Nishanth Gudapati
email: nishanth.gudapati@yale.edu

A NOTE ON THE DIMENSIONAL REDUCTION OF AXISYMMETRIC SPACETIMES

We investigate the dimensional reduction of 3+1 vacuum axisymmetric Einstein's equations to 2+1 dimensional Einstein-wave map system and observe that the resulting system is 1) not asymptotically flat, 2) its geometricmass diverges and 3) the energy of wave map also diverges. Subsequently, we discuss the consequences of these issues.

Background and Preliminaries

Let (M, g) be a 2+1 dimensional globally hyperbolic, Lorentzian manifold and further suppose that (M, g) admits a 2+1 decomposition such that it is foliated by a family of Cauchy hypersurfaces (Σ t , q t ). Let us assume that each (Σ t , q t ) is asymptotically flat in the sense of Ashtekar-Varadarajan [START_REF] Ashtekar | A striking property of the gravitational Hamiltonion[END_REF]:

q ab = r -m AV (δ ab + O(r -1 )), (1)
in the asymptotic region where δ ab is the 2d flat metric. It was long known that the notion of energy in general relativity is most reasonable for an isolated self-gravitating system that converges to the trivial solution asymptotically. In 1960s, following contributions from Dirac, ADM have demonstrated that energy at an instance can be measured in terms of the deviation from this trivial solution in the asymptotic region. The underlying notion is that the curvature of the asymptotically flat 3-manifold introduces a deviation from flat space which can be measured canonically at infinity with the trivial solution as the reference. Several fundamental results have followed which established positivity, consistency and convergence of this energy [START_REF] Schoen | On the proof of the positive mass conjecture in general relativity[END_REF][START_REF] Schoen | Proof of the positive mass theorem II[END_REF][START_REF] Witten | A new proof of the positive energy theorem[END_REF][START_REF] Bartnik | The mass of an asymptotically flat manifold[END_REF]. In another foundational result, which shall play a crucial role in our analysis, Ashtekar and Varadarajan [START_REF] Ashtekar | A striking property of the gravitational Hamiltonion[END_REF] have established that m AV is the natural Hamiltonian analogue of ADM-mass for the 2+1 case and that m AV is non-negative with m AV = 0 iff q is flat. Now suppose we are interested in the Einstein-wave map system in 2+1 dimensions. As we shall see later, the reason we are considering 2+1 Einstein-wave system is that it occurs in vacuum 3+1 Einstein's equations.

E µν =T µν on (M, g)

g U i + (h) Γ i jk (U )g µν ∂ µ U j ∂ ν U k =0 on (M, g), ( 2
)
where U is the wave map E is the Einstein-tensor of (M, g), T is the energy-momentum tensor for wave maps:

U : (M, g) → (N, h), (3
T µν = ∂ µ U, ∂ ν U h - 1 2 g µν ∂ σ U, ∂ σ U
g is the covariant wave operator, Γ's are the Christoffel symbols of the target (N, h). For demonstration purposes, let us restrict to the case where (N, h) is a hyperbolic 2-plane, which can also be interpreted a surface of revolution with the generating function f . As in [START_REF] Gudapati | The Cauchy problem for energy critical self-gravitating wave maps[END_REF], let us further restrict to the case where (M, g) is rotationally symmetric and the map U is an equivariant wave map, which roughly means that the map commutes with the rotational symmetries of (M, g), (N, h) and thus partially decouples the wave maps equation to give the system:

E µν =T µν g u = f u (u)f (u) r 2 . ( 4
)
Suppose (M, g) can be expressed in the following form in polar coordinates (t, r, θ):

ds 2 g = -e 2Ω(t,r) dt 2 + e 2γ(t,r) dr 2 + r 2 dθ 2 (5)
with γ, ∂ r γ and Ω = 0 on this axis Γ. The spatial part is

q = e 2γ dr 2 + r 2 dθ 2 (6) with r ∈ [0, ∞) and θ ∈ [0, 2π). Suppose γ ∞ = lim r→∞ γ
In view of our assumption on asymptotic flatness of q the limit above converges. Therefore, the form of the metric at the spatial asymptotic region is given by

q ∞ : = e 2γ∞ dr 2 ∞ + r 2 ∞ dθ 2 (7)
where r ∞ is the 2d Euclidean radius function.

Angle-deficit. In order to compare the asymptotic behaviour of our metric at spatial infinity to the flat metric let us introduce the variables ρ : = e γ∞ r ∞ and ϑ : = e -γ∞ θ with 0 ≤ ϑ < 2πe -γ∞ . [START_REF] Heusler | Stationary black holes: Uniqueness and beyond[END_REF] So that q ∞ can be represented in the form of the flat metric in the asymptotic region as

q ∞ = dρ 2 + ρ 2 dϑ 2 , 0 ≤ ϑ < 2πe -γ∞ . (9)
Thus the angle deficit at infinity is given by ∂θ ∞ : = 2π(1 -e -γ∞ ) [START_REF] Schoen | On the proof of the positive mass conjecture in general relativity[END_REF] As we shall see next, this notion is streamlined by Ashtekar-Varadarajan [START_REF] Ashtekar | A striking property of the gravitational Hamiltonion[END_REF].

AV-Mass. In order the deduce the AV-mass of our system, we need to rewrite our form of the metric in the AV-form. Let us define the asymptotic Cartesian coordinates x = r ∞ cos θ and y = r ∞ sin θ, so that the flat metric is

δ ab = ∇ a x∇ b x + ∇ a y∇ b y (11)
Let us introduce the variable such that our metric q can be expressed in the form

q ab = -m AV (δ ab + O( -1 )) in the asymptotic region. Suppose 2-m AV = r 2 ∞ (12) so that dr ∞ = 2-m AV 2 - m AV 2 d which implies that e 2γ∞ = 2 2-m AV , so m AV = 2(1 -e -γ∞ ). ( 13
)
We shall demonstrate later that γ, which is 0 on the axis Γ, is a monotonically increasing function with respect to r. This means that the AV-mass is also bounded from above m AV < 2.

Energy. Let N is the unit time-like normal of the (Σ, q) → (M, g) embedding, then the Einstein-Hamiltonian constraint is formally given by

R q + (K a a ) 2 -K ab K ab = T(N , N ) which is essentially the equation, E(N , N ) = T(N , N )
reduced by the Gauss-Kodazzi equation. The Einstein-Hamiltonian constraint shall play an important role in our analysis as it captures the energy in the system. But before we move on to the energy let us briefly discuss more details about wave maps. Wave maps are critical points of a functional that is a natural geometric generalization of harmonic maps:

S W M : = M L W M μg : = 1 2 M g µν h ij ∂ µ U i ∂ ν U j μg . (14)
In the case that we considered in (4), we have a wave map that is evolving with respect to a time function on the manifold (M, g) which has the wave map itself as the source. As the wave map is propagating in time its canonical stress energies are encloded in the energy-momentum tensor

T µν : = ∂L W M ∂g µν -g µν L W M . (15)
Then the energy is given by

E(t) : = Σt T(N , N ) μq (16) If we were in the Cartesian Minkowski space R 2+1 , E(t) = 1 2 R 2 ∂ t U 2 h + ∇ x U 2 h d 2 x (17)
In our case,

E(t) = Σt T(N , N ) √ q drdθ = 1 2 Σt N U h + RU h + 1 r ∂ θ U h √ q drdθ, R : = e -γ ∂ r = 1 2 Σt e -2Ω (∂ t u) 2 + e -2γ (∂ r u) 2 + f 2 (u) r 2 √ q drdθ (18)
Crucially, It should be noted that a priori we can impose conditions on the Cauchy slice so that E(t) converges. Now let us return to the Hamiltonian constraint: A calculation shows that the Einstein tensor is

E(N , N ) = e -2γ r -1 ∂ r γ (19)
Thus the Hamiltonian constraint gives

e -2γ r -1 ∂ r γ = 1 2 e -2Ω (∂ t u) 2 + e 2γ (∂ r u) 2 + f 2 (u) r 2 (20)
This means

e -γ ∂ r γ = re γ 1 2 e -2Ω (∂ t u) 2 + e 2γ (∂ r u) 2 + f 2 (u) r 2 -∂ r (e -γ ) = re γ 1 2 e -2Ω (∂ t u) 2 + e 2γ (∂ r u) 2 + f 2 (u) r 2 (21)
Integrating with respect to r, we shall obtain

1 -e γ = 1 4π r 0 e -2Ω (∂ t u) 2 + e -2γ (∂ r u) 2 + f 2 (u) r 2 √ q drdθ (22)
The equation shows that γ is monotonically increasing with respect to r

0 ≤ γ ≤ γ ∞ Further,(22) implies 1 -e γ∞ = 1 2π E(t) (23)
So that

e γ∞ = 1 - 1 2π E(t) -1 (24)
Combining them all,

∂θ ∞ = πm AV = E(t). ( 25
)
Thus we have established that there is a neat correlation between all these quantities. Weaker conservation laws associated to these quantities were useful in [START_REF] Gudapati | The Cauchy problem for energy critical self-gravitating wave maps[END_REF] to obtain non-concentration for the system for large data.

In the last few decades there has been an impressive progress in the technical aspects of dispersive and geometric PDE. Given this background, during his PhD work the author had observed and reported to the wider community that the rich structure due to the scaling symmetries in the 2+1 Einstein-wave map system can be used to obtain hitherto inaccessible results in the context of the initial value problem of Einstein's equations for general relativity, especially for large data. To this end, global existence for large data was obtained for equivariant Einstein-map system by building on the preceding nonconcentration result in [START_REF] Gudapati | The Cauchy problem for energy critical self-gravitating wave maps[END_REF], which marked the first instance of a global regularity result for large data for noncompact Cauchy hypersurfaces with one Killing field. In theory one can hope to use this formulation in the dynamical black hole stability problem, particularly since the Kerr-family admits a similar dimensional reduction. However, in this note we shall demonstrate that the Einstein-wave map approach to study dynamical axisymmetric spacetimes is fraught with significant foundational issues. These issues are in contrast with the translational symmetry case. The details are discussed in the next section.

