Philippe Poullain

Pierre Mounanga

Guy Bastian

Roger Coue

R Coué Gem

Determination of the thermophysical properties of evolutive porous media: application to Civil Engineering materials

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

In the field of Civil Engineering, the materials either natural (wood, rock) or artificial (concrete, mortar) are often multiphase materials. Some are partially water-saturated porous media and exhibit complex behaviour when exposed to chemical, mechanical or physical solicitations. In particular, building materials can undergo temperature gradients during their lifetime, which can alter their mechanical or aesthetic performances by generating cracks. Predicting the thermomechanical behaviour of the structure [START_REF] Acker | Comportement du béton au jeune âge[END_REF][START_REF] Bournazel | [END_REF][START_REF] Coussy | Congrès Français de Thermique, SFT 2004, Presqu'île de Giens[END_REF] requires the determination of the material thermophysical properties.

The present paper describes the theoretical bases, the development and the validation of two different measurement methods usable to determine the heat transfer parameters of evolutive materials (thermal conductivity and volumetric heat capacity).

The first technique is based on the method of the "heated and non-heated wires" (HnHW method). It is validated on wet clay and then applied to the study of hydrating cementitious matrices maintained at 294 ± 1 K under endogenous conditions (no water exchange with the exterior) during the first 24 hours of hydration. A peripheral thermal regulation system enables to maintain the matrix a e-mail: pierre.mounanga@univ-nantes.fr in quasi-constant temperature conditions since the hydration heat release can perturb the measurement [1]. The theoretical bases of the HnHW method consider the transient temperature field around a continuous line source dissipating a constant power in an infinite medium. The fact that the material sample is a finite medium demands to correct the temperature measurement before further treatment. This correction is evaluated by means of the image method.

The second technique is adapted from that of the "heating film". It allows, thanks to a result analysis based on inverse treatment, to study materials that do not release heat and whose properties evolve over time and through space. Teledeltos paper strips play the role of the heating film. This experimental method and the use of Teledeltos paper allow a short time of measurement and thus a weak disturbance of the medium. Temperature profiles, recorded at different locations inside the medium, are used as input data for a forward calculation program, exploiting a minimization function implemented in MatLab r (MathWorks inc.). The "heating film" method coupled with inverse calculation, noted HFIC, is first validated on reference materials (hardened cement pastes and mortars). Then the functions describing the spatial evolutions of the thermal properties of wet sand submitted to drying are determined.

Before presenting the experimental devices, a brief reminder of the theoretical background of heat and mass transfers in porous media is exposed in Section 2.

Heat and mass transfers in porous media

Let us consider a non-deformable medium whose porosity contains liquid water, vapour and air. Assuming the following hypothesis:

• constant and uniform pressure of the gaseous phase in the material, • slow changes in the material microstructure as compared with the time of observation (<5 minutes in the present study), • weak rate of chemical reactions, • radiative heat transfer neglected, • heat of wetting neglected, • hysteresis effects neglected, heat and mass transfers can be modelled by the following set of equations [START_REF] Crausse | [END_REF]:

⎧ ⎪ ⎨ ⎪ ⎩ ρC P ∂T ∂t = ∇ • λ ∇T + ρ l L v D θv ∇θ ∂θ ∂t = ∇ • D θ ∇θ + D T v ∇T + ∂K l ∂z (1)
-T : temperature [K] θ: water content [%] t: time [s] z: ascending vertical coordinate [m] -ρC P : volumetric heat capacity [J m -3

K -1 ] -λ: apparent thermal conductivity [W m -1 K -1 ] -L v : latent heat of water vaporisation [J kg -1 ] -D θ : global hydric diffusivity [m 2 s -1 ] -D T v : vapour diffusivity due to temperature gradient [m 2 s -1 ] -D θv : vapour diffusivity due to water content gradient [m 2 s -1 ] -K l : hydraulic conductivity [m s -1 ] -ρ l : density of the liquid [kg m -3 ].
Assuming that moisture diffusion is a much slower process than heat diffusion, it is possible to develop fast transient measurement methods that do not lead to modifications in the water content [8]. In this case, the heat flux -ρ l L v D θv ∇ θ is much lower than -λ∇ T and system 1 can thus be simplified in:

ρC P ∂T ∂t = ∇ • λ ∇T (2)
where λ = λ 0 + ρ l L v D T v , the "apparent" thermal conductivity takes into account the pure heat conduction phenomenon described by Fourier's law (λ 0 : medium pure thermal conductivity [W m -1 K -1 ]) and the internal evaporation-condensation mechanism due to the diffusion of water vapour inside the pores and induced by temperature gradients [9].

