Laurent Perrier

Gilles Pijaudier-Cabot

David Grégoire
email: david.gregoire@univ-pau.fr

G Pijaudier

Extended poromechanics for adsorption-induced swelling prediction in double porosity media: modeling and experimental validation on activated carbon

Keywords: Adsorption, swelling, double porosity media, poromechanical modelling

Natural and synthesised porous media are generally composed of a double porosity: a microporosity where the fluid is trapped as an adsorbed phase and a meso or a macro porosity required to ensure the transport of fluids to and from the smaller pores. Zeolites, activated carbon, tight rocks, coal rocks, source rocks, cement paste or construction materials are among these materials.

In nanometer-scale pores, the molecules of fluid are confined. This effect, denoted as molecular packing, induces that fluid-fluid and fluid-solid interactions sum at the pore scale and have significant consequences at the macroscale, such as instantaneous deformation, which are not predicted by classical poromechanics. If adsorption in nanopores induces instantaneous deformation at a higher scale, the matrix swelling may close the transport porosity, reducing the global permeability of the porous system. This is important for applications in petroleum oil and gas recovery, gas storage, separation, catalysis or drug delivery.

This study aims at characterizing the influence of an adsorbed phase on the instantaneous deformation of micro-tomacro porous media presenting distinct and well-separated porosities. A new incremental poromechanical framework with varying porosity is proposed allowing the prediction of the swelling induced by adsorption without any fitting parameters. This model is validated by experimental comparison performed on a high micro and macro porous activated carbon. It is shown also that a single porosity model cannot predict the adsorption-induced strain evolution observed during the experiment. After validation, the double porosity model is used to discuss the evolution of the poromechanical properties under free and constraint swelling.

Introduction

Following the IUPAC recommendation [START_REF] Sing | Reporting physisorption data for gas/solid systems with special reference to the deter-mination of surface area and porosity[END_REF][START_REF] Thommes | Physisorption of gases, with special reference to the evaluation of surface area and pore size distribution (IUPAC Technical Report)[END_REF], the pore space in porous materials is divided into three groups according to the pore size diameters: macropores of widths greater than 50 nm, mesopores of widths between 2 and 50 nm and micropores (or nanopores) of widths less than 2 nm. Zeolites, activated carbon, tight rocks, coal rocks, source rocks, cement paste or construction materials are among these materials. In recent years, a major attention has been paid on these microporous materials because the surface-to-volume ratio (i.e., the specific pore surface) increases with decreasing characteristic pore size. Consequently, these materials can trap an important quantity of fluid molecules as an adsorbed phase. This is important for applications in petroleum and oil recovery, gas storage, separation, catalysis or drug delivery.

For these microporous materials, a deviation from standard poromechanics [START_REF] Biot | General theory of three-dimensional consolidation[END_REF][START_REF] Coussy | Poromechanics[END_REF], is expected. In nanometer-scale pores, the molecules of fluid are confined. This effect, denoted as molecular packing, induces that fluid-fluid and fluid-solid interactions sum at the pore scale and have significant consequences at the macroscale, such as instantaneous deformation. A lot of natural and synthesised porous media are composed of a double porosity: the microporosity where the fluid is trapped as an adsorbed phase and a meso or a macro porosity required to ensure the transport of fluids to and from the smaller pores. If adsorption in nanopores induces instantaneous deformation at a higher scale, the matrix swelling may close the transport porosity, reducing the global permeability of the porous system or annihilating the functionality of synthesised materials. In different contexts, this deformation may be critical. For instance, in situ adsorption-induced coal swelling has been identified [START_REF] Larsen | The effects of dissolved CO2 on coal structure and properties[END_REF][START_REF] Pan | A theoretical model for gas adsorption-induced coal swelling[END_REF][START_REF] Sampath | Ch4co2 gas exchange and supercritical co2 based hydraulic fracturing as cbm production-accelerating techniques: A review[END_REF] as the principal factor leading to a rapid decrease in CO 2 injectivity during coal bed methane production enhanced by CO 2 injection. Conversely, gas desorption can lead to matrix shrinkage and microcracking, which may help oil and gas recovery in the context of unconventional petroleum engineering [START_REF] Levine | Model study of the influence of matrix shrinkage on absolute permeability of coal bed reservoirs[END_REF]. The effects of adsorbent deformation on physical adsorption has also been identified by [START_REF] Thommes | Physical adsorption characterization of nanoporous materials: Progress and challenges[END_REF] as one of the next major challenges concerning gas porosimetry in nano-porous non-rigid materials (e.g. metal organic framework). In conclusion, there is now a consensus in the research community that major attention has to be focused on the coupled effects appearing at the nanoscale within microporous media because they may have significant consequences at the macroscale.

Experimentally, different authors tried to combine gas adsorption results and volumetric swelling data (see e.g. [START_REF] Gor | Adsorption-induced deformation of nanoporous materials -A review[END_REF] for a review). The pioneering work of [START_REF] Meehan | The Expansion of Charcoal on Sorption of Carbon Dioxide[END_REF] showed the effect of carbon dioxyde sorption on the expansion of charcoal but only mechanical deformation was reported and adsorption quantities were not measured. Later on, different authors [START_REF] Briggs | Expansion and contraction of coal caused respectively by the sorption and discharge of gas[END_REF][START_REF] Levine | Model study of the influence of matrix shrinkage on absolute permeability of coal bed reservoirs[END_REF][START_REF] Day | Swelling of australian coals in supercritical co2[END_REF][START_REF] Ottiger | Competitive adsorption equilibria of co2 and ch4 on a dry coal[END_REF][START_REF] Pini | Role of adsorption and swelling on the dynamics of gas injection in coal[END_REF][START_REF] Hol | Competition between adsorption-induced swelling and elastic compression of coal at co 2 pressures up to 100mpa[END_REF][START_REF] Espinoza | Measurement and modeling of adsorptive-poromechanical properties of bituminous coal cores exposed to co2: Adsorption, swelling strains, swelling stresses and impact on fracture permeability[END_REF] performed tests on bituminous coal, because it is of utmost importance in the context of CO 2 geological sequestration and coal bed reservoirs exploitation. However, most results were not complete in a sense that adsorption and swelling experiments were not measured simultaneously [START_REF] Meehan | The Expansion of Charcoal on Sorption of Carbon Dioxide[END_REF][START_REF] Robertson | Measuring and modeling sorption-induced coal strain[END_REF] or performed on exactly the same coal samples [START_REF] Ottiger | Competitive adsorption equilibria of co2 and ch4 on a dry coal[END_REF]. Other authors presented simultaneous in situ adsorption and swelling results but the volumetric strain was extrapolated from a local measurement -using strain gauges [START_REF] Levine | Model study of the influence of matrix shrinkage on absolute permeability of coal bed reservoirs[END_REF][START_REF] Harpalani | Influence of matrix shrinkage and compressibility on gas production from coalbed methane reservoirs[END_REF][START_REF] Battistutta | Swelling and sorption experiments on methane, nitrogen and carbon dioxide on dry selar cornish coal[END_REF] or LVDT sensors [START_REF] Chen | Method for simultaneous measure of sorption and swelling of the block coal under high gas pressure[END_REF][START_REF] Espinoza | Measurement and modeling of adsorptive-poromechanical properties of bituminous coal cores exposed to co2: Adsorption, swelling strains, swelling stresses and impact on fracture permeability[END_REF] -or by monitoring the silhouette expansion [START_REF] Day | Swelling of australian coals in supercritical co2[END_REF]. [START_REF] Perrier | A novel experimental set-up for simultaneous adsorption and induced deformation measurements in microporous materials[END_REF] presented an experimental setup providing simultaneous in situ measurements of both adsorption and deformation for the same sample in the exact same conditions, which can be directly used for model validation. Gas adsorption measurements are performed using a custom-built manometric apparatus and deformation measurements are performed using a digital image correlation set-up. This set-up allows full-field displacement measurements, which may be crucial for heterogeneous, anisotropic or cracked samples.

As far as modeling is concerned, molecular simulations are the classical tools at the nanoscale. Important efforts have been involved in molecular simulations in order to characterise adsorption-induced deformation in nanoporous materials [START_REF] Vandamme | Adsorption and strain: The CO2-induced swelling of coal[END_REF]Brochard et al., 2012a;[START_REF] Hoang | Couplings between swelling and shear in saturated slit nanopores : A molecular simulation study[END_REF] and these investigations showed on few configurations that pressures applied on the pore surfaces may be very high (few hundred of MPa), depending on the thermodynamic conditions and on the pore sizes. Note that an alternative approach based on a non-local density functional theory can be used to obtain highly resolved evolutions of pore pressure versus pore widths and bulk pressure in slit-shaped pores for a large spectrum of thermodynamic conditions on the whole range of micropore widths, even for complex fluids [START_REF] Grégoire | Estimation of adsorption-induced pore pressure and confinement in a nanoscopic slit pore by a density functional theory[END_REF]. However, if macroscopic adsorption isotherms may be reconstructed in a consistent way from molecular simulations through the material pore size distribution [START_REF] Khaddour | A fully consistent experimental and molecular simulation study of methane adsorption on activated carbon[END_REF], molecular simulation tools are not tractable to predict resulting deformation at a macroscale due to the fluid confinement in nanopores (pore sizes below 2 nm). Note that [START_REF] Kulasinski | Impact of hydration on the micromechanical properties of the polymer composite structure of wood investigated with atomistic simulations[END_REF] proposed a molecular dynamic study where macroscopic swelling may be reconstructed from water adsorption in mesoporous wood (pore sizes in [4 -10] nm). If adsorption is essentially controlled by the amount and size of the pores, the mechanical effect of the pressure build up inside the pores due to fluid confinement requires some additional description about the topology and spatial organization of the porous network which is not easy to characterize, for sub-nanometric pores especially. Such a result motivates the fact that swelling is usually related to the adsorption isotherms instead of the pore pressure directly, the mechanical effect of the pore pressure being hidden in the poromechanical description.