3+1 Axisymmetric Spacetimes

Suppose ( M , ḡ) is a 3+1 dimensional globally hyperbolic Lorentzian spacetime that admits a foliation of 3d (Riemannian) Cauchy hypersurfaces which are asymptotically flat and its ADM-mass converges:

m ADM : = 1 16π lim r→∞ S 2 (r) δkl (∂ k qil -∂ i qlk ) x i r μS 2 (r) (26)
Now suppose ( Σ, q) is spatially rotationally symmetric: ∃ a space-like rotational Killing vector that has invariant points (Γ) and closed integral curves.

In the presence of Killing vectors, the Einstein's equations admit a reduction to 2+1 Einstein-wave map system. This is also applicable for a rotational Killing vector in the regular region. For demonstration purposes, let us go ahead and perform the dimensional reduction. Consider the following ansatz for ( M , ḡ) :

ḡµν dx µ dx ν = g µν dx µ dx ν + e 2u (dφ + A ν dx ν ) 2 for µ, ν = 0, 1, 2, 3. ( 27
)
where ∂ φ is the rotational Killing vector, g is the metric of (M, g) : ( M , ḡ) \ SO [START_REF] Bartnik | The mass of an asymptotically flat manifold[END_REF] where the axis is identified by ∂ φ ḡ = 0 (with the convention e -∞ : = 0.) In the following ∇ denotes the covariant derivative defined in (M, g).

With the above ansatz, the 3+1 vacuum Einstein's equations for ( M , ḡ) Rµν = 0 (28) can be rewritten as

0 = Rµν =R µν -∇ µ u∇ ν u -∇ µ ∇ ν u - 1 2 e 2u F µσ F σ ν (29a) 0 = Rµ3 = - 1 2 e -u ∇ σ (e 3u F σ µ ) (29b) 0 = R33 = -e 2u (g µν ∇ µ ∇ ν u + g µν ∇ µ u∇ ν u - 1 4 e 2u F µν F µν ). ( 29c
)
where

F µν : = ∇ µ A ν -∇ ν A µ
is the Faraday tensor in the chosen coordinate frame. Before we move on, we shall need the following formulas concerning the conformal transformations of Ricci tensor and wave operator. Suppose g : = e 2ψ g

We have,

-g =e 3ψ √ -g (30a) g u = 1 √ -g ∂ ν -g gµν ∂ µ u = e -2ψ ( g u + g µν ∂ ν ψ ∂ µ u) (30b) Rµν =R µν -g µν ∇ σ ∇ σ ψ -∇ µ ∇ ν ψ + ∇ µ ψ∇ ν ψ -g µν ∇ σ ψ∇ σ ψ. (30c) µ, ν, σ = 0, 1, 2.
If we examine the equation (29b) carefully, we can interpret this as the closure of a 1-form. Define

G : = e 3ψ * F Then (29b) for Rµ3 = 0 implies d G = 0.
This in turn implies there exists a twist potential v such that

G = d v
The equation d F = 0 for the twist potential v can be transformed into the wave maps equation in the conformally transformed manifold ( M , g) with ψ = u.

∇ µ (e -3u g µν ∂ ν v) = 0 (31)
This equation constitutes one of the wave maps equations. Subsequently, a linear combination of the equations (29a) and (29c) rewritten in the conformally transformed ( M , g) result in the Einstein's equations and the other wave maps equation:

Ẽµν = Tµν (32a) ∇µ ∂ µ u + 1 2 e -4u g µν ∂ µ v∂ ν v = 0 (32b) ∇µ ∂ µ v -4 g µν ∂ µ u∂ ν v = 0. (32c)
To simplify our notation from now on, we shall denote ( M , g) by (M, g) itself.

So,

E µν =T µν on (M, g) g U i + (h) Γ i jk (U )g µν ∂ µ U j ∂ ν U k =0 on (M, g), ( 33a
)
where U is now the specific wave map

U : (M, g) → (H 2 , h). (34)
Therefore, using the rotational Killing vector ∂ φ we have superficially arrived at the elegant 2+1 Einstein-wave map system. In 1960s Ernst had arrived at a similar system but with a crucial difference (more later). Now let us perform the reduction for some well-known spacetimes.

Dimensional Reduction of Minkowski. Consider the 3+1 dimensional Minkowski spacetime:

ds 2 = -dt 2 + dr 2 + r 2 dθ 2 + r 2 sin 2 θdφ 2 , (35) θ ∈ [0, π], φ ∈ [0, 2π). The spatial part is q = dr 2 + r 2 dθ 2 + r 2 sin 2 θdφ 2 (36)
This can be rewritten in the dimensional reduction ansatz: q = e -2u q + e 2u Φ 2 (37) where e 2u : = r 2 sin 2 θ. To illustrate our main points, let us cut out the axis i.e., u = log(r sin θ) and Φ in hypersurface othogonal case is dφ q = r 2 sin 2 θ(dr 2 + r 2 dθ 2 )

The first immediate observation we have arrived at a manifold that has curvature from the flat Minkowski space using the dimensional reduction procedure. Furthermore, we note that this manifold is not asymptotically flat and has infinite curvature at statial infinity. To prove that its geometric-mass also diverges, consider the following equations resulting from the dimensional reduction

E µν =T µν on (M, g) g u =0 on (M, g) (38)
where g = r 2 sin 2 θ(-dt 2 + dr 2 + r 2 dθ 2 ) and

T µν = ∇ µ u∇ ν u - 1 2 g µν ∇ σ u∇ σ u.
The unit normal of the Σ → M embedding is N = (r sin θ) -1 ∂ t and √ q = r 3 sin 2 θ. Also note that

∂ σ u∂ σ u =g rr ∂ r u∂ r u + g θθ ∂ θ u∂ θ u = 1 r 4 sin 4 θ (39)
and

T tt = - 1 2 g tt ∂ σ u∂ σ u = 1 2r 2 sin 2 θ (40) thus T(N , N ) = 1 2r 4 sin 4 θ (41)
The energy of the resulting wave equation is

E(t) = Σt T(N , N ) √ q drdθ → ∞. ( 42
)
The fact that the simplest solution arising out of the axisymmetric dimensional reduction of R 3+1 exhibits such a rampant asymptotic behaviour, that neither correlates with its parent R 3+1 nor with R 2+1 when it is supposed to help in our construction as a ground-state solution, shall have important implications in what is to follow. Let us now consider Schwarzschild.

Dimensional Reduction of Schwarzschild. As an example also consider the Schwarzschild metric

ḡsh = -1 - 2m r dt 2 + 1 - 2m r -1 dr 2 + r 2 dω 2 S 2 . (43) For simplicity suppose f (r) = 1 -2m r , then the spatial part is qsh = f -1 dr 2 + r 2 dω 2 S 2 (44)
Subsequently, the reduced q sh can be expressed as

qsh = r 2 sin 2 θ f -1 dr 2 + r 2 dθ 2 . ( 45
)
The Schwarzschild spacetime is also an example of a ∂ φ hypersurface orthogonal spacetime, therefore the reduced equations are:

E µν =T µν on (M, g) g u =0 on (M, g) (46) where g = r 2 sin 2 θ(-f dt 2 + f -1 dr 2 + r 2 dθ 2 )
and

T µν = ∇ µ u∇ ν u - 1 2
g µν ∇ σ u∇ σ u, u = log r sin θ, like before. The parameter m coincides with m ADM defined earlier.

The curvature of (Σ sh , q sh ) also blows up at infinity and its geometric-mass diverges. Again consider T(N , N ) for N = (r sin θ √ f ) -1 ∂ t for r sufficiently large. We have,

∇ σ u∇ σ u =g rr ∂ r u∂ u r + g θθ ∂ θ u∂ θ u =f 1 r 4 sin 2 θ + cos 2 θ r 4 sin 4 θ (47) = 1 r 4 sin 2 θ f + cot 2 θ (48) = 1 r 4 sin 2 θ csc 2 θ - 2m r (49)
Subsequently,

E(t) = Σ T(N , N ) μq → ∞. ( 50
)
Dimensional Reduction of Kerr. In Boyer-Lindquist coordinates Kerr solution can be represented as

ḡk = -1 - 2M r A dt 2 - 4M ra sin 2 θ A dtdφ + B sin 2 θ A dφ 2 + A C dr 2 + A dθ 2 . ( 51
)
where

A = r 2 + a 2 cos 2 θ, B = r 2 + a 2 2 -Ca 2 sin 2 θ, C = r 2 -2M r + a 2 (52)
As ∂ φ is a rotational Killing vector of the Kerr metric, it also admits the ∂ φ dimensional reduction. In addition, it also admits stationarity in the asymptotic region, so in this region the time dependence of the wave maps equation drops out. In tune with our discussion, the wave map arising out of ∂ φ dimensional reduction of Kerr has infinite energy

E(U ) = Σ e μq (53)
This behaviour is irrespective of the issue at the axis and is unaffected by the ergoregion. Indeed, one can prove in general that the AV-mass of (Σ, q) obtained by the aforementioned dimensional reduction from ( Σ, q) shall always be divergent even though m ADM of ( Σ, q) converges.

Starting from Minkowski to the Kerr family, we have generated infinite energy (divergent) 2+1 systems. This is already an awry situation but let us analyze this further.

Firstly, we would like to point out the reason: by construction, the norm of the rotational Killing vector blows up at spatial infinity whereas there is no such restriction on the norm of the translational isometry. Indeed, a calculation shows that the equivalent dimensional reduction of R 3+1 using a translational Killing vector does not have any of these issues.

From a PDE perspective, wave maps are considered to be tractable structures as they are natural geometric generalizations of harmonic maps and linear wave equations. In addition, wave maps have better null structure than Einstein's equations. Since the Kerr family admits the ansatz (27), in principle the Kerr family is also a 'solution' to the 2+1 Einstein wave map system. Therefore, in theory one can consider the dynamical perturbations within the 2+1 Einstein-wave map class. However, in view of the previously illustrated divergences, this consideration becomes quite subtle. The details are explained below:

(1) When faced with such divergences, a natural resolution that comes to mind is to introduce weights to induce the desired 2+1 asymptotic behaviour. Importantly, we would like to emphasize that any geometrically consistent way of inducing the desired asymptotics is bound to disturb the wave map structure, thus undermining the whole construction centered around their elegance. (2) In another marked distinction with the equivariant Einstein-wave map system, the divergent Einstein-wave map energies mean that the 2+1 wave map stress-energy fluxes are not the honest carriers of the true stress-energy fluxes propagating in the parent 3+1 axisymmetric system.