Presentation of the experimental methods

The experimental determination of wet porous media thermal properties is a complex problem. This complexity is related to the state of pore water: it is important to ensure that the measurement method does not induce internal movements of the liquid phase and/or liquid/vapour phase change. Those phenomena can indeed have a significant influence on heat transfers and consequently on the thermophysical properties of the material [9,10]. Transient methods are thus preferred because they allow to shorten the test duration and therefore to avoid internal moisture movements [START_REF] Flynn | National Institute of Standards and Technology[END_REF]. In these methods, the heat transfer parameters are determined from the temperature response of a sample submitted to a short thermal solicitation. In this paper, two techniques are used:

• the "heated and non-heated wire" method (HnHW),

• the "heating film" method (HF).

HnHW method

Experimental device

In this measurement method, the follow-up of the temperature evolution of two stainless-steel wires, whose one is heated during a short time lapse, makes it possible to determine the thermal conductivity and the volumetric heat capacity of the surrounding medium. The medium temperature must be initially uniform. In order to fulfil this condition, the assembly is equipped with a system of temperature control. In the case of hydrating cementitious matrices, this system eliminates the influence of the hydration heat and ensures that the material properties evolve only over time.

Figure 1 shows a schematic diagram of the experimental device. The assembly is composed of a rigid, parallelepipedal mould, whose walls are covered with a 50 mmthick layer of thermal insulator (polyurethan foam) (see Fig. 1). The stainless-steel wire-sensors, covered with a thermo-retractable sheath, are tightened between two rigid elements situated at both ends of an aluminium rule embedded in the bottom of the mould. The sheath prevents electric conduction effect between the wires and the wet material. Bastian and Khelidj [START_REF] Bastian | [END_REF] showed that the influence of the thermo-retractable sheath on experimental values of λ and ρC P can be neglected.

The material is cast on the higher face of the first water circulation box placed at the bottom of the mould. The second water circulation box is then placed on the top of the sample. The weight of this box is transferred on the peripheral insulator layer through the outlet and inlet pipes of the water circuit, avoiding the crushing of the sample.

A thermocouple is embedded in the medium in order to follow the temperature evolution.

At the end of casting, the electric circuit providing the power to the heated wire is closed. This electric circuit is shown at Figure 2. It is composed of two meshes. During the heating phase (t h = 120 s), an alternative current (AC; I = 4 A) passes through the first mesh. Taking into account the stainless-steel wire resistance, this current delivers an electric power of 20 W m -1 . A direct current (DC; i = 80 mA), provided by a stabilized power supply passes continuously through the second mesh. The DC, too weak to overheat the stainless-steel wire-sensors, enables to measure their electric voltage. A relay, controlled by a data acquisition station, allows the closing of the first mesh during the heating phase and the closing of the second mesh during the measurement phase. The data acquisition station records automatically the evolutions of the stainless-steel wire electric voltage and the embedded thermocouple signal. The observation time is 3 minutes and the measurement cycle period is 15 minutes. The first measurement cycle is carried out 15 minutes after casting.

The knowledge of the "resistance vs. temperature" law of the stainless-steel wires makes it possible to convert the electric voltage evolution into temperature evolution.

Theoretical description

The HnHW method can be used to determine the thermophysical properties of a material from the temperature time evolution of wires embedded in the medium [START_REF] Bastian | [END_REF]13].