In this context, different enhanced thermodynamical or poromechanical frameworks have been proposed within the last ten years to link adsorption, induced deformation and permeability changes (e.g [START_REF] Pan | A theoretical model for gas adsorption-induced coal swelling[END_REF] 2014)). For instance, Brochard et al. (2012b) (resp. Vermorel and[START_REF] Vermorel | Enhanced continuum poromechanics to account for adsorption induced swelling of saturated isotropic microporous materials[END_REF]) proposed enhanced poromechanical frameworks where swelling volumetric deformation may be estimated as a function of the bulk pressure and a coupling (resp. a confinement) coefficient, which may be deduced from adsorption measurements. However, if these models are consistent with experimental results from the literature, they cannot be considered as truly predictive because the model parameters have to be identified to recover the experimental loading path. An incremental poromechanical framework with varying porosity has been proposed by [START_REF] Perrier | Poromechanics of adsorption-induced swelling in microporous materials: a new poromechanical model taking into account strain effects on adsorption[END_REF] allowing the full prediction of the swelling induced by adsorption for isotropic nano-porous solids saturated with a single phase fluid, in reversible and isothermal conditions. This single porosity model has been compared with experimental data obtained by [START_REF] Ottiger | Competitive adsorption equilibria of co2 and ch4 on a dry coal[END_REF] on bituminous coal samples filled with pure CH 4 and pure CO 2 at T = 45 o C and a fair agreement was observed for these low porosity coals but these types of models have to be enhanced to take into account the intrinsic double porosity features of such materials.

This study aims at characterizing the influence of an adsorbed phase on the instantaneous deformation of microto-macro porous media presenting distinct and well separated porosities in term of pore size distribution. A model accounting for double porosity is proposed and validated by in-situ and simultaneous experimental comparisons.

The novelty of the approach is to propose an extended poromechanical framework taking into account the intrinsic double porosity features of such materials capable of predicting adsorption-induced swelling for high porous materials without any fitting parameters.

1. An incremental poromechanical framework with varying porosity for double porosity media In this section, an incremental poromechanical framework with varying porosity proposed by [START_REF] Perrier | Poromechanics of adsorption-induced swelling in microporous materials: a new poromechanical model taking into account strain effects on adsorption[END_REF] for single porosity media is extended to double porosity media.

We consider here a double porosity medium with distinct and separated porosities. The small porosity is called adsorption porosity (φ µ ) and the larger one transport porosity (φ M ). This medium is considered as isotropic with a linear poro-elastic behaviour and it is immersed and saturated by a surrounding fluid at bulk pressure P b under isothermal conditions. Confinement effects may change the thermodynamic properties of the interstitial fluids in both porosities. The adsorption porosity is saturated by an interstitial fluid of density ρ µ at pressure P µ . The transport porosity is fully saturated by an interstitial fluid (single-phase) of density ρ M at pressure P M (see Fig. 1).

For saturated isotropic porous solids, in reversible and isothermal conditions and under small displacementgradient assumptions, classical poromechanics may be rewritten for double porosity media [START_REF] Coussy | Poromechanics[END_REF] :

d Gs = dΨ s + dW s (1) = σ i j : dε i j + P M dφ M + P µ dφ µ dΨ s + d(-P M φ M -P µ φ µ ) dW s (2) = σ i j : dε i j -φ M dP M -φ µ dP µ .
(3) of the skeleton. The state variables (ε i j , φ M , φ µ ) are respectively the infinitesimal strain tensor, the transport porosity and the adsorption porosity. The associated thermodynamical forces (σ i j , P M , P µ ) are respectively the Cauchy stress tensor and the fluid pore pressures in both porosities.

For an isotropic linear poro-elastic medium, the state equations are then given by:

                       σ i j = ∂ Gs ∂ε i j φ M = -∂ Gs ∂P M φ µ = -∂ Gs ∂P µ
, and then:

                       dσ = K(φ M , φ µ )dε -b M (φ M , φ µ )dP M -b µ (φ M , φ µ )dP µ dφ M = b M (φ M , φ µ )dε + dP M N MM (φ M ,φ µ ) - dP µ N Mµ (φ M ,φ µ ) dφ µ = b µ (φ M , φ µ )dε -dP M N µM (φ M ,φ µ ) + dP µ N µµ (φ M ,φ µ ) . (4)
In Eq. 4, (σ = σ kk /3) and (ε = ε kk ) are respectively the total mean stress and the volumetric strain. (K, b M , b µ , N MM , N Mµ , N µM , N µµ ) are respectively the apparent modulus of incompressibility and six poromechanical properties, which depends on the two evolving porosities φ M and φ µ and on the constant skeleton matrix modulus.

Considering a single cylindrical porosity2 , homogenization models [START_REF] Halpin | The Halpin-Tsai Equations: A Review[END_REF] yield to:

           K(φ) = K s G s (1-φ) G s +K s φ , G s = 3K s (1-2ν s ) 2(1+ν s ) b(φ) = 1 -K(φ) K s , N(φ) = K s b(φ)-φ .
(5)

In Eq. 5, φ is the porosity and (G s , ν s ) are respectively the shear modulus and the Poisson ratio of the skeleton matrix.

Practically and for high porosity media, an iterative process of homogenization is chosen to avoid discrepancies in the apparent properties estimation as noticed by [START_REF] Barboura | Modélisation micromécanique du comportement de milieux poreux non linéaires : Applications aux argiles compactées[END_REF]. The iterative process of homogenization for a cylindrical porosity is detailed in Appendix A. Full details on the iterative processes for both spherical and cylindrical porosities are presented in [START_REF] Perrier | Poromechanics of adsorption-induced swelling in microporous materials: a new poromechanical model taking into account strain effects on adsorption[END_REF].

Considering that the two porosities are distinct, well separated and both cylindrical, the iterative process can be used in two successive steps to determine the different modulus of incompressibility. Note that this two step homogenization process may be reversed as well to estimate the skeleton properties knowing the apparent ones:

(K, G) = F n F n (K s , G s , φ µ ), φ M and (K s , G s ) = R n (R n (K, G, φ M ), φ µ ) . (6)
In Eq. 6, (F n , R n ) stand as the standard and the reverse iterative processes of homogenization defined in Eq. A.1 and A.2 respectively. K s and G s (resp. K and G) are the skeleton (resp. apparent) incompressible and shear modulii.

Based on stress/strain partitions [START_REF] Coussy | Poromechanics[END_REF] and on the response of the medium saturated by a non-adsorbable fluid [START_REF] Nikoosokhan | CO2 Storage in Coal Seams: Coupling Surface Adsorption and Strain[END_REF], the six poromechanical properties (b M , b µ , N MM , N Mµ , N µM , N µµ ) may be identified:

                       b M = 1 -K K µ , b µ = K( 1 K µ -1 K s ) 1 N MM = b M -φ M K µ , 1 N Mµ = 1 N µM = (b M -φ M )( 1 K µ -1 K s ) , 1 N µµ = (b µ -φ µ ) K s + (b M -φ M )( 1 K µ -1 K s ) with K µ = F n (K s , G s , φ µ ) . (7)
For a porous medium saturated by a fluid under isothermal conditions (isotropic surrounding/bulk pressure: P b , density: ρ b ), dσ = -dP b and Eq. 4 yield to:

                     dε = -dP b K s dφ M = -φ M K s dP b dφ µ = -φ µ K s dP b . (8)
Therefore, classical poromechanics predicts a shrinkage of the porous matrix and a decrease of the porosity under bulk pressure. This has been confirmed by experimental measurements (e.g. [START_REF] Reucroft | Gas-induced swelling in coal[END_REF] on a natural coal with a non-adsorbable gas).

Considering that the fluid is confined within both porosities, the thermodynamic properties (pressures: (2015) for single porosity media:

P M , P µ , densities: ρ M , ρ µ ) of
           dP M = ρ M dP b ρ b = dP b 1-χ M dP µ = ρ µ dP b ρ b = dP b 1-χ µ . (9)
In Eq. 9, (χ M = 1 -ρ b ρ M ) and (χ µ = 1 -ρ b ρ µ ) are the confinement degrees in the transport and in the adsorption porosities respectively, which characterize how confined is the interstitial fluid due to the number of adsorbate moles n ex M and n ex µ that exceeds the number of fluid moles at bulk conditions in porosities φ M and φ µ respectively:

             χ M = n ex M n tot M with n tot M = n ex M + ρ b V φ M M χ µ = n ex µ n tot µ with n tot µ = n ex µ + ρ b V φµ M . ( 10
)
In Eq. 10, (V φ M , V φ µ ) are the connected porous volume corresponding to the transport porosity φ M and to the adsorption porosities φ µ respectively, n ex M and n ex µ are the number of adsorbate moles that exceeds the number of fluid moles at bulk conditions and n tot M and n tot µ are the total number of moles of interstitial fluid in porosities φ M and φ µ respectively.

Generally, there is no way to link separately the two confinement degrees χ M and χ µ to quantities that can be measured experimentally because the partition of the excess number of adsorbate moles n ex , which can be measured experimentally, within the two porosities is unknown (n ex = n ex M + n ex µ ). However, assuming that the two scales of porosities are well separated, one can consider that most of the adsorption phenomenon occurs in the adsorption porosity (n ex µ >> n ex M ) and that the interstitial fluid is not confined in the transport porosity:

           χ µ ≈ n ex n tot µ and dP µ = dP b 1-χ µ χ M ≈ 0 and dP M = dP b . (11)
Finally, a new incremental poromechanical framework with varying porosities for double porosity media is proposed:

                                                                             dε = ( b µ 1-χ µ + b M -1) dP b K dφ µ =                      b µ 1 -χ µ + b M -1 b µ K T 1 + 1 N µµ 1 -χ µ T 2 - 1 N µM T 3                      dP b dφ M =                      b µ 1 -χ µ + b M -1 b M K T 4 - 1 N µM 1 -χ µ T 5 + 1 N MM T 6                      dP b χ µ = n ex n tot µ with n tot µ = n ex + ρ b V φµ M = n ex + m s M ρ b ρ s φ µ 1-φ µ -φ M (12)
In Eq. 12, K(φ M , φ µ ) is given by Eq. 6, (b M , b µ , N MM , N µM , N µµ ) all depend on (φ M , φ µ ) and are given by Eq. 7, (n ex , P b , ρ b ) are experimentally measurable and (m s , M, ρ s ) are respectively the adsorbent sample mass, the molar mass of the adsorbed gas and the density of the material composing the solid matrix of the porous adsorbent.