(3) An issue that is somewhat independent of the aforementioned issues is the general issue of proving the decay of waves in 2+1 dimensions. This can be traced back to the failure of classical Huygens principle for waves in R 2+1 and the associated weaker decay. In view of the fact that the major open problems on stability of blackholes are directly related to optimal decay rates, this issue also assumes significance.

Ernst Equations

Separately, we would like to emphasize that the dynamical axisymmetric dimensional reduction has to be strictly contrasted with a related system of equations that admit the Schwarzschild and Kerr spacetimes as solutions, but arrived via stationarity asumption and subsequent imposition of axisymmetry. Suppose ( M , ḡ) is a 3+1 dimensional globally hyperbolic Lorentzian spacetime. (M, g) is called a stationary spacetime if there exits a timelike Killing vector T outside a compact set of the Cauchy hypersurfaces Σ S , qS . In our construction, in order to consider the physically relevant cases we would of course omit the case with orbiting or closed integral curves of T . Then consider the ansatz

1 ḡ = qS + e 2u S T 2 (54)
where T : = dt + (A S ) a dx a , a = 1, 2, 3.

The vacuum Einstein's equations in this case Rµν = 0 on (M, g) (55) can also be reduced in essentially a similar way, but now by using the time-like Killing vector from the Lorentzian 4-manifold to a Riemannian 3-manifold ( Σ, q) (see Geroch's projection formalism [START_REF] Geroch | A method for generating solutions of Einstein's equations I[END_REF][START_REF] Geroch | A method for generating solutions of Einstein's equations II[END_REF]) and a subsequent conformal transformation qS : = e 2u S qS As before, we shall denote qS by qS itself for simplicity. Now, given the reduced stationary Einstein's equations on ( ΣS , qS ), we can impose axisymmetry: Suppose there exists a rotational Killing vector ∂ φ , with closed orbits and fixed points. Then consider (Σ S , q S ) : = (Σ S , qS )\SO(2), we shall arrive at the following system of equations ∆ q S Ψ i + (h) Γ i jk q ab S ∂ a Ψ j ∂ b Ψ k = 0, (56) Ψ : (Σ S , q S ) → (H 2 , h), where (H 2 , h) is again the hyperbolic 2-plane, Γ's are its Christoffel symbols and ∆ q : is the Laplace-Beltrami operator corresponding the covariant derivative of (Σ, q). Ψ = (u S , v S ) where u S is related to the lapse and the twist v S is constructed from the shift vector.

Aside from the expected issue at the axis, this harmonic map is a well-defined PDE object and one can easily specify conditions so that it is of finite energy. Historically, these equations are called Ernst equations [START_REF] Ernst | New formulation of the axially symmetric gravitational field problem I[END_REF][START_REF] Ernst | New formulation of the axially symmetric gravitational field problem II[END_REF][START_REF] Kramer | The Ernst equation in general relativity[END_REF][START_REF] Heusler | Stationary black holes: Uniqueness and beyond[END_REF] A = r 2 + a 2 cos 2 θ, B = r 2 + a 2 2 -Ca 2 sin 2 θ, C = r 2 -2M r + a 2 , (58) one can see that the Kerr spacetime also admits the stationary reduction in the spatial asymptotic region.

In other words, even though the Kerr solution fits the ansätze for both the dimensional reductions, we have arrived at maps with very different properties. This means that the dimensional reduction procedures are not commutable in general.

These stationary equations, apart from their role during the discovery of Kerr spacetime, have been instrumental in several mathematical and physical results of much value and delight throughout the history. In addition, one can establish a beautiful relation between m AV of (Σ S , q S ) and m ADM of (Σ S , qS ). Therefore, it is perhaps advisable to clearly distinguish the Ernst equations with (33a).

  ) 2010 Mathematics Subject Classification. Primary: 83, Secondary 35. (NG) is supported by a Deutsche Forschungsgemeinschaft (DFG) Postdoctoral Fellowship GU1513/1-1 to Yale University.

  2 . If one recalls the Kerr metric 1 variants of this ansatz are called Weyl coordinates and Papapetrou coordinates 2 sometimes more generally without axisymmetry

	(57)	ḡk = -1 -	2M r A	dt 2 -	4M ra sin 2 θ A	dtdφ +	B sin 2 θ A	dφ 2 +	A C	dr 2 + A dθ 2 ,

Acknowledgements The author gratefully acknowledges the conversations with Abhay Ashtekar.


P E Vende

F Trinquet

S Lacour

A Delahaye

L Fournaison

INFLUENCE OF POSITION AND ORIENTATION OF WATER SPRAYING ON THE EFFICIENCY OF A HEAT EXCHANGER

Keywords: atomizer, spray cooling, impacted surface, low pressure, position

Spraying water in the upstream air flow of a heat exchanger improves the heat transfer. Two effects are part of this phenomenon: the spray evaporation that decreases the air temperature by adiabatic cooling and the spray impingement on the heat exchanger surface that increases the heat exchanger coefficient. In general, those effects don't apply on the entire heat exchanger surface. So, an experimental study has been carried out in order to compare the influence of several positions and orientations of water spraying on the efficiency of heat transfer. A pilot has been built allowing moving easily the atomizer injecting droplets in air while controlling flow rates, temperatures and humidity. Among the experimental conditions, it has been chosen a low pressure atomizer that produces a hollow cone spray. This choice has been done to keep in the meantime a good atomization (Sauter mean diameter of 50 µm), a low pressure (5 bar) and low water consumption (1 L/h). For each configuration tested, a global and a local study of the system have been made. The global study consists in an overall energy balance. Until now, the best overall energy balance reveals an increase of the heat transfer of 10% for a water flow rate injected of 1 L/h with a heat exchanger power of 5 kW. The local study consists, on one hand, in an energy balance on a pass of the heat exchanger and, on another hand, on the characterization of the impacted surface. The energy balance is based on the microchannels temperatures of the heat exchanger pass. The impacted surface is characterized by the local cooling in the downstream flow of the heat exchanger and a qualitative measure with an infrared camera. With all these results, an optimization of the system has been done and gave the best position and orientation of the atomizer to obtain the maximal heat transfer.

Introduction

According to the Intergovernmental Panel on Climate Change (IPCC), worldwide power consumption for air conditioning alone is forecast to surge 33 times by 2100 as developing world incomes rise and urbanisation advances [START_REF] Arent | Key economic sectors and services[END_REF]. Research by the Netherlands Environmental Assessment Agency predicts that the amount of energy used worldwide in cooling will overtake that used in heating by about 2060 [START_REF] Bollen | Co-benefits of climate policy[END_REF]. Moreover, the demand for residential air conditioning alone will rise from 300 TWh in 2000, to about 4,000 TWh in 2050 and more than 10,000 TWh in 2100 [START_REF] Arent | Key economic sectors and services[END_REF]. As a comparison, it is about all the electricity produced in 2013 by the United States and the Republic of China combined [START_REF]Key World Energy Statistics 2015[END_REF]. Solutions must be finding to increase the efficiency of air conditioning and refrigeration systems in order to reduce the cold domain consumption.

For several years now, one solution is studied and already used in industry: the water spraying on the condenser of a refrigeration unit. Indeed, several works [START_REF] Hajidavalloo | Performance improvement of air-cooled refrigeration system by using evaporatively cooled air condenser[END_REF][START_REF] Sarntichartsak | Modeling and experimental study on the performance of an inverter air conditioner using R-410A with evaporatively cooled condenser[END_REF][START_REF] Youbi-Idrissi | Numerical model of sprayed air cooled condenser coupled to refrigerating system[END_REF][START_REF] Yu | Improved energy performance of air-cooled chiller system with mist pre-cooling[END_REF] have shown that there is an improvement of heat exchanges due whether on the spray evaporation that decreases the air temperature upstream the condenser by adiabatic cooling or the spray impingement on the heat exchanger wall surface that increases locally the heat exchanger coefficient. Those two effects have their own advantages and drawbacks. On one hand, total spray evaporation upstream the heat exchanger avoids stagnant water and so any environment where bacteria and microbial organisms can develop. It also can improve the COP up to 50 % [START_REF] Sarntichartsak | Modeling and experimental study on the performance of an inverter air conditioner using R-410A with evaporatively cooled condenser[END_REF] and reduce the electric cost by around 18% [START_REF] Yu | Improved energy performance of air-cooled chiller system with mist pre-cooling[END_REF]. However, this solution requires the formation of a very fine spray with droplets diameters less than 20 µm [START_REF] Alkhedhair | Numerical simulation of water spray for pre-cooling of inlet air in natural draft dry cooling towers[END_REF]. Thus, it needs a high injection pressure (at least 50 bar) and a high energy demand. On the other hand, the spray impingement doesn't require much energy, but can cause some bacteriological problems with run-off and stagnating water. Several models and experimental measurements [START_REF] Freund | Investigation of local heat transfer coefficients in plate heat exchangers with temperature oscillation IR thermography and CFD[END_REF][START_REF] Naphon | Study on the heat transfer characteristics of an evaporative cooling tower[END_REF][START_REF] Webb | A Unified Theoretical Treatment for Thermal Analysis of Cooling Tower, Evaporative Condensers and Fluid Coolers[END_REF] have been done on wet wall heat transfer coefficients.

Irstea is one of the research institutes that is working on water application on the air flow upstream air condenser. Until now, works focus on maximizing the spray evaporation. Therefore, it has been shown numerically [START_REF] Youbi-Idrissi | Numerical model of sprayed air cooled condenser coupled to refrigerating system[END_REF] and validated experimentally [START_REF] Tissot | Experimental study on air cooling by spray in the upstream flow of a heat exchanger[END_REF] that the spray should be injected in a counter-flow situation and involves droplets with mean diameter below 25 µm. It has also been assessed that, for a counter flow injection, atomizers can be located at 5 cm away from the heat exchanger and increases the heat exchange by a factor 3. Nevertheless, unlike the actual work, the energy and water consumption were not primordial parameters for industrial applications.

In the present work, the spraying system must be water self-sufficient, thanks to the condensate water of the refrigerator-unit evaporator, in order to work in all atmospheric conditions. A numerical model has established that, in a French climate, condensate water of the evaporator produces an average of 1 L/h. Furthermore, the spraying system must consume as little energy as possible to reduce more the electric cost of air conditioning. For these reasons, the spraying system chosen work under low water flow and pressure conditions. With these parameters, the evaporation rate is limited (around 20% at 10 cm of the heat exchanger in the co-flow configuration) and so the spray impingement on the heat exchanger surface is predominant. Finally, the atomizer orifice has to be placed close to the surface of the heat exchanger (less than 5 cm) because of space constraint. As the surface of the heat exchanger is large (52 × 40 cm), the water spray will impact only partially the heat exchanger. The aim of this study is to determine the best position and orientation of the atomizer to obtain the maximal heat transfer.