Continuous source line in an homogeneous infinite medium

Let us consider a continuous source line, materialized by an infinitely thin wire, dissipating a constant linear power P l [W m -1 ] from the moment t = 0 to t h, in an infinite medium whose temperature is initially uniform. This medium is characterized by its thermal conductivity λ [W m -1 K -1 ] and its volumetric heat capacity ρC P [J m -3 K -1 ]. In these conditions, the heat conduction equation is:

ρC P ∂T ∂t = ∇ • λ ∇T (3)
The resolution of equation ( 3) in cylindrical coordinates gives the following relation:

T (r, t) = P l 4 πλ E 1 r 2 4at ( 4
)
where r is the distance to the source [m], a = λ ρCP the medium thermal diffusivity [m 2 s -1 ] and t the time [s]. E 1 (x) is the exponential integral of x:

E 1 (x) = +∞ x e -v v dv ( 5
)
Equation ( 4) gives the temperature evolution of a nonheated wire embedded in the material, parallel to the heated wire at a distance r, but also describes the thermal evolution of the heated wire [START_REF] Martinet | Éléments de Thermocinétique : Conduction de la Chaleur (Technique et Documentation-Lavoisier[END_REF]. As soon as heating is stopped, the temperature has to be calculated from the following relation:

T (r, t) = P l 4πλ E 1 r 2 4at -E 1 r 2 4a (t -t h ) ( 6
)
This latter is obtained from the superimposition of the temperature field relative to the heat source and the temperature field of a heat sink.

The temperature evolution of the heated and nonheated wires from t = 0 to t > t h can be seen in Fig- ures 19 and 20 respectively (Appendix A). For the heated wire and for t >> t h , equation ( 6) can be approximated by [START_REF] Abramowitz | Handbook of Mathematical Functions[END_REF]:

T (r, t) = P l 4 πλ ln 4at r 2 -ln 4a (t -t h ) r 2 (7)
which yields, after transformation:

T (r, t) = P l 4πλ ln t t -t h ( 8
)
λ can thus be written as (where ∆ represents a time variation between two points of the curve, see Fig. 3):

λ = P l 4π ∆ ln t t-t h ∆T (9)
For the non-heated wire, the expression of the theoretical temperature curve slope at point I, indicated in Figure 20 (Appendix A) [13], can be derived from equation ( 4). The development of the exponential integral for very small values of x is [START_REF] Abramowitz | Handbook of Mathematical Functions[END_REF]:

E 1 1 x ≈ x e 1/x 1 + A 1 x + A 2 x 2 1 + B 1 x + B 2 x 2 ≈ x e 1/x (10)
At the point of inflexion I (see Fig. 4), where 4atI r 2 = 1, equation (4) can thus be written:

T (r, t I ) = P l 4πλ
4at I er 2 with e= 2.718... and finally:

ρC P = P l eπr 2 ∆t ∆T I (11)

Continuous source line in a finite medium -Image method

In the case of a finite medium, equations (3)[START_REF] Acker | Comportement du béton au jeune âge[END_REF][START_REF] Bournazel | [END_REF][START_REF] Coussy | Congrès Français de Thermique, SFT 2004, Presqu'île de Giens[END_REF][START_REF] Crausse | [END_REF](8)(9)(10)[START_REF] Flynn | National Institute of Standards and Technology[END_REF] can not be applied directly. Considering a slab, for example, the boundary conditions on its upper and lower faces influence the time evolution of the wire temperature. Therefore, the determination of the thermal properties from equations (9, 11) lays on a preliminary correction of the temperature profiles: the influence of the boundary conditions is quantified by the "image method" [START_REF] Carslaw | Conduction of Heat in Solids[END_REF].

According to this method, the real configuration, shown in Figure 5a, is equivalent to the configuration in an infinite space, given in Figure 5b.

In the second configuration (Fig. 5b), the problem finally consists in determining the temperature change of the heated and non-heated wires submitted to the influence of a heat source (heated wire) and to that of an infinity of alternated heat sinks and heat sources (imagewires), in an infinite medium, with an uniform initial temperature field.

Prior to the application of equations (9,[START_REF] Flynn | National Institute of Standards and Technology[END_REF] for the calculation of λ and ρC P , the thermal effects of the image heat sources and sinks have to be quantified and subtracted from the temperature time evolution of the wires. Complete development of the "image method" is given in Appendix A.

HF and HFIC methods

Experimental device

The geometry of study is composed of parallelepipedal test samples of square section.

The sample receives a heat flux Φ [W m -2 ] through one of its faces. The other faces are thermally insulated.