Validation by experimental comparisons on a double porosity synthetic activated carbon

The experimental results obtained by [START_REF] Perrier | A novel experimental set-up for simultaneous adsorption and induced deformation measurements in microporous materials[END_REF] on a double porosity synthetic activated carbon (Chemviron) are used in this study for validation purpose. The main advantage of the proposed method is to provide simultaneous in situ measurements of both adsorption and deformation for the same sample in the exact same conditions.

The material and the adsorption-induced strain measurements are briefly recalled in section 2.1. The model input parameters are identified in section 2.2 and finally comparisons between experimental and model results are performed in section 2.3.

Material description and adsorption-induced strain measurements

In this section, the experimental results obtained by [START_REF] Perrier | A novel experimental set-up for simultaneous adsorption and induced deformation measurements in microporous materials[END_REF] on a double porosity synthetic activated carbon are briefly recalled. Full details may be found in [START_REF] Perrier | A novel experimental set-up for simultaneous adsorption and induced deformation measurements in microporous materials[END_REF].

An activated carbon (Chemviron) is used as adsorbent material. The sample is a cylinder and its main characteristics are collected in Table 1. The geometrical dimensions have been measured with a caliper, the mass has been measured with a Precisa scale (XT 2220 M-DR), the specific pore surface has been measured with a gas porosimeter (Micromeretics ASAP 2020) according to the BET theory [START_REF] Brunauer | Adsorption of Gases in Multimolecular Layers[END_REF]. The specific micropore volume has been estimated according to the IUPAC classification [START_REF] Thommes | Physisorption of gases, with special reference to the evaluation of surface area and pore size distribution (IUPAC Technical Report)[END_REF] (pore diameter below 2 nm) based on a pore size distribution deduced from a low-pressure adsorption isotherm (N 2 at 77 K from 8.10 -8 to 0.99 P/P 0 in relative pressure range) measured with the same gas porosimeter according to the HK theory [START_REF] Horvath | Method for the calculation of effective pore size distribution in molecular sieve carbon[END_REF]. The specific macropore volume has been estimated according to the IUPAC classification [START_REF] Thommes | Physisorption of gases, with special reference to the evaluation of surface area and pore size distribution (IUPAC Technical Report)[END_REF] (pore diameter above 50 nm) based on a pore size distribution deduced from mercury intrusion porosimetry.

Both porosimetry techniques show that there are almost no pore of diameters between 2 nm and 50 nm in this material.

The two porosities are well separated in term of pore size distribution.

The adsorbates, CO 2 and CH 4 , as well as the calibrating gas, He, are provided with a minimum purity of 99.995%, 99.995% and 99.999% respectively. Fig. 3.a presents the results in term of excess adsorption/desorption isotherms. CO 2 and CH 4 gas sorption in activated carbon is a reversible phenomenon and no hysteresis is observed between adsorption and desorption paths as previously reported in the literature in [START_REF] Khaddour | A fully consistent experimental and molecular simulation study of methane adsorption on activated carbon[END_REF]. Noting that adsorbed quantity amount increases when temperature decreases, Fig. 3.a shows that CO 2 is preferentially adsorbed in carbon compare to CH 4 as previously reported in the literature [START_REF] Ottiger | Competitive adsorption equilibria of co2 and ch4 on a dry coal[END_REF][START_REF] Battistutta | Swelling and sorption experiments on methane, nitrogen and carbon dioxide on dry selar cornish coal[END_REF]. This is the reason why CO 2 injection is used to increase CH 4 recovery in Enhanced Coal Bed Methane production.

Fig. 3.b presents the results in term of adsorption-induced volumetric strain. CO 2 and CH 4 gas adsorption-induced deformation is a reversible phenomenon but a small hysteresis is observed between the adsorption and the desorption paths. This hysteresis is not linked to the adsorption-deformation couplings but is due to an elastic compaction of the carbon matrix grains [START_REF] Perrier | Coupling between adsorption and deformation in microporous media[END_REF]. Cycling effect and material compaction are detailed in [START_REF] Perrier | A novel experimental set-up for simultaneous adsorption and induced deformation measurements in microporous materials[END_REF]. For a given pressure, CO 2 adsorption produces more volumetric deformation than CH 4 adsorption, which is the source of the rapid decrease in CO 2 injectivity during coal bed methane production enhanced by CO 2 injection.

Identification of model input parameters

The input parameters of the incremental poromechanical framework with varying porosities for double porosity media presented in Eq. ( 12) are:

• The adsorbent sample mass (m s ), the molar mass of the adsorbed gas (M CO 2 , M CH 4 ), the density of the material composing the solid matrix of the porous adsorbent (ρ s ), the initial transport porosity (φ 0 M ) and the initial adsorption porosity (φ 0 µ ), which are all given in Table 1.

• The surrounding fluid bulk pressure (P b ), the excess adsorbed quantities (n ex ) and the bulk density (ρ b ), which are both experimentally measured or deduced. From the experimental measurements of the excess adsorption isotherm (Fig. 3.a), a power-law fit is identified and used as an input in the incremental estimation of Eq. ( 12).

From the bulk pressure and the temperature, the bulk density (ρ b ) of the surrounding fluid is estimated by its state equation using the AGA8 software [START_REF] Starling | Compressibility and super compressibility for natural gas and other hydrocarbon gases[END_REF].

• The skeleton incompressibility (K s ) and shear (G s ) modulii are deduced from the apparent ones from the two step homogenization reversed process presented in Eq. ( 6). The apparent properties are experimentally measured using an ultra-sonic technique where longitudinal and transverse waves are generated by a piezo-electric source and detected by a laser Doppler vibrometer [START_REF] Shen | Seismic wave propagation in heterogeneous limestone samples[END_REF]:

K = ρ s (V 2 p -4 3 V 2 s ) G = ρ s V 2 s . ( 13
)
In Eq. 13, (V p , V s ) are respectively the velocities of the longitudinal and the transverse waves. With V p = (302 ± 2) m.s -1 and V s = (176 ± 1) m.s -1 , we get K = (120 ± 15) MPa and G = (75 ± 8) MPa and then K s = (6.0 ± 0.6) GPa and G s = (3.5 ± 0.4) GPa.

Note that the experimental technique developed by [START_REF] Perrier | A novel experimental set-up for simultaneous adsorption and induced deformation measurements in microporous materials[END_REF] and allowing simultaneous measurements of adsorption-induced swelling may also be used to characterize K s directly as previously reported by [START_REF] Hol | Competition between adsorption-induced swelling and elastic compression of coal at co 2 pressures up to 100mpa[END_REF]. Indeed, if a non-adsorbable gas (such as helium) is used, the skeleton incompressible modulus may be deduced from bulk pressure and volumetric shrinkage strain measurements using Eq. 8. Figure 2 presents a typical result of K s direct identification. An experimental value of K s = (6 ± 1) GPa is then obtained, which is in good agreement with the latter one. Note that dynamic and static mechanical properties may differ for a lot of materials so a perfect match is not expected here. However, for this material with a low rigidity, the difference between the static and the dynamic properties is relatively not important and it may stand within the measurement uncertainty.

As discussed in [START_REF] Perrier | A novel experimental set-up for simultaneous adsorption and induced deformation measurements in microporous materials[END_REF], activated carbon is subjected to cycling effect and material compaction. The process to produce the active carbon is composed of three phases: first the carbon is grinded, then it is activated, and finally it is compacted to obtain a cylindrical sample. During the first cycle of gas adsorption, there is a competition between the grain compaction shrinkage and the adsorption-induced volumetric swelling and a large hysteresis is observed because of the material compaction. This compaction is mostly irreversible and after the first cycle, the second and the third cycles are reversible. Fig. 3 presents the results in term of adsorptioninduced volumetric strain obtained during the third cycle when the activated carbon is fully compacted and swelling strain fully reversible. The two other cycles are presented in [START_REF] Perrier | A novel experimental set-up for simultaneous adsorption and induced deformation measurements in microporous materials[END_REF]. However, the ultra-sonic technique used to identify the apparent elastic properties has been performed on the sample before compaction and the skeleton elastic properties may differ after compaction. Therefore, a second direct K s identification has been performed after the adsorption-induced swelling test and the value of K s = 7.0 ± 0.8 GPa is obtained. Assuming that the shear skeleton modulus is affected by the compaction in the same proportion of the incompressible onei.e. assuming that the Poisson's ratio is not affected by the compaction -the following skeleton modulii are identified and further used in the comparisons with experimental data:

K s = (7.0 ± 0.8) GPa G s = (4.1 ± 0.4) GPa . ( 14
)

Comparisons between experimental and model results

Fig. 3 presents also the results obtained with the double porosity adsorption-induced deformation model presented in part 1. All the parameters being identified in section 2.2, the volumetric strain induced by gas adsorption is estimated step by step as well as the evolutions of the transport and adsorption porosities and the poromechanical properties without any fitted parameters.

Fig. 3.b shows that the double porosity adsorption-induced deformation model presented in part 1 is capable to predict swelling induced by both CH 4 and CO 2 gas adsorption without any additional fitting parameters. The entry parameters in the model are those collected in Table 1, the skeleton elastic modulii corrected as explained in the latter section, and the adsorption isotherms. For this activated carbon, a swelling strain of ≈ 2% is recovered for a CO 2 bulk pressure up to 46 bar and a swelling strain of ≈ 1.5% is recovered for a CH 4 bulk pressure up to 107 bar.