The paper is organized as follows: the experimental setup will be described first with all the measurement devices and configurations tested, then experimental data will be discussed. The system will be first studied globally with an overall energy balance. Then, a discussion will be focused on the local heat exchanges at the pass level and impacted surface.

Experimental setup 2.1 Devices

Figure 1 -Schematic view of the experimental setup

The experimental setup is illustrated in Figure 1. It aims to optimize the heat exchange obtained with spraying by varying the air flow, the water flow and the position and orientation of the spraying device beside the heat exchanger wall. The heat exchanger used is a microchannels condenser in aluminium with four passes. As the work is mainly interested in the heat exchanges on the condenser surface, only this component of the refrigeration unit is used. Thus, a water flow is generated inside the condenser by a thermo regulator in a closed loop. As water is a single phase fluid, there is no phase change in the condenser. Inlet water is heated at 80°C much above the condensation temperature of the correspondent refrigerant fluid. In this way, like the real condenser, there is a hot spot at the top of the exchanger, on the first pass. A flowmeter measures the inlet flow rate in the heat exchanger and two Pt100 measure the inlet and outlet temperatures: these three sensors allows doing an overall energy balance of the system. The inlet flow rate is fixed at about 200 L/h. To cool the fluid condenser, an air flow is sucked through the heat exchanger by the downstream cooling fan. This air flow is measured thanks to a flow air sensor fan which frequency rotation is count by a tachymeter. The whole device is located in a room where the temperature and the humidity are controlled. All the experiments are realized at the same atmospheric set point: (25°C, 40%). Two hygrometers are located upstream and downstream the heat exchanger to measure and control air humidity with and without water spraying. Thermocouples are used in two situations. On one hand, they measure microchannels wall temperatures of the heat exchanger. For each pass of the condenser, there is one thermocouple at the inlet and one thermocouple every two microchannels at the outlet (see Figure 1). It allows doing a local energy balance on passes. On the other hand, thermocouples are used to measure air temperature downstream the heat exchanger. These thermocouples are on a grid that can move up and down, and left and right. They are used to know the local cooling downstream the heat exchanger caused by spraying, as well as the heating through the heat exchanger. The spraying system is composed of an atomizer and a pump. The pump brings water from a tank to the atomizer that pulverizes it into a spray. A continuous measurement of the tank weight is done with a precision balance in order to deduce the water flow. An infrared camera is used in order to know the impacted surface by the water spray. This camera takes pictures of the entire heat exchanger surface, with and without spraying. The images have a resolution of 320×240 pixels. Thus, 5 pixels correspond to 1 cm in both directions. The picture without spraying is used as a blank test and subtracted to the pictures with spraying to highlight the impacted surface and do pattern recognition by imaging processing with Python. This pattern recognition is done by binarizing grey-level images with the Otsu method (see Figure 2). In all these configurations, atomizer is parallel or with an angle of 45° to the surface of the heat exchanger. These orientations have been chosen as a compromise between counter-flow and co-flow configurations. Indeed, even if studies show that counter-flow configuration is the best way to inject water in the air, the spray takes too much space and goes back too far in the air flow for the application. The co-flow configuration is not a better way, because the spray angle is narrowed by the air flow and impacted a small surface. Atomizer must be closed to the surface of the heat exchanger. As a consequence, all the configurations are located on the edges of the heat exchanger: thus, the atomizer doesn't disturb air flow and the surface impacted by the spray can be easily seen by the infrared camera. For each of these eleven configurations, the influence of water flow and air flow rate on the heat exchange is studied. Two air flow speed and two water flow values are tested. For air flow, the two set points are named v1 and v2 in the rest of the paper and have values of respectively 420 m 3 /h and 775 m 3 /h. These set points correspond to the extreme operating conditions of the cooling fan. The water spraying is tested with two pump pressures: at 5 bar and 10 bar. These pressures correspond to water flow rate of respectively 1 L/h and 1.4 L/h and are named p5 and p10 in the rest of the paper.

Treatment

In the overall energy balance, a heat gain coefficient evaluates the global gain of the system due to spraying device. This spraying heat gain 𝜂 𝑒𝑥𝑝 (in %) is the ratio of the additional spraying heat flux and the heat exchanger heat flux without spraying. These quantities are given by relations:

𝑄 ℎ𝑒𝑥 𝑜𝑓𝑓 = 𝑚̇𝑓𝐶𝑝 𝑓 (𝑇 𝑖𝑛,𝑓 -𝑇 𝑜𝑢𝑡,𝑓 𝑜𝑓𝑓 ) (1)
∆𝑄 ℎ𝑒𝑥 𝑜𝑛 = 𝑚̇𝑓𝐶𝑝 𝑓 (𝑇 𝑜𝑢𝑡,𝑓 𝑜𝑓𝑓 -𝑇 𝑜𝑢𝑡,𝑓 𝑜𝑛 ) (2)
So the spraying heat gain 𝜂 𝑒𝑥𝑝 is:

𝜂 𝑒𝑥𝑝 = 𝑇 𝑜𝑢𝑡,𝑓 𝑜𝑓𝑓 -𝑇 𝑜𝑢𝑡,𝑓 𝑜𝑛 𝑇 𝑖𝑛,𝑓 -𝑇 𝑜𝑢𝑡,𝑓 𝑜𝑓𝑓 (3)
This experimental gain is compared, on one hand, to the total evaporation gain 𝜂 𝑒𝑣𝑎𝑝 obtained if all water is evaporated (and so all the latent energy is exploited) and on the other hand, to the saturation gain 𝜂 𝑠𝑎𝑡 obtained if the wall temperature of the heat exchanger on the impacted surface is equal to the air saturation temperature. These gains can be expressed as follow:

𝜂 𝑒𝑣𝑎𝑝 = 𝑚̇𝑤𝐿 𝑣 𝑄 ℎ𝑒𝑥 𝑜𝑓𝑓 (4)
𝜂 𝑠𝑎𝑡 = 𝑚̇𝑎𝐶𝑝 𝑎 𝑆 𝑖𝑚𝑝 𝑆 ℎ𝑒𝑥 • 𝑇 𝑎,𝑠𝑎𝑡 -𝑇 𝑎 𝑄 ℎ𝑒𝑥 𝑜𝑓𝑓 (5)
Finally, the water mass efficiency 𝜀 𝑤 is defined as the ratio of the spraying heat gain 𝜂 𝑒𝑥𝑝 on the water flow rate 𝑚̇𝑤 . This ratio allows knowing if it's better to spray a little or a lot of water.

𝜀 𝑤 = 𝜂 𝑒𝑥𝑝 𝑚̇𝑤 (6)
3 Results

44 configurations have been experimented: indeed, for the eleven positions chosen (see Figure ), two air and water flows have been tested. First of all, let's see how the mean impacted surfaces and water efficiencies of these eleven positions are influenced by the air and water flow rates. 1, the mean impacted surfaces 𝑆 𝑖𝑚𝑝 ̅̅̅̅̅̅ and the mean water efficiencies 𝜀 𝑤 ̅̅̅ for all atomizer positions are given for the two air and water flow rates. It can be noticed that when the pressure is low (5 bar), the impacted surface is independent of air flow rate. On the contrary, at high pressure (10 bar), the mean impacted surface decreases from v1 to v2 by about 20%. This difference can be explained by the air flow speed that narrows the spray angle. When the water pressure injection varies, the impacted surface increases from low to high pressure. This rise is more important at low air flow (+55%) than at high air flow (+21%). This can be explained by two competitive effects: the augmentation of water pressure injection increases the spray angle but the augmentation of air flow decreases the spray angle. As regards the water efficiency, it can be observed that pressure doesn't change significantly the water efficiency: more water is even better. By contrast, increase air flow causes a drop of water efficiency by about 70%. No obvious correlation is found between impacted surface and water efficiency (or heat gain): it means that another key parameter has to be taken into account.

Figure 4 -Experimental, total evaporation and saturation gain at two air flow rates and p = 10 bar

The experimental heat gain 𝜂 𝑒𝑥𝑝 , the total evaporation gain 𝜂 𝑒𝑣𝑎𝑝 , and the saturation gain 𝜂 𝑠𝑎𝑡 obtained for the two air flow rates tested at p = 10 bar are presented in Figure . At v1, 𝜂 𝑒𝑥𝑝 varies between 4% and 8% depending on the position of the atomizer. Compared to the ideal gain 𝜂 𝑒𝑣𝑎𝑝 , 𝜂 𝑒𝑥𝑝 is divided by a factor 2 to 3. This is because there is only 10 to 20% of evaporation upstream the heat exchanger. On abscissa there is the heat exchanger heat flux without spraying. This heat flux is on average 6.25 kW at air flow speed v1 compared to 8 kW at v2. It means that the only increase of air flow speed allows increasing heat transfers at about 30%. In comparison, the maximum value of the theoretical evaporation gain at v1 is at about 15%. In theory, it would need about twice more water (~ 3 L/h) to evaporate to have the gain corresponding to double the air speed. In practice, as the maximal experimental heat gain is at about 8% at v1, it would need at least four times more water. Moreover, 𝜂 𝑒𝑥𝑝 is above 𝜂 𝑠𝑎𝑡 : it means that, experimentally, the gain is not only due to a decrease of the local air temperature to its saturation value. So, an additional benefit is provided by the formation of a water film: it increases the local heat exchange coefficient and adds to the adiabatic cooling effects. On the contrary, at v2, 𝜂 𝑒𝑥𝑝 is near 𝜂 𝑠𝑎𝑡 : the formation of the water film has quite no impact on the heat exchange. This difference is due to a smaller impacted surface at v2 (Table 1): as the water flow is the same than at v1, there is more water deposit by unit surface and so the water film thickness is greater. With this higher thickness, the film is less conductive: it acts like a thermal resistance that clogs the surface of the heat exchanger. This effect of thickness and clogging is also present in Figure . These positions correspond to different wall temperatures from the bottom of the heat exchanger (the coldest) to the top (the hottest). For p = 5 bar, 𝜂 𝑒𝑥𝑝 increases with the wall temperature. At low water flow (1 L/h), it's better to point hottest spots. On the contrary, for p = 10 bar, when the water flow is increased by 40%, 𝜂 𝑒𝑥𝑝 decreases with the wall temperature. So, with high water flow, it's better to point coldest spots. This contradiction is due to a different water deposit. At low pressure, the water film is thin and evaporates quickly. At high pressure the film is thick, and acts like a thermal resistance. So, for the hottest spots, it's better to evaporate as quickly as possible the water deposit contrary to the coldest spots where the presence of a water thermal resistance allows increasing heat exchanges.

