authors acknowledge the support of Agence Nationale de la Recherche via the research project OTARIE, of the Université Franco-Italienne via the mobility program Galilée "Allocation et Exploitation et Evolution Optimales des Ressources: réseaux, points et densités, modèles discrets et continus" as well of the Università di Pisa through the program Cooperazione Accademica Internazionale "Optimal trasportation and related topics". The first author has been partially supported by the European Research Council under FP7, Advanced Grant n. 226234 "Analytic Techniques for Geometric and Functional Inequalities".

Proof. We first observe that thanks to the results of the previous section, for every µ 0 , µ 1 ∈ P(Ω) we get

for a suitable (ρ, q) admissible in the formulation (2.5), with q = ρv. Moreover, using once more the inequality (3.1) the right-hand side in the previous expression can be estimated as

and finally, using the fact that (ρ, q) is solution of the continuity equation, we can infer (see [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measure[END_REF], Theorem 8.3.1)

where in the last inequality we just estimated the length of a curve by the distance between its endpoints. Thus we have obtained (4.5), concluding the proof.

In order to prove the other inequality, first of all we have to introduce some notations: we set Q = [0, 1) d and Q L = [0, L) d , for every j ∈ N we consider the following subset of multi-indexes

and observe that #(B j ) = 2 jd , then we make a partition of the cube Q L by dyadic cubes having edge length L/2 j , i.e.

For every µ ∈ P(Ω) such that Ω ⊂ Q L , its dyadic approximation is given by

where µ i j = µ(Q i j ) and x i j is the center of Q i j . We shall always assume that Ω ⊂ Q L for a suitable L, then the following estimate is well-known (Proposition 6.6, [START_REF] Bernot | Optimal transportation networks. Models and theory[END_REF]). Proposition 4.6. Let α ∈ (1 -1/d, 1], then for every µ ∈ P(Ω) we have

The main tool if one wans to estimate d α from above by a power of W p is to show that the distance d α between two dyadic approximations can be estimated in terms of their 1/α-Wasserstein distance: this is the content of the next result. Lemma 4.7. Let α ∈ (1 -1/d, 1], then for every µ 0 , µ 1 ∈ P(Ω) we get d α (a j (µ 0 ), a j (µ 1 )) ≤ C W 1/α (a j (µ 0 ), a j (µ 1 )) 2 jd(1-α) , (4.7)

with C depending only on N and α.