Fig. 4 shows the same results in term of excess adsorption quantities versus adsorption-induced swelling. One can note that the relationship between excess adsorbed quantities and resulting swelling is not linear. Moreover, the two evolutions for the two different gases are close together showing that the volumetric swelling is directly linked to the excess adsorbed quantity.

Fig. 5 shows that for this challenging high micro and macro porous activated carbon, a single porosity model, as the one presented in [START_REF] Perrier | Poromechanics of adsorption-induced swelling in microporous materials: a new poromechanical model taking into account strain effects on adsorption[END_REF], highly overestimates the swelling deformation induced by gas adsorption in this activated carbon. The coupling appearing between the evolving adsorption porosity and the evolving transport porosity limits the macroscopic swelling of the material. This can only be captured with a double porosity model.

Evolution of the poromechanical properties under free swelling

The proposed double porosity model being validated by experimental comparisons in the latter section, we study here the evolution of the poromechanical properties under this free swelling.

Fig. 6 presents the evolution of the confinement degree in the adsorption porosity under free swelling for an activated carbon filled with pure CO 2 and pure CH 4 at T = 318.15 K and T = 303.15 K respectively. At the early adsorption stage, the confinement degree is high (≥ 0.9) for both CO 2 and CH 4 . This is due to the fact that at the onset of adsorption, the interstitial fluid density is much higher that the bulk density (ρ b << ρ µ ). Upon swelling, 0 5 -0.14 -0.13 -0.12 -0.11 -0.1 -0.09 -0.08

Pressure [bar]

Volumetric strain [%] Experiment Fit

Figure 2: Activated carbon K s direct identification based on Eq. 8 thanks to helium bulk pressure and volumetric shrinkage strain measurements (the slope provides directly K s ). (1) CO 2 -adsorption (1) CO 2 -desorption (1) CH 4 -adsorption

(1) CH 4 -desorption

(2) Model -mean

(2) Model -dispersion

(1) Experiment [START_REF] Perrier | A novel experimental set-up for simultaneous adsorption and induced deformation measurements in microporous materials[END_REF], (2) Model (this study). (1) CO 2 -adsorption

(1) CO 2 -desorption (1) CH 4 -adsorption

(1) CH 4 -desorption

(2) Model -mean (2) Model -dispersion

(1) Experiment [START_REF] Perrier | A novel experimental set-up for simultaneous adsorption and induced deformation measurements in microporous materials[END_REF], (2) Model (this study). the confinement degree is decreasing for both CO 2 and CH 4 . This may due to two main reasons: firstly, the ratio of the bulk density over the interstitial fluid density is increasing; secondly, the adsorption porosity is increasing due to the active carbon swelling, and therefore, the confinement of the fluid decreases. That way the functional

φ µ 1-φ µ -φ M in
Eq. 12 is increasing even faster. Therefore the total number of adsorbed gas moles increases faster than the excess number of moles and the confinement degree is decreasing, the CO 2 interstitial fluid being more confined than the CH 4 one.

Figs. 7 presents the evolution of the poromechanical properties in term of relative variations under free swelling. Fig. 7.a shows that all porosities are increasing with increasing bulk pressure under free swelling and, for this activated carbon, a relative variation of total porosity of ≈ 2% is recovered for a CO 2 bulk pressure up to 46 bar and a a relative variation of ≈ 1.5% is recovered for a CH 4 bulk pressure up to 107 bar. Due to the increase of porosities, Fig. Even if it may be counter-intuitive, Fig. 7.a shows that, under free swelling, the transport porosity is not decreasing even if the adsorption porosity increases. It simply increases less than the adsorption porosity. Note that this cannot be generalised for other materials or thermodynamic conditions because this is due to the complex couplings appearing between the transport and the adsorption porosities in Eq. 12. For the conditions considered here, Fig. 8 shows that, whatever the bulk pressure, the different contributions of the terms (T 1 to T 6 ) in Eq. 12 lead to a positive derivative of both porosities in respect to the bulk pressure, and therefore both porosities increases upon swelling (see Eq. 12 for

T 1 to T 6 term expressions).

δ r K µ δ r K (c) (d) -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0 2 4 6 8 10 12 CO 2 CH 4 Relative variation [%] Bulk pressure (P b ) [MPa] δ r b µ δ r b M -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0 2 4 6 8 10 12 CO 2 CH 4 Relative variation [%] Bulk pressure (P b ) [MPa] δ r N µ µ δ r N M M δ r N µ Μ
where δ r X = X-X 0 X 0 is the relative variation of X. with

n tot µ = n ex + ρ b V φµ M = n ex + m s M ρ b ρ s φ µ 1-φ µ -φ M (15)
Figs. 9 presents the evolution of the poromechanical properties in term of relative variations under constrained swelling. Fig. 9.a shows that the total porosity and the transport porosity are now decreasing whereas the adsorption porosity still increases. Indeed, and for the conditions considered here, Fig. 10 shows that, whatever the bulk pressure, T 7 and T 8 in Eq. 15 lead to a positive derivative of the adsorption porosity in respect to the bulk pressure, whereas T 9 and T 10 in Eq. 15 lead to a negative derivative of the transport porosity. Figs. 9.b to 9.d show the corresponding evolution of the other poromechanical properties with increasing bulk pressure. Fig. 11 shows the evolution of the total mean stress under constraint swelling. The volumetric strain being imposed equal to zero, the continuum is submitted to compressive total mean stresses.

Concluding remarks

• A new incremental poromechanical framework with varying porosity has been proposed allowing the prediction of the swelling induced by adsorption. Within this framework, the adsorption-induced strain are incrementally estimated based on experimental adsorption isotherm measurements only. The evolution of the porosity and the evolutions of the poromechanical properties, such as the apparent incompressible modulus, the apparent shear modulus, the Biot modulus and the Biot coefficient, are also predicted by the model.

• A double porosity model has been proposed for which the adsorption porosity and the transport porosity are distinguished. These two scales of porosity are supposed to be well separated and a two steps homogenization process is used to estimate incrementally the evolution of the poromechanical properties, which couple the evolutions of both porosities.

• An existing custom-built experimental set-up has been used to test the relevance of this double porosity model.

A challenging high micro and macro porous activated carbon has been chosen for this purpose. An adsorption porosity of 32.2±0.2% and a transport porosity of 41.6±0.2% have been characterized as well as its apparent and skeleton elastic properties. In situ adsorption-induced swelling has been measured for pure CO 2 and pure CH 4 at T = 318.15 K and T = 303.15 K respectively and the corresponding model responses have been estimated.

It has been shown that the double porosity model is capable to predict accurately the swelling induced by both CH 4 and CO 2 gas adsorption without any fitting parameters. Conversely, it has been shown that a single porosity model highly overestimates the swelling deformation induced by gas adsorption for this high micro and macro porous activated carbon. The coupling appearing between the evolving adsorption porosity and the evolving where δ r X = X-X 0 X 0 is the relative variation of X. 18 transport porosity limits the macroscopic swelling of the material. This can only be captured with a double porosity model.

• After validation, the double porosity model has been used to discuss the evolution of the poromechanical properties under free and constraint swelling. The case-study of constraint swelling consists here in assuming a global volumetric strain equal to zero. It has been shown that for the considered material, all porosities are increasing with increasing bulk pressure under free swelling, whereas the total porosity and the transport porosity are decreasing when the adsorption porosity still increases under constraint swelling.

(K s , G s ) knowing the homogenized properties (K m , G m ) and a given number of increment n: i) , K (i-1) , G (i-1) ) i) , K (i-1) , G (i-1) )

(K s , G s ) = R n (K m , G m , φ) with:                                K (0) = K m , G (0) = G m , ∆φ = φ n , φ (i) = ∆φ 1-φ+i∆φ          K (i) = H K (φ (
G (i) = H G (φ (
K s = K (n) , G s = G (n)
, where:

             H K (φ, K s , G s ) = K s + φK s 1-(1-φ)×( Ks Ks +Gs ) H G (φ, K s , G s ) = G s + φG s 1-(1-φ)×( Ks+2Gs 2Ks +2Gs )
.

(A.2) Bibliography

Figure 1 :

 1 Figure 1: Schematic of a double porosity media.

  the interstitial fluid in the two porosities (φ M , φ µ ) differ from the surrounding ones (P b , ρ b ) but the thermodynamic equilibrium imposes that the three fluids are chemically balanced (equality of the chemical potentials µ b , µ M and µ µ ). Assuming that the Gibbs-Duhem equation (dP = ρdµ) still applies for both the surrounding fluid and the interstitial ones, a macroscopic relation between the interstitial pore pressures and the surrounding one may be derived similarly to the relation initially proposed by[START_REF] Vermorel | Enhanced continuum poromechanics to account for adsorption induced swelling of saturated isotropic microporous materials[END_REF] and used inPerrier et al.

Fig. 3

 3 Fig.3presents the results of these simultaneous measurements for an activated carbon filled with pure CO 2 and pure CH 4 at T = 318.15 K and T = 303.15 K respectively. Full-field deformation maps and collected experimental data are reported in[START_REF] Perrier | A novel experimental set-up for simultaneous adsorption and induced deformation measurements in microporous materials[END_REF].

Figure 3 :

 3 Figure 3: Simultaneous adsorption and induced swelling measurements for an activated carbon filled with pure CO 2 and pure CH 4 at T = 318.15 K and T = 303.15 K respectively: (a) experimental excess adsorption isotherms; (b) comparison between experimental and modeling adsorptioninduced swelling results.

  adsorbed quantity [mmol.g -1 ]

Figure 4 :Figure 6 :

 46 Figure 4: Comparison between experimental and modeling results in term of excess adsorption quantities versus adsorption-induced swelling for an activated carbon: (a) CO 2 at T = 318.15 K; (b) CH 4 at T = 303.15 K.