In Figure , the temperature profiles are compared with and without spraying, for the positions 3 and 11 (see Figure ). They are plotted as function of the curvilinear abscissa. The definition of the curvilinear abscissa is also presented in Figure . On these figures, the curve named "off" correspond to the evolution of wall temperature without spraying 𝑇 𝑤𝑙 𝑜𝑓𝑓 , the one named "on" to the evolution with spraying 𝑇 𝑤𝑙 𝑜𝑛 . The difference "off -on" ∆𝑇 𝑤𝑙 is also reported. First of all, it can be noticed that the wall temperature decreasing is high for the two first passes and lower for the latter two: it has an exponential profile typical of microchannels heat exchangers. This is due to a difference between wall temperature and air temperature higher at the top of the heat exchanger: there is more heat exchanges at the top than at the bottom. For the configuration 3, the atomizer sprays water on the first pass: as a consequence, ∆𝑇 𝑤𝑙 rises on the first pass. Its value maintains on the second pass. But, at the third and four passes, ∆𝑇 𝑤𝑙 decreases slightly. This is due to the closer temperature approach between wall and air at the end of the heat exchanger. This phenomenon is also observed for the configuration 11 at the fourth pass. For this last configuration, the spraying zone is on the second and the third passes. The cooling is better than in the configuration 3. This can be explained because the fluid has less distance to travel between the spraying zone and the outlet of the heat exchanger. Thus, in configuration 3, fluid has to cross 3 passes after spraying zone before the outlet. In configuration 11, there is only one pass. So, the fluid warming due to less effective heat exchanges because of a smaller temperature pinch between wall and air is lower. This difference of cooling is reflected in the heat gain: for the configuration 3, it is only 5% against 8% in the 11.

Conclusion

An experimental study has been carried out in order to compare the influence of several positions and orientations of water spraying on the efficiency of heat transfer. A pilot has been built allowing moving easily the atomizer injecting droplets in air while controlling flow rates, temperatures and humidity. With this setup, it has been shown that the impacted surface by the spray depends mainly on the water pressure injection. The water efficiency depends on the air flow. But there is no obvious correlation between impacted surface and water efficiency. The position of the atomizer has an impact on the heat gained by spraying. Thus, with a low water flow, it's better to point the atomizer on the hottest spots, i.e. at the top on the heat exchanger. On the contrary, with a high water flow, it's better to point the atomizer on the coldest spots, at the bottom on the heat exchanger. This contradiction is due to a different water deposit: at low flow, the water film is thin and evaporates quick, unlike at high flow where it is thick, and acts like a thermal resistance. However, at the local scale, it has been observed that the temperature pinch between wall and air increases in the spraying zone, and then decreases along the remaining distance. This decrease is higher when the distance between the spraying zone and the outlet of the heat exchanger is more important. This loss of cooling is reflected in the heat gain with only 5% in the third configuration against 8% in the eleventh. The next step of this work will be to study the effect of several atomizers on the heat exchange, and also to assess the interest of intermittent spraying for reducing thermal resistance of film. 𝐶𝑝 Heat capacity (J • kg -

Figure 2 -

 2 Figure 2 -Thermal images without and with spraying, difference in grey scale levels and binarized

Figure 3 -

 3 Figure 3 -Atomizer positions and orientations on the heat exchanger

Figure 5 -

 5 Figure 5 -Experimental gain vs wall temperature at air flow rate v = v1 In Figure , 𝜂 𝑒𝑥𝑝 is plotted in function of the wall heat exchanger temperature at air flow rate v1 for the two pressures tested. The four lateral positions of atomizer (1,2,10,3) have been taken to study the wall temperature impact (see Figure ). The wall temperature heat exchanger temperatures considered are the inlet or outlet passes temperatures without spraying at the level of the corresponding atomizer.These positions correspond to different wall temperatures from the bottom of the heat exchanger (the coldest) to the top (the hottest). For p = 5 bar, 𝜂 𝑒𝑥𝑝 increases with the wall temperature. At low water flow (1 L/h), it's better to point hottest spots. On the contrary, for p = 10 bar, when the water flow is increased by 40%, 𝜂 𝑒𝑥𝑝 decreases with the wall temperature. So, with high water flow, it's better to point coldest spots. This contradiction is due to a different water deposit. At low pressure, the water film is thin and evaporates quickly. At high pressure the film is thick, and acts like a thermal resistance. So, for the hottest spots, it's better to evaporate as quickly as possible the water deposit contrary to the coldest spots where the presence of a water thermal resistance allows increasing heat exchanges.

Figure 6 -

 6 Figure 6 -Evolution of wall temperature of heat exchanger microchannels passes with and without spraying in the configurations 3 (a) and 11 (b) at v = v1 and p = 10 bar

Nomenclature𝑄

  Heat flux (W) ∆𝑇 Difference of temperature (°C) ∆𝑄 Additional heat flux (W) 𝑆 Exchange surface (cm 2 )

Table 1 -

 1 Mean impacted surfaces and water efficiencies

		𝑺 𝒊𝒎𝒑 ̅̅̅̅̅̅ (𝒄𝒎 𝟐 )	𝜺 𝒘 ̅̅̅̅(% • 𝑳 -𝟏 • 𝒉)
		v1	v2	v1	v2
	p5	141	143	4.9	1.3
	p10	219	173	4.7	1.2
	In Table

Acknowledgements

The authors are grateful to the ADEME for the financial support.

P.E. Vende, F. Trinquet, S. Lacour, A. Delahaye, L. Fournaison


Mohammad Rouhani

Florent Lafarge

Pierre Alliez

Semantic Segmentation of 3D Textured Meshes for Urban Scene Analysis

Keywords: Semantic Segmentation, Textured Meshes, Urban Scene, Markov Random Fields, Random Forest, Joint Label Prediction, Contextual Information

Classifying 3D measurement data has become a core problem in photogrammetry and 3D computer vision, since the rise of modern multiview geometry techniques, combined with affordable range sensors. We introduce a Markov Random Field-based approach for segmenting textured meshes generated via multi-view stereo into urban classes of interest. The input mesh is first partitioned into small clusters, referred to as superfacets, from which geometric and photometric features are computed. A random forest is then trained to predict the class of each superfacet as well as its similarity with the neighboring superfacets. Similarity is used to assign the weights of the Markov Random Field pairwise-potential and accounts for contextual information between the classes. The experimental results illustrate the efficacy and accuracy of the proposed framework.

Introduction

The recent advances on Multi-View Stereo (MVS) imagery [START_REF] Seitz | A comparison and evaluation of multi-view stereo reconstruction algorithms[END_REF][START_REF] Vu | Towards high-resolution large-scale multiview[END_REF]) make it possible to generate in routine dense meshes from airborne images acquired on large-scale urban scenes. Several commercial solutions such as ContextCapture (Acute3D/Bentley) and Pix4Dmapper (Pix4D) generate meshes with greater geometric accuracy and completeness than the common digital surface models. Contrary to LIDAR scans, such dense meshes represent 2-manifold surfaces, and do not require the interpolation of sampled points. As depicted by Figure 1, these meshes exhibits geometric details both on roofs and facades as MVS systems can altogether deal with oblique and vertical airborne images.

Dense meshes from multiview stereo imagery are relevant 3D representations for visualization-based applications such as navigation or augmented reality. They are however too raw for applications that require additional structure and semantic information to interpret the represented scenes. There is a dire need to recover the nature of the objects composing the scenes.

We propose a dense classification algorithm that infer the class of urban objects in such dense meshes. Departing from previous work, our approach leverages both radiometric and geometric information in a supervised manner. In addition, we learn the contextual knowledge required to recover additional coherence between the urban classes of interest.

Related Work

We review below the most recent and related works on semantic segmentation, with focus on methods dealing with meshes of urban scenes.

Classification

Many different classification approaches have been used in photogrammetry and computer vision in order to partition images or point clouds, and to identify the nature of each area. Classification approaches differ mainly in the level of supervision: Supervised algorithms require a training set to learn how to correctly classify data, whereas unsupervised approaches require tuning the model parameters. Classification approaches also differ in the use of spatial dependencies and contextual information. In Markov Random Fields (MRFs) and Conditional Random Fields (CRFs) for instance, a datum is not classified independently from the rest of the data: the classification decision relies upon non-local information that accounts for spatial consistency between neighboring areas.

Supervised learning. Among the many classification approaches, Texton Boost and Texton Forest can be directly applied to the input data without requiring any type of feature descriptor [START_REF] Shotton | Semantic texton forests for image categorization and segmentation[END_REF]. In addition, these methods are extremely fast as they avoid computing filter-bank responses. A multi-feature variant of TextonBoost [START_REF] Sengupta | Urban 3d semantic modelling using stereo vision[END_REF] is used for labeling each image of the stereo pair and fuse them into a scene. Both the image and geometric features have been used to train a JointBoost classifier [START_REF] Valentin | Mesh based semantic modelling for indoor and outdoor scenes[END_REF] for segmenting RGB-D images. [START_REF] Xiao | Multiple view semantic segmentation for street view images[END_REF] use a series of one-vs-all AdaBoost classifiers to perform multiview semantic segmentation. In contrast, [START_REF] Kalogerakis | Learning 3d mesh segmentation and labeling[END_REF] learn a label compatibility function between the neighboring segments of an input mesh through training a JointBoost classifier on the pairwise geometric features. Randomized decision forests (or Random Forests) are popular parametric classifiers for the segmentation and regression tasks (Zhang et al., 2010a). They are relevant for real-time applications as they generate label predictions very efficiently through performing few simple tests on the query data. [START_REF] Kähler | Efficient 3d scene labeling using fields of trees[END_REF] employ random forests for segmenting RGB-D images; the feature vectors describe photometric and geometric information for every segment and pair of segments.