For materials with uniform thermophysical properties, only two thermocouples, centred on the heating film and on the opposite insulated face are sufficient to determine λ and ρC P (HF method - [1]). Analytical relations indeed link the difference between the temperature of the two thermocouples to λ and the slope of the temperature time evolutions to ρC P .

In the case of materials with non-uniform properties (for example drying materials exhibiting internal water content gradients) no analytical equation is available to determine the functions describing these parameters. The thermocouples are then placed into the material at different locations and the temperature time evolutions are used as input data for the forward calculation program. The experimental device for the HFIC method is shown in Figure 6 and examples of temperature time evolutions are presented in Figure 7. In the experiments, the temperature increase is always lower than 4 K.

Theoretical description

Temperature profiles measured at different locations in the material are analysed with an inverse calculation program to determine the medium thermal properties. Forward calculation lays on the numerical resolution of the heat transfer equation (Eq. ( 2)). The thermophysical parameters leading to the same time evolutions of temperature are sought numerically. This method is particularly adapted when space variations of the thermophysical properties are observed, especially during the drying of materials.

Direct problem

Heat conduction equation together with the thermophysical properties of the material, the initial and boundary conditions can give the temperature field in a material. In order to solve the problem numerically, the partial derivative equations are discretized applying central difference approximation and the ADI method (Alternating Direction Implicit - [START_REF] Özışık | Heat Conduction[END_REF]). A system of linear equations is then built in the shape of two easily inversible tridiagonal matrices (see Appendix B).

For two-dimensional problems, ADI method consists in using an implicit scheme in one direction of space and an explicit one in the perpendicular direction at one time step, and then inverting the implicit and explicit directions at the next time step.

Flux boundary conditions are written using "fictitious nodes" [START_REF] Özışık | Heat Conduction[END_REF]. The temperature derivatives are then expressed as the difference between a temperature at a node inside the material and a temperature at a virtual node out of the material.

The ADI method yields the following systems of equations:

A x T 2n = B z T 2n-1
+C z for the even moments (explicit in z and implicit in x)

A z T 2n+1 = B x T 2n + C x for the odd moments (explicit in x and implicit in z) where A x , A z , B x and B z are tridiagonal matrices and C x and C z are vectors expressing the boundary conditions. Thus, the temperature field can be obtained by calculating the inverse matrices of A x and A z :

T 2n = A -1 x B z T 2n-1 + C z T 2n+1 = A -1 z B x T 2n + C x 3.2.

Inverse problem

The resolution requires successive computations of the direct problem. Test function has to be minimized, i.e. the difference between the measured experimental data and the calculated numerical data has to be made as small as possible. Minimization is done by the conjugate gradient method [START_REF] Press | Numerical Recipes: The Art of Scientific Computing[END_REF]. In this work, the test function has the following form:

ε i (λ i ; (ρC P ) i ) = t=tmax t=0 N n=1 T exp n,t -T num n,t (λ i ; (ρC P ) i ) 2 ( 12
)
where t is the discrete time Figure 8 shows the principle of the inverse method. Initial values are given to the thermal conductivity and to the heat capacity and constitute the initial guesses of the problem. Those parameters are used to calculate the temperature field and to obtain a first estimate of the test function from a comparison with the experimental data (Eq. ( 12)). Calculation of the temperature field for parameters close to the initial guesses makes it possible to know the local shape of the test function and to guide the research towards the optimal parameters. Computation is terminated as soon as the difference between two consecutive values of ε i (λ i ; (ρC P ) i ) becomes lower than 10 -6 .

Experiments

Studied materials

Cement

The cement used during the tests is CPA-CEM I 52.5 PM ES from Le Teil (France). Its chemical composition is given in Table 1. The characterization tests were carried out in the section "Characterization and Study of Materials" at the LCPC (Laboratoire Central des Ponts et Chaussées, Paris).

Clay

Clay is used to validate the HnHW method. It is chosen because of its consistency and its thermophysical characteristics close to those of fresh cement pastes: it is a homogeneous sample of natural clay, already studied by Bastian [START_REF] Bastian | [END_REF]. Saturated clay is mixed during fifteen minutes. Distilled water is gradually added in order to obtain desired consistency. An impermeable plastic film is then placed on the mixer bowl. Drying is thus prevented during the few minutes that separate the end of mixing and the casting into the experimental device. The clay water content, defined as the mass ratio between water and dry material, is 123%.