  7.b shows that incompressible modulii are decreasing under free swelling, K decreasing faster than K µ . Consequently, b M is increasing and b µ upon free swelling as shown in Fig. 7.c. The evolution of the coupling poromechanical properties N MM , N µµ , N µM are more difficult to anticipate but Fig. 7.d shows that they are all decreasing upon swelling.

Figure 7 :Figure 8 :

 78 Figure 7: Evolution of the poromechanical properties in term of relative variations under free swelling for an activated carbon filled with pure CO 2 and pure CH 4 at T = 318.15 K and T = 303.15 K respectively.

Figure 9 :Figure 10 :Figure 11 :

 91011 Figure 9: Evolution of the poromechanical properties in term of relative variations under constrained swelling for an activated carbon filled with pure CO 2 and pure CH 4 at T = 318.15 K and T = 303.15 K respectively.

Table 1 :

 1 Main characteristics of the adsorbent and the adsorbates. 2 and CH 4 excess adsorption isotherms are built step by step from gas adsorption measurements performed using a custom-built manometric set-up. Simultaneously adsorption-induced swelling strain are measured based on digital image correlation. Full details are provided in[START_REF] Perrier | A novel experimental set-up for simultaneous adsorption and induced deformation measurements in microporous materials[END_REF].

	Property	Unit	Symbol	Value
	Height	(cm)	h	1.922 ± 0.004
	Diameter	(cm)	d	2.087 ± 0.002
	Volume	(ml)	V ech	6.57 ± 0.03
	Adsorbent sample mass	(g)	m s	4.137 ± 0.001
	Solid matrix density	(kg/L)	ρ s	2.4 ± 0.8
	Specific pore surface	(m 2 .g -1 )	S BET	1090 ± 10
	Specific micropore volume (cm 3 .g -1 )	v φ µ	0.51 ± 0.01
	Adsorption porous volume	(cm 3 )	V φ µ	2.115 ± 0.001
	Adsorption porosity Specific macropore volume (cm 3 .g -1 ) (%)	φ 0 µ v φ M	32 ± 1 0.66 ± 0.01
	Transport porous volume	(cm 3 )	V φ M	2.712 ± 0.001
	Transport porosity Total porosity	(%) (%)	φ 0 M φ 0	41 ± 1 73 ± 2
	CO 2 molar mass CH 4 molar mass	(g.mol -1 ) (g.mol -1 )	M CO 2 M CH 4	44.01 16.04

CO

This assumption is discussed in[START_REF] Perrier | Poromechanics of adsorption-induced swelling in microporous materials: a new poromechanical model taking into account strain effects on adsorption[END_REF] where both spherical and cylindrical porosities are considered.

Acknowledgements

Financial supports from the Région Aquitaine through the grant CEPAGE ( 20121105002), from the Conseil Départemental 64 through the grant CEPAGE2 (2015 0768), from the Insitut Carnot ISIFoR and from the Université de Pau et des Pays de l'Adour through the grant Bonus Qualité Recherche are gratefully acknowledged. We also gratefully acknowledge Dr. Frédéric Plantier and Dr. Christelle Miqueu for their advices and our different discussions and Dr. Valier Poydenot for his help concerning the ultra-sonic technique and the apparent properties identification. D. Grégoire and G. Pijaudier-Cabot are fellows of the Institut Universitaire de France.

(1) CO 2 -adsorption

(1) CO 2 -desorption (1) CH 4 -adsorption

(1) CH 4 -desorption

(2) Double porosity model

(3) Single porosity model

(1) Experiment [START_REF] Perrier | A novel experimental set-up for simultaneous adsorption and induced deformation measurements in microporous materials[END_REF], (2) Model (this study),

(3) Model [START_REF] Perrier | Poromechanics of adsorption-induced swelling in microporous materials: a new poromechanical model taking into account strain effects on adsorption[END_REF].

Figure 5: Comparison between the adsorption-induced swelling results provided by the single and the double porosity models for an activated carbon filled with pure CO 2 and pure CH 4 at T = 318.15 K and T = 303.15 K respectively.

Case-study of constrained swelling

In this section, one case-study of constrained swelling is considered. The volumetric strain is then assumed to be equal to zero and Eq. 12 may be rewritten as:

Iterative process of homogenization

Following Smaoui-Barboura [START_REF] Barboura | Modélisation micromécanique du comportement de milieux poreux non linéaires : Applications aux argiles compactées[END_REF], an iterative homogenization process may be applied for the linear homogenization functions used for a cylindrical porosity (Eq. 5) 3 .

Considering a given cylindrical porosity φ, the local skeleton properties (K s , G s ) and a given number of increment n, the global homogenized properties (K m , G m ) are determined step by step using the following scheme: i) , K (i-1) , G (i-1) )

, where:

.

(A.1)

Moreover, the latter iterative process may be reversed to determine step by step the local skeleton properties

3 For a full description on the iterative processes for both spherical and cylindrical porosities see [START_REF] Perrier | Poromechanics of adsorption-induced swelling in microporous materials: a new poromechanical model taking into account strain effects on adsorption[END_REF].


P Biasutti
email: pierre.biasutti@math.u-bordeaux.fr

J-F Aujol

M Brédif

A Bugeau

RANGE-IMAGE: INCORPORATING SENSOR TOPOLOGY FOR LiDAR POINT CLOUD PROCESSING

This paper proposes a novel methodology for LiDAR point cloud processing that takes advantage of the implicit topology of various LiDAR sensors to derive 2D images from the point cloud while bringing spatial structure to each point. The interest of such a methodology is then proved by addressing the problems of segmentation and disocclusion of mobile objects in 3D LiDAR scenes acquired via street-based Mobile Mapping Systems (MMS). Most of the existing lines of research tackle those problems directly in the 3D space. This work promotes an alternative approach by using this image representation of the 3D point cloud, taking advantage of the fact that the problem of disocclusion has been intensively studied in the 2D image processing community over the past decade. Using the image derived from the sensor data by exploiting the sensor topology, a semi-automatic segmentation procedure based on depth histograms is presented. Then, a variational image inpainting technique is introduced to reconstruct the areas that are occluded by objects.

Experiments and validation on real data prove the effectiveness of this methodology both in terms of accuracy and speed.

INTRODUCTION

Over the past decade, street-based Mobile Mapping Systems (MMS) have encountered a large success as the onboard 3D sensors are able to map full urban environments with a very high accuracy. These systems are now widely used for various applications from urban surveying to city modeling [START_REF] Serna | Urban accessibility diagnosis from mobile laser scanning data[END_REF][START_REF] Hervieu | Road marking extraction using a Model&Data-driven RJ-MCMC[END_REF][START_REF] El-Halawany | Detection of road curb from mobile terrestrial laser scanner point cloud[END_REF][START_REF] Hervieu | Semi-automatic road/pavement modeling using mobile laser scanning[END_REF][START_REF] Goulette | An integrated onboard laser range sensing system for on-the-way city and road modelling[END_REF]. Several systems have been proposed in order to perform these acquisitions. They mostly consist in optical cameras, 3D LiDAR sensor and GPS combined with Inertial Measurement Unit (IMU), built on a vehicle for mobility purposes [START_REF] Paparoditis | Stereopolis II: A multi-purpose and multi-sensor 3D mobile mapping system for street visualisation and 3D metrology[END_REF][START_REF] Geiger | Vision meets robotics: The KITTI dataset[END_REF]. They provide multi-modal data that can be merged in several ways, such as LiDAR point clouds colored by optical images or LiDAR depth maps aligned with optical images. Although these systems lead to very complete 3D mapping of urban scenes by capturing optical and 3D details (pavements, walls, trees, etc.), providing billions of 3D points and RGB pixels per hour of acquisition, they often require further processing to suit their ultimate usage.

For example, MMS tend to acquire mobile objects that are not persistent to the scene. This often happens in urban environments with objects such as cars, pedestrians, traffic cones, etc. As LiDAR sensors cannot penetrate opaque objects, those mobile objects cast shadows behind them where no point has been acquired (Figure 1, left). Therefore, merging optical data with the point cloud can be ambiguous as the point cloud might represent objects that are not present in the optical image. Moreover, these shadows are also largely visible when the point cloud is not viewed from the original acquisition point of view. This might end up being distracting and confusing for visualization. Thus, the segmentation of mobile objects and the reconstruction of their background remain strategic issues in order to improve the understanding of urban 3D scans.

We argue that working on simplified representations of the point cloud enables specific problems such as disocclusion to be solved not only using traditional 3D techniques but also using techniques brought by other communities (image processing in our case). Exploiting the sensor topology also brings spatial structure into the point cloud that can be used for other applications such as segmentation, remeshing, colorization or registration.

The main contribution of this paper is a novel methodology for point cloud processing by exploiting the implicit topology of various LiDAR sensors that can be used to infer a simplified representation of the LiDAR point cloud while bringing spatial structure between every points. The utility of such a methodology is here demonstrated by two applications.

First, a fast segmentation technique for dense and sparse point clouds to extract full objects from the scene is presented (Figure 1, center). Then, we introduce a fast and efficient variational method for the disocclusion of a point cloud using the range image representation while taking advantage of a horizontal prior without any knowledge of the color or texture of the represented objects (Figure 1, right). This paper is an extension of [START_REF] Biasutti | Disocclusion of 3D LiDAR point clouds using range images[END_REF] with improved technical details of the methodology as well as a complete validation of the proposed applications and a discussion about its limitations.

The paper is organized as follows: after a review on the state-of-the-art of the two application scenarios (Section 2), we detail how the topology of various sensors can be exploited to turn a regular LiDAR point cloud into a range image (Section 3). In Section 4, a point cloud segmentation model using range images is introduced with corresponding results and a validation on several datasets. Then, a disocclusion method for point clouds is presented in Section 5 as well as results and validation on various datasets. Finally conclusions are drawn and potential future work is identified in Section 6.