MRFs and CRFs. MRF or CRF formulations usually rely on an energy minimization problem. The energy is commonly composed of two terms: a data term that measures the coherence of each datum with respect to a label, and a pairwise potential that favors label smoothness. A supervised classifier can be used as prediction function to model the unary data term of MRFs and CRFs models. The contextual information provides relevant clues for improving the results of semantic segmentation. Co-occurrence statistics are modeled in [START_REF] Ladicky | Inference methods for crfs with co-occurrence statistics[END_REF] through a global potential function recording which pairs of classes are likely to occur in the same image. [START_REF] Galleguillos | Object categorization using co-occurrence, location and appearance[END_REF] model the co-occurrence and relative locations through the following pairwise term: Four different types of relative locations are considered (above, bellow, inside and around ) to capture the spatial context through frequency matrices that record the likelihood of two labels to appear in a relative position. [START_REF] Myeong | Learning object relationships via graph-based context model[END_REF] propose a pairwise cost function based on a similarity graph that encodes the relationship between two regions through context links. A context learning algorithm estimates the strength of a context link in the query image. [START_REF] Yao | Describing the scene as a whole: Joint object detection, scene classification and semantic segmentation[END_REF] introduces auxiliary variables to consider different terms altogether, such as segmentation energy, object-reasoning as well as scene and class presence potentials; the relation between these potentials are formulated in a general CRF model. Global and local contexts have been modeled in [START_REF] Mottaghi | The role of context for object detection and semantic segmentation in the wild[END_REF] to improve both semantic segmentation and object detection. The primer considers the presence or absence of a class in the scene, while the latter refers to the classes appearing in the vicinity of the object.

Formulating the segmentation problem as MRFs makes it possible to leverage many efficient inference algorithms to find the optimal labeling. Simulated annealing [START_REF] Kähler | Efficient 3d scene labeling using fields of trees[END_REF], range/swap-based approaches [START_REF] Liu | GRSA: generalized range swap algorithm for the efficient optimization of mrfs[END_REF], or mean-field approximation [START_REF] Krähenbühl | Efficient inference in fully connected crfs with gaussian edge potentials[END_REF] are relevant inference approaches when the configuration space is large. Inferring on global co-occurrences is commonly formulated as integer programming, and solved via linear relaxation [START_REF] Ladicky | Inference methods for crfs with co-occurrence statistics[END_REF]. [START_REF] Yao | Describing the scene as a whole: Joint object detection, scene classification and semantic segmentation[END_REF] relies upon a message-passing algorithm to solve a holistic CRF that includes contextual terms such as scene and class-presence potentials.

Mesh Segmentation

Mesh segmentation is still a research challenge in geometry processing, robotic, and computer vision [START_REF] Shamir | A survey on mesh segmentation techniques[END_REF][START_REF] Chen | A Benchmark for 3D Mesh Segmentation[END_REF][START_REF] Theologou | A comprehensive overview of methodologies and performance evaluation frameworks in 3d mesh segmentation[END_REF]. In its simplest form mesh segmentation may be seen as an unsupervised clustering problem based on specific geometric criteria [START_REF] Shlafman | Metamorphosis of polyhedral surfaces using decomposition[END_REF]. Region growing [START_REF] Page | Perception-based 3d triangle mesh segmentation using fast marching watersheds[END_REF] and spectral analysis (Zhang et al., 2010b) are other instances of such deterministic approaches for mesh segmentation. The probabilistic approaches such as MRFs or CRFs provide an efficient means to enforce spatial consistency [START_REF] Lafarge | Insertion of 3D-Primitives in Mesh-Based Representations: Towards Compact Models Preserving the Details[END_REF]. Designing the energy terms for such approaches ranges from totally unsupervised [START_REF] Verdie | LOD generation for urban scenes[END_REF] to supervised [START_REF] Van Kaick | Prior knowledge for part correspondence[END_REF], through semi-supervised [START_REF] Lv | Semi-supervised mesh segmentation and labeling[END_REF].

A key aspect of the mesh segmentation approaches is the design of feature vectors encoding geometric information such as normals, curvatures or planarity. For textured meshes, additional features based on photometric information such as colors or texture histograms, provide relevant clues to design the energy terms of MRFs and CRFs. [START_REF] Verdie | LOD generation for urban scenes[END_REF] design three geometric attributes to define the unary term of a MRF. In our approach we instead adopt a supervised approach to learn an effective classifier.

In MVS contexts, some methods first perform classification directly from the images before mapping it to the output 3D model [START_REF] He | Nonparametric semantic segmentation for 3d street scenes[END_REF]. In [START_REF] Sengupta | Urban 3d semantic modelling using stereo vision[END_REF], each image is labeled by a supervised CRF before mapping and fusing the sets of labels with the mesh. [START_REF] Lafarge | A hybrid multi-view stereo algorithm for modeling urban scenes[END_REF] propose a hybrid approach in which the output model is progressively refined while detecting regular urban entities. [START_REF] Kundu | Joint semantic segmentation and 3d reconstruction from monocular video[END_REF] proposes a joint CRF model defined in 3D (volumetric) space and infers altogether the semantic label and voxel occupancy. [START_REF] Savinov | Discrete optimization of ray potentials for semantic 3d reconstruction[END_REF] and [START_REF] Blaha | Large-scale semantic 3d reconstruction: an adaptive multi-resolution model for multi-class volumetric labeling[END_REF] define the unary cost function along rays by considering both the semantic label and the depth of the first occupied voxel along the ray. Referred to as semantic 3D reconstruction by [START_REF] Haene | Joint 3D scene reconstruction and class segmentation[END_REF], such approaches are typically memory intensive and require complex inference approaches. The incremental approach proposed by [START_REF] Vineet | Incremental dense semantic stereo fusion for large-scale semantic scene reconstruction[END_REF] operates in near real time and delivers a rough reconstruction with a street-based semantics. [START_REF] Xiao | Multiple view semantic segmentation for street view images[END_REF] propose a larger MRF that includes all the views, and models all connections between the associated areas. The smoothness term between two views is defined either based on the color similarity or using the number of common feature tracks between the two associated images. These image-based methods are compute-intensive and insufficiently exploit the geometric properties of the observed scene.

Literature on the segmentation of textured meshes is marginal. [START_REF] Chauve | Transductive segmentation of textured meshes[END_REF] proposed an interactive approach based on transductive segmentation. In this approach the classifier is learned from basic photometric and geometric features derived from a set of strokes sketched on the input textured mesh.

Urban Modeling

Urban modeling is a notoriously hard challenge in photogrammetry and computer vision [START_REF] Musialski | A survey of urban reconstruction[END_REF], and extracting semantics from raw input data is a major step of this challenge. In an urban context, the classes of interest mainly correspond to the stationary objects such as buildings, roads and trees. We refer the reader to two comprehensive surveys [START_REF] Rottensteiner | The ISPRS benchmark on urban object classification and 3d building reconstruction[END_REF][START_REF] Schindler | An overview and comparison of smooth labeling methods for land-cover classification[END_REF]. From images [START_REF] Volpi | Semantic segmentation of urban scenes by learning local class interactions[END_REF][START_REF] Montoya-Zegarra | Semantic segmentation of aerial images in urban areas with class-specific higher-order cliques[END_REF] or from 3D point clouds [START_REF] Lai | Unsupervised feature learning for 3D scene labeling[END_REF][START_REF] Niemeyer | Contextual classification of lidar data and building object detection in urban areas[END_REF], the non-local approaches that exploit spatial and contextual dependencies of urban scenes have shown highly effective. Some approaches as [START_REF] Cabezas | Semantically-aware aerial reconstruction from multi-modal data[END_REF] exploit multiple sources of data for reconstructing urban scenes with more semantic information. In realworld scenarios, the availability of these different types of data is however mostly unlikely.

Literature on classification of urban meshes is more marginal. [START_REF] Verdie | LOD generation for urban scenes[END_REF] segment 3D urban scenes into different semantic parts such as ground, facade, roof and vegetation. In addition, the scene is represented with different levels of details derived from the classification. In this approach several geometric attributes based on elevation, planarity and verticality are used to design a labeling cost function. In contrast, [START_REF] Martinovic | 3d all the way: Semantic segmentation of urban scenes from start to end in 3d[END_REF] exploits a supervised classifier trained to define a labeling cost function. Some weak architecture rules are then enforced as a post-processing step by iterative quadratic programming.

Positioning and Contributions

MVS meshes are a recent type of input data in city modeling, and the semantic segmentation of these meshes has been overlooked. Most of the few existing approaches such as [START_REF] Verdie | LOD generation for urban scenes[END_REF], suffer from several issues, including (i) frequent labeling errors between some urban classes, in particular between roof and vegetation, (ii) trial-and-error processes required to tune the parameters, and (iii) ad-hoc post-processing operations required to exploit the local context.

Departing from the work of [START_REF] Verdie | LOD generation for urban scenes[END_REF], we propose a series of technical improvements to alleviate some of the aforementioned issues. Although the use of (i) region decomposition for reducing the algorithmic complexity, and (ii) a general MRF formulation for imposing spatial consistency is kept, we propose three important contributions:

• Joint Photometric and geometric analysis. Both geometric and photometric features are used for local analysis and description of the input meshes. In addition, some geometric features are extracted at different scales to increase robustness to the diversity of urban landscapes. Contrary to [START_REF] Verdie | LOD generation for urban scenes[END_REF] that exploit some mono-scale geometric features only, this richer set of features allows urban classes to be better discriminated, in particular parts of trees and pieces of ground from hilly areas mislabeling as roof.

• Supervision. A supervised approach based on random forests is used for building a local classifier in reasonable computational times. Contrary to the unsupervised approach of [START_REF] Verdie | LOD generation for urban scenes[END_REF], our approach has a limited number of parameters, and in particular, does not require tuning weighting coefficients between the different features. This design choice brings both more flexibility to the classification procedure and more robustness to data variety.

• Joint labeling. Departing from common labeling approaches in which a label corresponds to one of the classes of interest, our label space also takes into account the class transition between neighboring areas. This provides us with a means to account for contextual knowledge in an intuitive and effective way. In particular, the weights of the pairwise potential are directly learned from a training set to favor certain configurations of adjacent classes. For instance, roof is more likely to be adjacent to facade than to ground or vegetation.

As illustrated by Figure 2, the input textured mesh is first partitioned into small segments, referred to as superfacets. A set of geometric and photometric features relevant for discriminating urban objects are then computed from each superfacet. The urban class of each superfacet is then predicted using a random forests classifier within a MRF formulation.