Cement pastes and mortars

Cement pastes and mortars are prepared by first mixing the solid phase during thirty seconds in order to break the cement agglomerates and to homogenize the sand-cement mixture. Distilled and deaerated water is then added. Mixing is carried out during 3 minutes using a 5 L-mixer.

Sand

The sand comes from the region of Nantes (France), it has been sieved to keep the particles between 80 µm and 1 mm.

HnHW method -Calibration of the stainless-steel wires

The stainless-steel wires, embedded in the material, are used both as heating element and thermal sensors: their electric resistance R is indeed linearly related to their temperature T (Fig. 9) and can be modelled as:

R (T ) = R 0 (1 + αT ) ( 13
)
where R 0 is the resistance at 0

• C [Ω] and α is the tem- perature coefficient [K -1 ].
A calibration is carried out in a thermostated bath in order to determine α. The detailed method can be found in [START_REF] Bastian | [END_REF]. For the actual stainless-steel wire, the following value of α is obtained:

α = 2.6304 × 10 -3 K -1 ( 14
)
Knowing the material temperature that is presumed uniform before heating the wire, voltage measurements provide, with a good accuracy, the temperature of the wires [START_REF] Bastian | [END_REF].

HFIC method -Characteristics of the heating film

Teledeltos paper is used as the heating film. Its properties have already been given in [1]. The application of an electric potential difference between two points of a Teledeltos paper sheet creates an electric current able to heat the surface by Joule effect. In order to check the uniformity of heating on the paper sheet surface, a 50 V-electric potential difference is applied at the boundaries of the paper, painted with silver lacquer. This potential difference induces the circulation of a 50 mA-current through the film. The power dissipated by the film is then 2.5 W, which corresponds to a heat flux of 280 W m -2 . From voltage measurements in various points on the film surface, equipotential lines can be drawn. These lines are vertical, the power is thus uniformly dissipated on the whole surface of the heating film (see Fig. 10). Moreover, temperature measurements have proved that the power is only dissipated by the unpainted surface.

Validation tests

HF method, described by Mounanga et al. [1] and already validated on glass plates, is used as a reference technique.

The validation of the HnHW and HFIC methods consists in comparing the thermal conductivity and volumetric heat capacity values of the same material obtained from the different methods.

HFIC method

The tested materials are on one hand four 60 day-aged mortars with different water-to-cement mass ratios (W /C ratios) and sand-to-cement mass ratios (S/C ratios), and on the other hand glass and wet sand (mass water content: w = 10%). From casting to test, the mortars are kept in endogenous conditions in order to have their properties be uniform.

During the test, the samples are equipped with two thermocouples. One is in contact with the heating film; the second one is placed on the opposite insulated face.

The properties determined by the HF method are averages of four successive tests. For the inverse method, initial guesses of 1 W m -1 K -1 for λ and of 3 × 10 6 J m -3 K -1 for ρC P are taken.

Application to evolutive porous materials

HnHW method -Hardening cement pastes

In massive concrete structures, the hydration heat release of the cementitious matrix generates thermal stresses high enough to increase the risk of premature cracking of concrete. In most of the models proposed to predict the thermomechanical behaviour of concrete at early age, the thermophysical properties are considered as constant parameters, but very few studies [20,21] have been achieved to assess that hypothesis.

Experimental tests are performed to measure the thermal conductivity and the volumetric heat capacity of various W/C ratio cement pastes, during the first twenty four hours of hydration. The aim of the tests is to quantify the possible variations of λ and ρC P due to the microstructural changes occurring during the setting period of the matrices.

HFIC method -Drying sand

Tests are carried out on drying wet sand prismatic samples (4 × 4 × 16 cm 3 ) using the HFIC method. The initial water content of the sample is 10%. Only the upper square face is in contact with the air and allows moisture exchanges. Before commencing the test, drying is performed in a room at a temperature of 294 K and a relative humidity of 55% during 3 days. As the thermal properties evolve with the distance x to the drying face, the inverse method is chosen to identify functions λ (x) and ρC P (x). The water content w(x) is also measured. It is assumed to be uniform in a cross section of the sample located at x.