Related Works

The growing interest for MMS over the past decade has lead to many works and contributions for solving problems that could be tackled using range images. In this part, we present a state-of-the-art on both segmentation and disocclusion.

Point cloud segmentation

The problem of point cloud segmentation has been extensively addressed in the past years. Three types of methods have emerged: geometry-based techniques, statistical techniques and techniques based on simplified representations of the point cloud.

Geometry-based segmentation. The first well-known method in this category is regiongrowing where the point cloud is segmented into various geometric shapes based on the neighboring area of each point [START_REF] Huang | Automatic data segmentation for geometric feature extraction from unorganized 3D coordinate points[END_REF]. Later, techniques that aim at fitting primitives (cones, spheres, planes, cubes ...) in the point cloud using RANSAC [START_REF] Schnabel | RANSAC based out-of-core point-cloud shape detection for city-modeling[END_REF] have been proposed. Others look for smooth surfaces [START_REF] Rabbani | Segmentation of point clouds using smoothness constraint[END_REF]. Although these methods do not need any prior about the number of objects, they often suffer from over-segmenting the scene resulting in objects segmented in several parts.

Semantic segmentation.

The methods in this category analyze the point cloud characteristics [START_REF] Demantke | Dimensionality based scale selection in 3D LiDAR point clouds[END_REF][START_REF] Weinmann | Semantic point cloud interpretation based on optimal neighborhoods, relevant features and efficient classifiers[END_REF][START_REF] Landrieu | Comparison of belief propagation and graph-cut approaches for contextual classification of 3D LiDAR point cloud data[END_REF]. They analyze the geometric neighborhood of each point in order to perform a point-wise classification, possibly with spatial regularisation, which, in turn, yields a semantic segmentation. It leads to a good separation of points that belong to static and mobile objects, but not to the distinction between different objects of the same class.

Simplified model for segmentation. MMS LiDAR point clouds typically represent massive amounts of unorganized data that are difficult to handle. Different segmentation approaches based on a simplified representation of the point cloud have been proposed. [START_REF] Papon | Voxel cloud connectivity segmentationsupervoxels for point clouds[END_REF] propose a method in which the point cloud is first turned into a set of voxels which are then merged using a variant of the SLIC algorithm for super-pixels in 2D images [START_REF] Achanta | SLIC superpixels compared to state-of-the-art superpixel methods[END_REF]. This representation leads to a fast segmentation but it might fail when the scale of the objects in the scene is too different. [START_REF] Gehrung | An approach to extract moving objects from MLS data using a volumetric background representation[END_REF] propose to extract moving objects from MLS data by using a probabilistic volumetric representation of the MLS data in order to cluster points between mobile objects and static objects. However this technique can only be used with 3D sensors. Another simplified model of the point cloud is presented by [START_REF] Zhu | Segmentation and classification of range image from an intelligent vehicle in urban environment[END_REF]. The authors take advantage of the implicit topology of the sensor to simplify the point cloud in order to segment it before performing classification. The segmentation is done through a graph-based method as the notion of neighborhood is easily computable on a 2D image. Although the provided segmentation algorithm is fast, it suffers from the same issues as geometry-based algorithms such as over-segmentation or incoherent segmentation. Finally, an approach for urban objects segmentation using elevation images is proposed in [START_REF] Serna | Detection, segmentation and classification of 3D urban objects using mathematical morphology and supervised learning[END_REF]. There, the point cloud is simplified by projecting its statistics onto a horizontal grid. Advanced morphological operators are then applied on the horizontal grid and objects are segmented using a watershed approach. Although this method provides good results, the overall precision of the segmentation is limited by the resolution of the projection grid and leads to the occurence of artifacts at object borders.

Moreover, all those categories of segmentation techniques are not able to treat efficiently both dense and sparse LiDAR point clouds i.e. point clouds acquired with high or low sampling rates compared to the real-world feature sizes (e.g. macroscopic objects such as cars, pedestrians, etc.). For example, one sensor turn in the KITTI dataset [START_REF] Geiger | Vision meets robotics: The KITTI dataset[END_REF] corresponds to 10 5 points (sparse) whereas for a scene of similar size in the Stereopolis-II dataset [START_REF] Paparoditis | Stereopolis II: A multi-purpose and multi-sensor 3D mobile mapping system for street visualisation and 3D metrology[END_REF], the scene contains more than 4 • 10 6 points (dense). In this paper, we present a novel simplified model for segmentation based on histograms of depth in range images by leveraging grid-like topology without suffering from accuracy loss that is often caused by projection/rasterization.

Disocclusion

Disocclusion of a scene has only been scarcely investigated for 3D point clouds [START_REF] Sharf | Context-based surface completion[END_REF][START_REF] Park | Shape and appearance repair for incomplete point surfaces[END_REF][START_REF] Becker | LiDAR inpainting from a single image[END_REF]. These methods generally work on complete point clouds (with homogeneous sampling) rather than LiDAR point clouds. This task, also referred to as inpainting, has been much more studied in the image processing community. Over the past decades, various approaches have emerged to solve the problem in different manners. Patch-based methods such as the one proposed by Criminisi et al.

(2004) (and more recently [START_REF] Lorenzi | Inpainting strategies for reconstruction of missing data in VHR images[END_REF] and Buyssens et al. (2015b)) have proven their strengths. They have been extended for RGB-D images (Buyssens et al., 2015a) and to LiDAR point clouds [START_REF] Doria | Filling large holes in LiDAR data by inpainting depth gradients[END_REF] by considering an implicit topology in the point cloud. Variational approaches represent another type of inpainting algorithms [START_REF] Weickert | Anisotropic diffusion in image processing[END_REF][START_REF] Bertalmio | Image inpainting[END_REF][START_REF] Bredies | Total generalized variation[END_REF][START_REF] Chambolle | A first-order primal-dual algorithm for convex problems with applications to imaging[END_REF].

They have been extended to RGB-D images by taking advantage of the bi-modality of the data [START_REF] Ferstl | Image guided depth upsampling using anisotropic total generalized variation[END_REF][START_REF] Bevilacqua | Joint inpainting of depth and reflectance with visibility estimation[END_REF]. Even if the results of the disocclusion are quite satisfying, these models require the point cloud to have color information as well as the 3D data. In this work, we introduce an improvement to a variational disocclusion technique by taking advantage of a horizontal prior.

Range images derived from the sensor topology

In this paper, we demonstrate that a simplified model of the point cloud can be directly derived from it using the intrinsic topology of the sensing pattern during acquisition. This section introduces this sensor topology and how it can be exploited on various kinds of sensors. Examples of its usages are presented.

Sensor topology

Most of modern LiDAR sensors offer an intrinsic 2D topology in raw acquisitions.

However, this feature is rarely considered in recent works. Namely, LiDAR points may obviously be ordered along scanlines, yielding the first dimension of the sensor topology, linking each LiDAR pulse to the immediately preceding and succeeding pulses within the same scanline. For most LiDAR devices, one can also order the consecutive scanlines. It amounts to considering a second dimension of the sensor topology across the scanlines as it can be seen in Figure 2.

From sensor topology to range image

The sensor topology often varies with the type of LiDAR sensor that is being used. 2D

LiDAR sensors (i.e., featuring a single simultaneous scanline acquisition) such as the one used in [START_REF] Paparoditis | Stereopolis II: A multi-purpose and multi-sensor 3D mobile mapping system for street visualisation and 3D metrology[END_REF] generally send an almost constant number H of pulses per scanline (or per turn for 360 degree 2D LiDARs) where each pulse was emitted at a certain θ angle value. Therefore, any measurement of the sensor might be organized in an image of size W × H, where W is the number of consecutive scanlines and thus a temporal dimension. This is illustrated in Figure 3 in which one can see how the 2D image is spanned by the sensor topology. In this work, such images are only built using the range measurement as pixel intensity, later refered to as range images. Note that these range images differ from typical range images (Kinect, RGB-D) as the origin of acquisition is not the same for each pixel and the 3D directions of pixels are not regularly spaced along the image, but warped by the orientation changes of the sensor trajectory.

3D LiDAR sensors are based on multiple simultaneous scanline acquisitions (e.g. H = 64 fibers) such as in the MMS proposed in [START_REF] Geiger | Vision meets robotics: The KITTI dataset[END_REF]. Again, each scanline contains the same number of points and each scanline may be stacked horizontally to form the same type of structure, as illustrated in Figure 4. Note that Figures 3 and4 are simplified for better understanding, but that realistic cases can be more chaotic as discussed later in this section.

Whereas LiDAR pulses are emitted somewhat regularly, many pulses yield no range measurements due, for instance, to reflective surfaces, absorption or absence of target objects (e.g. in the sky direction) or an ignored measurement whenever the measure is too uncertain. Therefore the sensor topology is only a relevant approximation for emitted pulses but not for echo returns, such that the range image is sparse with undefined values where the sensor measured no echoes (or when further processing was performed on the acquisition, leading to the removal of points having a too incertain measurement). This is illustrated in Figure 5.b in which pulses with no echoes appear in dark. Note that considering multi-echo datasets as a multilayer depth image is beyond the scope of this paper, which only considers first returns.

This 2D sensor topology encodes an implicit neighborhood between LiDAR measurement pulses. Whereas the implicit topology of pixels in optical images is supported by a regular geometry of rays (shared origin and regular grid of directions if geometric distortion is neglected), the proposed 2D sensor topology for LiDAR point clouds is supported by the trajectory-warped geometry of 3D rays. However, it readily provides, with minimal effort, an approximation of the immediate 3D point neighborhoods, especially if the sensor moves or turns slowly compared to its sensing rate. We argue however that this approximation a.

b. is sufficient for most purposes, as it has the added advantage of providing pulse neighborhoods that are reasonably local both in terms of space and time, thus being robust to misregistrations, and being very efficient to handle (constant time access to neighbors).

Moreover, as LiDAR sensor designs evolve to higher sampling rates within and/or across scanlines, the sensor topology will better approximate spatio-temporal neighborhoods, even in the case of mobile acquisitions.