Proposed Approach

We now detail the three main steps of our approach, as illustrated by Figure 2. The third step, i.e., MRF-based labeling, constitutes our main contribution.

Superfacet Partitioning

As the input meshes are very dense, typically 12M facets representing 1km 2 of a urban scene, labeling each facet of such meshes is very computeintensive. To lower the computing times and improve scalability, we first group facets into small clusters referred to as superfacets. Such a clustering step is analogous to over-segmenting images into superpixels, a popular approach for image analysis.

Clustering is performed through a region growing approach based on two similarity measures: (i) a covariance-based shape operator for measuring the geometric similarities between the triangular facets [START_REF] Cohen-Steiner | Variational shape approximation[END_REF], and (ii) the L1-distance measured in color space for quantifying the photometric similarity. The former favors the grouping of coplanar facets whereas the latter helps preserving image discontinuities of the texture map. A region grows until one of the two similarity measures exceeds user-specified thresholds (for both geometry and photometry). In addition, the maximum area of a superfacet can be determined by an optional user-specified parameter, as in superpixel approaches. Figure 2-(b) illustrates the superfacets obtained through clustering.

Two superfacets are said adjacent if their borders share at least one edge from the input triangular mesh.

Feature Extraction

For each superfacet of the input mesh we compute a set of geometric and photometric features. These features, which correspond to local statistics over the geometry of the superfacets and their color distributions, are later used by the classification step to discriminate between the urban objects of interest.

Geometric Features. These features are mostly similar to the ones proposed by [START_REF] Verdie | LOD generation for urban scenes[END_REF]. We compute three features relating to elevation, planarity and horizontality:

• Elevation a e . This feature discriminates the ground from higher objects such as roofs. It measures the height of the superfacet with respect to the ground:

a e = z -z min z max -z min , (1)
where z denotes the mean Z-coordinate of the superfacet, and z min (resp. z max ) denotes the minimal (resp. maximal) Z-coordinate of the superfacets contained in a local window on the XY plane. The size of the local window is chosen so as to trade robustness to highly variable terrain height, for accuracy of the estimation for large buildings. More specifically, the size of the local window should be larger than the largest building in the scene. We consider multiple elevation features, each being associated to a window size. The idea is to let the subsequent classifier select the window size in accordance to the urban landscapes observed in the training set. We consider three sizes by default: 10 meters, 20 meters and 40 meters.

• Planarity a p . We measure the planarity of a superfacet through computing the variation of the covariance matrix of all facets of the superfacet:

a p = 1 - 3λ 0 λ 0 + λ 1 + λ 2 , ( 2
)
where λ 0 < λ 1 < λ 2 denote the eigenvalues of the covariance matrix.

For near-planar superfacets this feature is close to 1 as λ 0 is much smaller than the two other eigenvalues.

• Verticality a v . This feature is computed as the cosine of the angle between the normal n of the superfacet and the vertical axis n z :

a v = |n.n z |. (3)
The feature value is low for the vertical components that mainly correspond to facades.

In ideal settings the aforementioned geometric features are sufficient to discriminate ground, facade, roof and trees. On real-world datasets however, these three features are insufficient to resolve ambiguous cases, and often confuse complex roofs and trees with regular shapes, as discussed by [START_REF] Verdie | LOD generation for urban scenes[END_REF]. This motivates our proposal for using additional radiometric features to improve the classification performances.

Photometric Features. The photometric information contained in the textured meshes provide a complementary clue for superfacet classification. For each superfacet we analyze its color distribution, measured in HSV color space. We choose the HVS instead of RGB color space as it is more effective for discriminating objects with different reflectivity properties. We then derive three photometric features for each superfacet: (1) the average color, (2) the standard deviation of its color distribution (see Fig 3-bottom), and(3) its color distribution. For the third feature we discretize the distribution by clustering the HSV color values of the whole texture map into a color palette. Our experiments show that such features are more discriminant than pure geometric features ones for classifying the superfacets (see Table 1).

For each superfacet we then construct a feature vector by concatenating all geometric and photometric features. We denote by f i the feature vector of superfacet i.

Labeling

Label space. In previous work the label space is often limited to one label per area, associated to one of the classes of interest. We enlarge the label space so as to include the class transition between neighboring areas, in addition to the class of each area.

Given the set of classes of interest L = {1, 2, ..., N }, we consider a random variable l i ∈ L that associates a class to superfacet i. In our experiments N is set to 4 or 5, covering the classes of ground, trees, facades, roofs, and optionally superstructures. Our joint labeling system extends L into a larger set of classes L = {1, 2, ..., 2N } so that

∀ lij ∈ L, lij = l i + N.1 {l i =l j } , (4)
where j denotes the index of a superfacet adjacent to superfacet i, and 1 {•} is the characteristic function. The joint label lij records the class of superfacet i, and whether superfacet j has the same class. While previous approaches such as [START_REF] Kähler | Efficient 3d scene labeling using fields of trees[END_REF]preformed joint labeling with N 2 labels, they are very compute-intensive in both training and testing stages. given an input mesh (left, color code: HSV channels). Such an information is exploited in the MRF formulation to disambiguate complex cases, such as clusters mislabeled as facade on the vertical parts of some trees (right).

Classifier training. For training a classifier over our joint labeling approach, we concatenate feature vectors f i and f j into a single vector fij for all lij ∈ L.

For machine learning we use randomized decision trees. They enable efficient prediction since each feature vector undergoes only a few branching nodes before reaching the leaves of the tree. In our implementation we use by default 100 decision trees with 25 as maximum depth. Figure 4 illustrates the label prediction obtained by the classifier.

MRF formulation. The trained random forest is able to predict the joint labels locally. However, the quality of label prediction may become spatially incoherent in presence of geometric defects and texture noise as illustrated by Figure 5. We thus apply a common MRF formulation for smoothing the labels via the following energy minimization:

E(l) = i∈S ψ i (l i ) + γ (i,j)∈E ϕ ij (l i , l j ), ( 5
)
where S denotes the set of superfacets of the input mesh, and E denotes the set of pairs of neighboring superfacets. The unary data term ψ i measures the coherence of the class l i of superfacet i whereas the pairwise potential ϕ ij encourages similar classes for neighboring superfacets with similar features. The regularization parameter γ balances the importance of the latter with respect to the former. Figure 6 shows the impact of this parameter on different cases. Both data term and pairwise potential are designed using the aforementioned joint labeling approach. Data term. The unary data term of superfacet i is computed from the output probabilities of the trained decision trees:

ψ i (l i ) = - 1 |T | t∈T log(P t (l i )), ( 6
)
where T denotes the set of decision trees, and P t denotes the prediction probability of the class l i for the decision tree t defined as

P t (l i ) = 1 |N i | j∈N i p t ( lij = l i | fij ) + p t ( lij = l i + N | fij ) , (7)
where N i denotes the set of neighboring superfacets of superfacet i, and p t denotes the output probability function of the decision tree t. Intuitively, we average the prediction of the class of superfacet i over all its neighboring superfacets, without distinguishing whether the joint label must yield similar classes ( lij = l i ) or different classes ( lij = l i + N ).

Pairwise potential. This term introduces both spatial consistency between superfacets and contextual information between classes in the decision process. We express the pairwise potential term as

ϕ ij (l i , l j ) = 1 |T | t∈T ω ij • A t (l i , l j ), ( 8
)
where ω ij denotes the length of the common frontier between superfacets i and j, and A t denotes a function equating to 0 if l i = l j , and to a positive penalty otherwise. More specifically, A t is expressed through our joint labeling approach as

A t (l i , l j ) = 0 if l i = l j p t ( lij = l i + N | fij ) otherwise (9)
This term acts as a generalized Potts model in which the penalty weight is defined through the output probability function of random forests.

Inference. For inference through energy minimization we use an α-expansion approach [START_REF] Boykov | Fast approximate energy minimization via graph cuts[END_REF]. Although the input meshes are very dense, the size of our MRF formulation is moderate when applied to superfacets.

Experimental Results

We evaluated our approach on textured meshes generated by the Con-textCapture software (Acute3D) applied to multiview aerial images shot on urban landscapes (cities of Paris and Toulouse, France). Such textured meshes correspond to a planimetric area of approximately 300m × 300m, and contain around 1M triangular facets and two texture maps of size 8192×8192 pixels. Four classes of interest are considered in our experiments: roof, facade, vegetation and ground.

We used a unique 3D mesh of 50K superfacets for training our classifier, except for the experiments conducted in Figure 15. This mesh corresponds to a 300m × 300m tile whose superfacets have been manually labeled by an expert. As illustrated in Figure 7, this tile represents a part of Paris with a few building blocks, some isolated buildings and vegetation along roads. In particular, only 10% of the large scale scenes presented in Figures 17,18 and 19 overlap with the training set.

Parameters

Our algorithm takes as input several model parameters. To extract the superfacets, two thresholds are required to define the growing condition: the geometric threshold imposes a maximal angle between the normal of the region and the facet candidate, whereas as the photometric threshold guarantees a maximal L1-distance between the colors of the region and the candidate. In all experiments shown the geometric (respectively photometric) threshold has been set to 20 deg (respectively 30 over 256). We limit the size of superfacets to 100m 2 in order to avoid the propagation of potential classification errors, as illustrated by Figure 8. A single textured mesh (1M facets) is used to train a random forest of 100 decision trees. The recursive splits in the testing nodes are stopped when the maximum depth (set by default to 25) is reached or when the number of samples is smaller than 20. Figure 9 shows the impact of the size of the training set on the label prediction accuracy during the testing stage. Parameter γ is the only tradeoff parameter of the energy. By default we set γ to 0.5 except for Figure 6.

We consider four main classes of interest: Ground, Roof, Facades, Vegetation. In addition, and in order to demonstrate the flexibility of our algorithm, Figure 10 showcases an experiment where an additional class is used to detect roof superstructures such as chimneys or dormer-windows. Note that despite a very rough manual labeling for the training stage, the achieved labeling results are promising. These results can be improved by defining a more complete and accurate training set with more samples of chimneys.

Robustness to Urban Landscapes

We evaluate our approach on different types of urban scenes with a wide range of geometric and photometric features. Figures 11 -13 depict classification results for industrial, residential and dense urban landscapes, respec- tively. Our approach is sufficiently versatile to correctly classify these three types of urban landscapes.