We argue that most raw LiDAR datasets contain all the information (scanline ordering, pulses with no echo, number of points per turn...) to enable the access to a well-defined implicit sensor topology. However it sometimes occurs that the dataset received further processings (points were reordered or filtered, or pulses with no return were discarded) or that the sensor does not acquire neighbouring points consecutively. Therefore, the sensor topology may then only be approximated using auxilliary point attributes (time, θ, fiber id...) and guesses about acquisition settings (e.g. guessing approximate ∆time or ∆θ values between successive pulse emissions). Using this information, one can recreate the range map by stacking points even if some points were discarded. Defining a grid-like topology is a good approximation if the number of pulses per scanline/per turn is close to an integer constant with relatively stable rotation offsets between pulses.

Interest and applications

The use of range images as the simplified representation of a point cloud directly brings spatial structure to the point cloud. Therefore, retrieving neighbors of a point, which was formerly done using advanced data structures [START_REF] Muja | Scalable nearest neighbor algorithms for high dimensional data[END_REF], is now a trivial operation and is given without any ambiguities. This was proved to be very useful in applications such as remeshing since faces can be directly associated to the grid structure of the range image. As shown in this paper, considering a point cloud as a range image supported by its implicit sensor topology enables the adaptation of the many existing image processing approaches to LiDAR point cloud processing (e.g.: segmentation, disocclusion).

Moreover, when optical data was acquired along with LiDAR point clouds, the range image can be used for improving the point cloud colorization and the texture registration on the point cloud as the silouhettes present in the range image are likely to be aligned with the gradients of optical images.

In the following sections, the LiDAR measurements, equipped with this implicit 2D topology, are denoted as the sparse range image u R .

Application to point cloud segmentation

In this section, a simple yet efficient segmentation technique that takes advantage of the range image will be introduced. Results will be presented and a quantitative analysis will be performed to validate the model.

Range Histogram Segmentation technique

We now propose a segmentation technique based on range histograms. For the sake of simplicity, we assume that the ground is relatively flat and we remove ground points, which are identified by plane fitting.

Instead of segmenting the whole range image u R directly, we first split this image into S sub-windows u R s , s = 1 . . . S of size W s × H along the horizontal axis to prevent each sub-window from representing several objects at the same range. For each u R s , a depth histogram h s of B bins is built. This histogram is automatically segmented into C s classes using the a-contrario technique presented in [START_REF] Delon | A nonparametric approach for histogram segmentation[END_REF]. This technique presents the advantage of segmenting a 1D-histogram without any prior assumption, e.g.

the underlying density function or the number of objects. Moreover, it aims at segmenting the histogram following an accurate definition of an admissible segmentation, preventing over-and under-segmentation. An example of a segmented histogram is given in Figure 6.

Once the histograms of successive sub-images have been segmented, we merge together the corresponding classes by checking the distance between each of their centroids in order to obtain the final segmentation labels. Let us define the centroid C i s of the i th class C i s in the histogram h s of the sub-image u R s as follows:

C i s = b∈C i s b × h s (b) b∈C i s h s (b) (1)
a.

b. where b are all bins belonging to class C i s . The distance between two classes C i s and C j r of two consecutive windows r and s can be defined as follows:

d(C i s , C j r ) = |C i s -C j r | (2)
Finally, we can set a threshold such that if d(C i s , C j r ) ≤ τ , classes C i s and C j r should be merged (e.g. they now share the same label). If two classes of the same window are eligible to be merged with the class of an other window, then only the one with lower depth should be merged. Results of this segmentation procedure can be found in the next subsection.

The choice of W s , B and τ mostly depends on the type of data that is being treated (sparse or dense). For sparse point clouds (few thousand points per turn), B has to remain small (e.g. 50) whereas for dense point clouds (> 10 5 points per turn), this value can be increased (e.g. 200). In practice, we found out that good segmentations may be obtained on various kinds of data by setting W s = 0.5 × B and τ = 0.2 × B. Note that the windows are not required to be overlapping in most cases, but for very sparse point clouds, an overlap of 10% is enough to achieve good segmentation. For example in our experiments on the KITTI dataset [START_REF] Geiger | Vision meets robotics: The KITTI dataset[END_REF], for range images of size 2215 × 64px, W s = 50, B = 100, τ = 20 with no overlap.

Results & Analysis

Figure 7 shows two examples of segmentations obtained using our method on different point clouds from the KITTI dataset [START_REF] Geiger | Vision meets robotics: The KITTI dataset[END_REF]. Each object, of different scale, is correctly distinguished from all others as an individual entity. Moreover, both results appear to be visually plausible.

Apart from the visual inspection, we also performed a quantitative analysis on the IQmulus dataset [START_REF] Vallet | TerraMobilita/IQmulus urban point cloud analysis benchmark[END_REF]. The IQmulus dataset consists of a manually annotated point cloud of 12 million points in which points are clustered into several classes corresponding to typical urban entities (cars, walls, pedestrian, etc.). Our aim is to compare the quality of our segmentation on several objects to the ground truth provided by this dataset. First, the point cloud is segmented using our technique, using 100px wide windows with a 10px overlap and a threshold for merging set to 50. After that, we manually select labels that correspond to the wanted object (hereafter: cars). We then compare the result of the segmentation to the ground truth in the same area, and compute the Jaccard distance (Intersection over Union) between our result and the ground truth. Figure 8 presents the result of such a comparison. The overall distance shows that the segmentation matches 97.09% of the ground truth, for a total of 59021 points. Although the result is very satisfying, our result differs in some ways from the ground truth. Indeed, in the first zoom of Figure 8, one can see that our model better succeeds in catching the points of the cars that are close to the ground (we remind here that the ground truth on IQmulus was manually labelled and thus subject to errors). In the second zoomed-in part, one can see that points belonging to the windows of the car were not correctly retrieved using our model. This is due to the fact that the measure in areas where the beam was highly deviated (e.g. beams that were not reflected in the same direction as the one they were emitted along) is not reliable as the range estimation is not realistic. Therefore our model fails in areas where the estimated 3D point is not close to the actual 3D surface. Note that a similar case appears for the review mirror (Figure 8, on the left) which is made of specular material that leads to bad measurements.

In some extreme cases, the segmentation is not able to separate objects that are too close from the sensor point of view. Figure 9.a shows a result of the segmentation in a scene where two cars that are segmented with the same label (symbolised by the same color).

In order to better distinguish the different objets, one can simply compute the connected components of the points regarding their 3D neighborhood (that can be computed using K-NN for example). Figure 9.b shows the result of such post-processing on the same two cars. We can notice how both cars are distinguished from one other.

Application to disocclusion

In this section, we show that the problem of disocclusion in a 3D point cloud can be addressed using basic image inpainting techniques.

a.

b.

Range map disocclusion technique

The segmentation technique introduced above provides labels that can be manually selected in order to build masks. As mentioned in the beginning, we propose a variational approach to the problem of disocclusion of the point cloud that leverages its range image representation. By considering the range image representation of the point cloud rather than the point cloud itself, the problem of disocclusion can be reduced to the estimation of a set of 1D ranges instead of a set of 3D points, where each range is associated with the ray direction of the pulse. The Gaussian diffusion algorithm provides a very simple algorithm for the disocclusion of objects in 2D images by solving partial differential equations. This technique is defined as follows:

   ∂u ∂t -∆u = 0 in (0, T ) × Ω u(t = 0, x, y) = u R (x, y) in Ω (3)
having u an image defined on Ω, t being a time range and ∆ the Laplacian operator. As the diffusion is performed in every direction, the result of this algorithm is often very smooth.

Therefore, the result in 3D lacks of coherence as shown in Figure 10.b.

In this work, we show that the structures that require disocclusion are likely to evolve smoothly along the x W and y W axes of the real world as defined in Figure 11.a. Therefore, we set η for each pixel to be a unitary vector orthogonal to the projection of z W in the u R range image (Figure 11.b). This vector defines the direction in which the diffusion should be done to respect this prior. Note that most MLS systems provide georeferenced coordinates of each point that can be used to define η. For example, using a 2D LiDAR sensor that is orthogonal to the path of the vehicle, one can define η as the projection of the pitch angle of the aquisition vehicle. We aim at extending the level lines of u along η. This can be expressed as ∇u, η = 0.

Therefore, we define the energy F (u) = 1 2 ( ∇u, η ) 2 . The disocclusion is then computed as a solution of the minimization problem inf u F (u). The gradient of this energy is given by ∇F (u) = -(∇ 2 u) η, η = -u η η , where u η η stands for the second order derivative of u with respect to η and ∇ 2 u for the Hessian matrix. The minimization of F can be done by gradient descent. If we cast it into a continuous framework, we end up with the following equation to solve our disocclusion problem:

   ∂u ∂t -u η η = 0 in (0, T ) × Ω u(t = 0, x, y) = u R (x, y) in Ω (4)
using the notations introduced earlier. We recall that ∆u = u η η + u η T η T , where η T stands 280 for a unitary vector orthogonal to η. Thus, Equation (4) can be seen as an adaptation of the Gaussian diffusion equation (3) with respect to the diffusion prior in the direction η. Figure 10 shows a comparison between the original Gaussian diffusion algorithm and our modification. The Gaussian diffusion leads to an over-smoothing of the scene, creating an aberrant surface, whereas our modification provides a result that is more plausible.

The equation proposed in (4) can be solved iteratively. The number of iterations simply depends on the size of the area that needs to be filled in.

Results & Analysis

In this part, the results of the segmentation of various objects and the disocclusion of their background are detailed.

Sparse point cloud.

A first result is shown in Figure 12. This result is obtained for a sparse point cloud (≈ 10 5 pts) of the KITTI database. A pedestrian is segmented out of the scene using our proposed segmentation technique (using the parameters introduced in 4.1) and a manual selection of the corresponding label. This is used as a mask for the disocclusion of its background using our modified variational technique for disocclusion. Figure 12.a shows the original range image. In Figure 12.b, the dark region corresponds to the result of the segmentation step for the pedestrian. For practical purpose, a very small dilation is applied to the mask (radius of 2px in sensor topology) to ensure that no outlier points (near the occluder's silhouette with low accuracy or on the occluder itself) bias the reconstruction.

Finally, Figure 12.c shows the range image after the reconstruction. We can see that the disocclusion performs very well as the pedestrian has completely disappeared and the result is visually plausible in the range image. Notice how the implicit sensor topology of the range image has allowed here to use a standard 2D image processing technique from mathematical morphology to filter mislabelled and inaccurate points near silhouettes.

In this scene, η has a direction that is very close to the x axis of the range image and the 3D point cloud is acquired using a 3D LiDAR sensor. Therefore, the coherence of the reconstruction can be checked by looking how the acquisition lines are connected. Figure 13 shows the reconstruction of the same scene in three dimensions. This reconstruction simply consists in the projection of the depth of each pixel along the axis formed by each corresponding point and the sensor origin. We can see that the acquisition lines are properly retrieved after removing the pedestrian. This result was generated in 4.9 seconds using Matlab on a 2.7GHz processor. Note that a similar analysis can be done on the results presented in Figure 1. Dense point cloud. In this work, we aim at presenting a model that performs well on both sparse and dense data. Figure 14 shows a result of the disocclusion of a car in a dense point cloud. This point cloud was acquired using the Stereopolis-II system [START_REF] Paparoditis | Stereopolis II: A multi-purpose and multi-sensor 3D mobile mapping system for street visualisation and 3D metrology[END_REF] and contains over 4.9 million points. In Figure 14.a, the original point cloud is displayed with the color based on the reflectance of the points for a better understanding of the scene. Figure 14.b highlights the segmentation of the car using our model (with the same parameters as in Section 4.2), dilated to prevent aberrant points. Finally, Figure 14.c depicts the result of the disocclusion of the car using our method.

We can note that the car is perfectly removed from the scene. It is replaced by the ground that could not have been measured during the acquisition. Although the reconstruction is satisfying, some gaps are left in the point cloud. Indeed, in the data used for this example, pulses returned with large deviation values were discarded. Therefore, the windows and the roof of the car are not present in the point cloud before and after the reconstruction as no data is available. We could have added these no-return pulses in the inpainting mask as well to reconstruct these holes as well.

Quantitative analysis. To conclude this section, we perform a quantitative analysis of our various point clouds in order to reconstruct them using our model. Therefore, the original point clouds can serve as ground truth. Note that areas are removed while taking care that no objects are present in those locations. Indeed, this test aims at showing how the disocclusion step behaves when reconstructing backgrounds of objects. The size of the removed areas corresponds to an approximation of a pedestrian's size at 8 meters from the 335 sensor in the range image (20 × 20px).

The test was done on 20 point clouds in which an area was manually removed and then reconstructed. After that, we computed the MAE (Mean Absolute Error) between the ground truth and the reconstruction (where the occlusion was simulated) using both

Gaussian disocclusion and our model. We recall that the MAE is expressed as follows:

MAE(u 1 , u 2 ) = 1 N i,j∈Ω |u 1 (i, j) -u 2 (i, j)| (5)
where u 1 , u 2 are images defined on Ω with N pixels where each pixel intensity represents the depth value. Table 1 sums up the result of our experiment. We can note that our method provides a great improvement compared to the Gaussian disocclusion, with an average MAE lower than 3cm. These results are obtained on scenes where objects are located from 12 to 25 meters away from the sensor. The result obtained using our method is very close to the sensor accuracy as mentionned by the manufacturer ( 2cm). Overlapping objetcs. Although the proposed disocclusion method performs well in realistic scenarios as demonstrated above, in some specific contexts, the reconstruction quality can be debatable. Indeed, when two small objects (pedestrians, poles, cars, etc.) overlap in front of the 3D sensor (e.g. one object is in front of the other), the disocclusion of the closest object may not fully recover the farthest object. Figure 16.a shows an example of such a scenario where the goal is to remove the cyclist (highlighted in green). In this case, a pole (Figure 16.a, in orange) is situated between the cyclist and the background.

Conclusion

In this paper, we have proposed a novel methodology for LiDAR point cloud processing that relies on the implicit topology that is brought by most recent LiDAR sensors.

Considering the range image derived from the sensor topology has enabled a simplified formulation of the problem from having to determine an unknown number of 3D points to estimating only the 1D range in the ray directions of a fixed set of range image pixels.

Beyond simplifying drastically the search space, it also provides directly a reasonable sampling pattern for the reconstructed point set. Moreover, it also directly provides a robust estimation of the neighborhood of each point according to the acquisition, while improving the computational time and the memory usage.

To highlight the relevance of this methodology, we have proposed novel aproaches for the segmentation and the disocclusion of objects in 3D point clouds acquired using MMS.

These models take advantage of range images. We have also proposed an improvement of a classical imaging technique that takes the nature of the point cloud into account (horizontality prior on the 3D embedding), leading to better results. The segmentation step can be done online any time a new window is acquired, leading to great speed improvement, constant memory requirements and the possibility of online processing during the acquisition. Moreover, our model is designed to work semi-automatically with using very few parameters in reasonable computational time. We have validated both the segmentation and the disocclusion methods by visual inspection as well as quantitative analysis against ground truth and we have proved their effectiveness in terms of accuracy.

In the future, we will focus on extending the methodology to other point cloud processing tasks such as LiDAR point cloud colorization / registration using range images and optical images through variational models.

Acknowledgement

J-F. Aujol is a member of Institut Universitaire de France. This work was funded by the ANR GOTMI (ANR-16-CE33-0010-01) grant. We would like to thank the anonymous reviewer for his/her useful comments.

Figure 1 :

 1 Figure 1: Result of the segmentation and the disocclusion of a pedestrian in a point cloud using range images. (left) original point cloud, (center) segmentation using range image, (right) disocclusion using range image. The pedestrian is correctly segmented and its background is then reconstructed in a plausible way.

Figure 2 :

 2 Figure 2: Example of the intrinsic topology of a 2D LiDAR sensor built on a plane

Figure 5 :

 5 Figure 5: Example of a point cloud from the KITTI database (Geiger et al., 2013) (a) turned into a range image (b). Note that the dark area in (b) corresponds to pulses with no returns.

Figure 6 :

 6 Figure 6: Result of the histogram segmentation using the approach of Delon et al. (2007). (a) segmented histogram (bins of 50cm), (b) result in the range image using the same colors. We can see how well the segmentation follows the different modes of the histogram.

Figure 7 :

 7 Figure 7: Example of point cloud segmentation using our model on various scenes. We can note how each label stricly corresponds to a single object (pedestrian, poles, walls).

Figure 8 :

 8 Figure 8: Quantitative analysis of the segmentation of cars. Our segmentation result only slighly differs from the ground truth in areas close to the ground or for points that were largely deviated such as points through windows.

Figure 9 :

 9 Figure 9: Result of the segmentation of a point cloud where two objects end up with the same label (a), and the labeling after considering the connected components (b).

Figure 10 :

 10 Figure 10: Comparison between disocclusion algorithms. (a) is the original point cloud (white points belong to the object to be disoccluded), (b) the result after Gaussian diffusion and (c) the result with our proposed algorithm (1500 iterations). Note that the Gaussian diffusion oversmoothes the background of the object whereas our proposed model respects the coherence of the scene.

Figure 11 :

 11 Figure 11: (a) is the definition of the different frames between the LiDAR sensor (xL, yL, zL) and the real world (xW , yW , zW ), (b) is the definition and the visualization of η.

Figure 12 :

 12 Figure 12: Result of disocclusion on a pedestrian on the KITTI database (Geiger et al., 2013). (a) is the original range image, (b) the segmented pedestrian (dark), (c) the final disocclusion. Depth scale is given in meters. After disocclusion, the pedestrian completely disappears from the image, and its background is reconstructed accordingly to the rest of the scene.

Figure 13 :

 13 Figure 13: 3D representation of the disocclusion of the pedestrian presented in Figure 12. (a) is the original mask highlighted in 3D, (b) is the final reconstruction.

Figure 14 :

 14 Figure 14: Result of the disocclusion on a car in a dense point cloud. (a) is the original point cloud colorized with the reflectance, (b) is the segmentation of the car highlighted in orange, (c) is the result of the disocclusion. The car is entirely removed and the road is correctly reconstructed.

  disocclusion model on the KITTI dataset. The experiment consists in removing areas of a

Figure 15 :

 15 Figure 15: Example of results obtained for the quantitative experiment. (a) is the original point cloud (ground truth), (b) the artificial occlusion in dark, (c) the disocclusion result with the Gaussian diffusion, (d) the disocclusion using our method, (e) the Absolute Difference of the ground truth against the Gaussian diffusion, (f) the Absolute Difference of the ground truth against our method. Scales are given in meters.

Figure 15

 15 Figure 15 shows an example of disocclusion following this protocole. The result of our proposed model is visually very plausible whereas the Gaussian diffusion ends up oversmoothing the reconstructed range image which increases the MAE.

Figure 16 .

 16 Figure 16.b presents the disocclusion of the cyclist. The background is reconstructed in a plausible way, however, details of the occluded part of the pole are not recovered.

Figure 16 :

 16 Figure 16: Example of a scene where two objects overlap in the acquisition. (a) is the original point cloud colored with depth towards sensor with the missing part of a pole highlighted with dashed pink contour, (b) shows the two objects that overlap: a pole (highlighted in orange) and a cyclist (highlighted in green), (c) shows the disocclusion of the cyclist. Although the background is reconstructed in a plausible way, details of the occluded part of the pole are missing.

Table 1 :

 1 Comparison of the average MAE (Mean Absolute Error) on the reconstruction of occluded areas.

		Gaussian Proposed model
	Average MAE (meters)	0.591	0.0279
	Standard deviation of MAEs	0.143	0.0232