Our multi-scale approach for computing the elevation features is particularly relevant to classify hilly areas, as illustrated by Figure 14. However, extremely sharp hills challenges our approach that fails to correctly discriminate ground from roof and facade. Such cases that often correspond to quarries are quite marginal in practice.

We trained some classifiers on different urban areas in order to measure the impact of urban diversity on the learning phase. As illustrated in Figure 15, the most accurate results are obtained when the classifier is trained and tested on a similar urban landscape. Note that using an industrial landscape as training set gives the most generic classifier. This choice must be considered when the observed urban landscapes in the data are unknown.

Feature Importance

Five geometric features (elevation at 3 scales, planarity and verticality) and three photometric features (color histogram, HSV average intensity and variance) are considered to describe the input surfaces locally. These features, grouped into a single feature vector of 36 components for describing each superfacet, do not exhibit the same discriminative power, referred to as importance, when training the classifier. Table 1 highlights the important features by showing their frequencies during the training phase. The color histogram is the most relevant feature with an occurrence above 50%. The photometric features are on average twice more important than the geometric features. Figure 16 illustrates the impact of a joint use of geometric and photometric features.

Comparisons

We compare our method on large-scale urban scenes with the unsupervised approach of [START_REF] Verdie | LOD generation for urban scenes[END_REF] where the MRF data term relies upon only geometric attributes. We also perform experiment with a variant of our model that does not use the joint labeling system, but a standard system with N classes. In this model, no contextual information is taken into account. We provide qualitative comparisons in Figures 17,18 and 19, where the input meshes (3M facets, 1.5M vertices), cover a third of one kilometer square of urban scenes. Our supervised approach, combined with a joint labeling system and both geometric and photometric attributes, outperforms the two competing approaches. More specifically, our method discriminates more accurately the trees, complex roof structures, as well as the small common urban objects such as fences and cars.

Performance

Running times of our algorithm are given for different sizes of input mesh in Table 2. Training the classifier is the most time-consuming operation. It takes close to 3 minutes, but has to be done only once. Superfacet partitioning, feature extraction and MRF-labeling require each a few seconds for a 1Mfacet input mesh. Memory peak reached from a standard 300m × 300m tile is typically 3.2Gb.

Limitations

The cases that challenge our algorithm are typically hilly areas for which the relief is particularly sharp and sloppy. Our algorithm often mislabels vertical components of quarries into facades. Also, the classification accuracy decreases with low quality input meshes, typically when occluded parts in the MVS images locally generate over-smooth surfaces that tend to be classified as trees.

Conclusions

We proposed a supervised approach for classifying urban landscapes provided in the form of textured meshes. Our approach relies upon two main technical contributions: a joint labeling system that leverages contextual knowledge in an intuitive and effective way, and a joint use of photometric and geometric attributes to locally describe the input meshes. We demonstrated these contributions combined with a supervised strategy provide an Our approach (first and fourth row) alleviates these errors. By restricting our model to a standard labeling system, ie with N classes of interest, no contextual information is exploited anymore (third row). Some mislabeled patches typically appear on top of trees as tree superfacets adjacent to roof or facade ones are not penalized anymore. Small common urban objects such as low fences and cars often create mislabeling errors using either [START_REF] Verdie | LOD generation for urban scenes[END_REF] or our model with a standard labeling system. The insertion of contextual knowledge with the joint labeling system proves to be more flexible by correctly classifying these elements (see left column).

efficient solution that outclasses the approach proposed by [START_REF] Verdie | LOD generation for urban scenes[END_REF] in terms of correctness, ease-to-use, and robustness to data diversity. Globally speaking, such a two step strategy where mesh is first reconstructed before being partitioned into semantic classes remains more efficient that semantic reconstruction methods. Recovery 3D geometry and semantics simultaneously is methodologically more elegant but offers serious challenges in terms of algorithmic complexity.

In future work we would like to investigate on hierarchical classification of urban scenes where for instance the class roof would have some children classes as roof section or dormer-windows, and a parent class as building. we also plan to leverage the function of urban objects such as buildings to improve the accuracy of results. An open question is how to use such additional knowledge in a supervised classification approach.

Figure 1 :

 1 Figure 1: Textured meshes produced from MVS imagery. Our input is a textured mesh combining geometry (left) and radiometry (right).

Figure 2 :

 2 Figure 2: Overview. The input textured mesh generated from MVS images (a) is first partitioned into superfacets. (b) A set of geometric and photometric features are then extracted from the superfacets, (c) and combined into a MRF-based formulation to classify them at a non-local scale. (d) The class roof is depicted in blue, ground in brown, tree in green, and facade in yellow. This color code is used throughout the paper.

Figure 3 :

 3 Figure 3: Geometric and photometric features. We compute the verticality, planarity and elevation are geometric features (top). We complement them by photometric features measured in the HVS color space (bottom).

Figure 4 :

 4 Figure4: Joint labeling. In addition to the most probable class for each superfacet, our classifier records when a class transition is most likely to occur (see black lines, middle) given an input mesh (left, color code: HSV channels). Such an information is exploited in the MRF formulation to disambiguate complex cases, such as clusters mislabeled as facade on the vertical parts of some trees (right).

Figure 5 :Figure 6 :

 56 Figure 5: Local label prediction. The trained classifier alone produces decent results (right) given a superfacet partitioning (left). The subsequent use of the MRF formulation is often sufficient to resolve the small mislabeled areas on roofs and trees. The black lines underline the locations where the classifier advocates for a class transition.

Figure 7 :

 7 Figure 7: Training set. The training set used for our experiments corresponds to a 300m × 300m tile whose superfacets have been manually labeled by an expert (left). The trained classifier is globally consistent with the training set (right). Typical learning errors are shown in the close-ups.

Figure 8 :

 8 Figure 8: Size of superfacets. The use of large superfacets (top) enables fast processing. Conversely, small superfacets (bottom) often yield fewer mislabeling errors as the label propagation process operates at a more local scale. Notice the mislabeling on the hilly area.

Figure 9 :

 9 Figure 9: Classifier accuracy vs. size of training set. Using at least 10K superfacets in the training set allows the classifier to yield correct predictions for more than 85% of samples. Beyond 10K the gain obtained by increasing the number of samples is negligible.

Figure 10 :

 10 Figure 10: Adding a new class. Our formulation is flexible enough to define new classes of interest. When adding the new class superstructure that includes chimneys and dormerwindows, our algorithm can detect it reasonably well (see purple clusters) without adding any specific features.

Figure 11 :

 11 Figure 11: Classification of industrial areas. Very large industrial buildings are challenging to classify due to confusions between, e.g., roof and ground. Relying upon multi-scale elevation features yields satisfactory results on such buildings.

Figure 12 :

 12 Figure 12: Classification of residential areas. Despite the small size of buildings and the bad geometry of their facade, our algorithm correctly distinguishes the classes of interest, even when trees are merging with roof and facade components.

Figure 13 :

 13 Figure13: Classification of dense urban areas. In such urban landscapes, the main challenge is to distinguish trees from roofs and to identify ground in presence of narrow streets (see middle example). Our results are mostly correct except a few confusions for trees in inner courtyards.

Figure 14 :

 14 Figure 14: Classification of hilly areas. Using the elevation feature computed at different scales our classifier correctly detects sloppy ground in most cases (see left and middle). In presence of very abrupt relief changes, however, it tends to misclassify ground into facade or roof (right).

Figure 15 :

 15 Figure 15: Impact of diversity of urban landscapes on the learning process. As expected training and testing the classifier on similar urban landscape yields the best classification results (see diagonal). As dense and residential areas have distinctive urban signatures, the classifiers learned from these landscapes is insufficiently general. Industrial landscapes, with a more diverse signature, enable the associated classifier to also perform well on other landscapes (see bottom row).

Figure 16 :

 16 Figure 16: Joint use of geometric and photometric features. Relying upon geometry only often leads to confusions between vegetation and roof (c), whereas the use of photometry alone does not allow the ground and facades to be discriminated correctly (d). Combining geometry and photometry improves the classification by a significant margin, in particular by correcting the aforementioned confusion errors (b).

Figure 17 :

 17 Figure 17: Large scene classification. The results of Verdie et al. (2015) yield frequent confusions on vegetation with groups of trees mislabeled as facade and roof (second row).Our approach (first and fourth row) alleviates these errors. By restricting our model to a standard labeling system, ie with N classes of interest, no contextual information is exploited anymore (third row). Some mislabeled patches typically appear on top of trees as tree superfacets adjacent to roof or facade ones are not penalized anymore.

Figure 18 :

 18 Figure18: Large scene classification. For some complex buildings, none of the three methods achieve to correctly classify the roof superstructures as a few chimneys are typically labeled as facade. For such cases, our contextual information is not helpful as adjacent facade and roof superfacets are considered as plausible.

Figure 19 :

 19 Figure19: Large scene classification. Small common urban objects such as low fences and cars often create mislabeling errors using either[START_REF] Verdie | LOD generation for urban scenes[END_REF] or our model with a standard labeling system. The insertion of contextual knowledge with the joint labeling system proves to be more flexible by correctly classifying these elements (see left column).

Table 1 :

 1 Feature importance during the random forest training. The joint feature vector fij contains 72 channels for describing a superfacet i and one of its neighbors j. "dim." and "frq." show the dimension and the frequency of features, respectively.

		Superfacet i	Neighboring superfacet j	Pair of superfacets
	Features	dim. frq.	dim. frq.	dim. frq.
	Average HSV	3	3.72%	3	3.17%	6	6.89%
	Variance HSV	3	3.85%	3	2.86%	6	6.71%
	Color Histogram	25	27.36%	25	23.05%	50	50.41%
	Geometric Features 5	19.38%	5	16.61%	10	35.99%
	Total	36	54.31%	36	45.69%	72	100%

Table 2 :

 2 Running times. Timings (sec) are given for a Intel Core i7 processor clocked at 2GHz.

		Superfacet	Feature	Labeling Classifier
		partitioning extraction (testing) (training)
	training set (1Mfacets, Fig 7)	10	16	28	175
	large scene (3.4Mfacets, Fig 18)	34	45	85	-

Acknowledgments

We wish to thank Bentley/Acute3D and Interatlas for providing datasets.This work was supported by the European Research Council (ERC Starting Grant Robust Geometry Processing, Grant agreement 257474).