λ ρCP W/C S/C [W m -1 K -1 ] [10 6 J m -3 K -1 ] [W m -1 K -1 ] [10 6 J m -3 K -1 ] 50% 0
The water content is measured by a classical method described hereafter. Slices are cut in the sample and placed into different flasks. Each slice is then heated in a microwave oven and weighted until the mass is constant. The difference between the initial and final sand slice masses makes it possible to calculate its water content w:

w (%) = M i -M f M f -M flask 100% ( 15
)
where M flask is the flask mass [g], M i is the initial mass of the flask filled with the wet sand [g] and M f the final mass of the flask filled with dried sand [g].

Results and discussions

Validation of the experimental methods

HnHW method

The experimental results are given in Table 2.

A good repeatability of the measurements achieved with the HnHW device is observed: the relative difference between the measurements and the average value is lower than 2%. Moreover, the results obtained from both methods are very close, the relative variations are lower than 3 and 4% for the thermal conductivity and for the volumetric heat capacity, respectively.

HFIC method

Table 3 presents the thermophysical property values measured by the HF method and calculated by the HFIC method. A good agreement between the data measured by the HF method and the inverse method can be noted. The values of λ obtained by the inverse method are inside the zone of uncertainty of the HF method. The values of ρC P determined by the inverse method are slightly lower than the HF method results. However, the agreement is satisfactory. A better estimation of the parameters is possible by refining the grid of calculation. Indeed, although the discretization scheme is unconditionally stable, the solution can evolve according to the coarseness of the grid.

HnHW method -Hardening cement pastes

Temperature control of the samples

Application of equations (9,[START_REF] Flynn | National Institute of Standards and Technology[END_REF] used for the HnHW method requires initial constant temperature before each measurement cycle. It is thus necessary to dissipate the heat generated by the cement hydration. With this aim, a peripheral system of thermal regulation is integrated into the experimental device (see Sect. 3

.1.1).

A thermocouple is embedded in each sample in order to check the efficiency of this thermal regulation system (see Fig. 1).

Figure 13 reveals that, during the tests, the sample temperature is not constant; however, the thermal increase remains lower than 1 K and the maximum rate of the temperature evolution is lower than 0.3 K h -1 . An elementary calculation shows that these variations have a negligible influence on the thermal evolution of the stainless-steel wires.

Maturity and equivalent age

The hydration of cement is a thermoactivated chemical process: it is accelerated by a high ambient temperature. It means that a cementitious matrix maintained at constant temperature follows a different "thermodynamic path" than the same material submitted to another thermal history. The concept of maturity has been introduced to take into account the effect of temperature on the cement hydration rate. This concept needs the knowledge of the real temperature history of the material and consists in transforming the real age t of concrete into the equivalent age t eq using a suitable maturity function [START_REF] Carino | Proceedings of the 2001 Structures Congress & Exposition[END_REF]. The maturity function suggested by Arrhenius is best qualified for a proper description. With this function, the equivalent age can be derived:

t eq = t 0 exp -E a R 1 T (τ ) - 1 T ref dτ ( 16
)
with E a the apparent activation energy [J mol -1 ] and R the perfect gas constant [J mol -1 K -1 ]. In the following sections, the evolution of λ and ρC P of the cement pastes are expressed as a function of their equivalent age, considering the apparent activation energy of cement to be equal to 34.0 kJ mol -1 [23]. The reference temperature T ref is 293 K, in agreement with European standard.

Evolutions of λ and ρC P at 294 ± 1 K

The experimental results obtained for the thermal conductivity and the volumetric heat capacity of early age cement pastes are shown in Figures 14 and15.

During the first twenty-four hours, the thermal conductivity slightly increases (see Fig. 14), rising by 6, 11, 14 and 4% for the cement pastes with W/C ratio = 25, 30, 35 and 40%, respectively.

The influence of the W/C ratio on the evolution of λ remains weak. Indeed, the curves with W/C ratio = 30, 35 and 40% are superimposed over almost the whole investigation period. Although the curve with W/C ratio = 25% shows higher values, the differences with the other curves are lower than 7%. Thermal conductivity values measured at 24 hours are close for the four W/C ratios (see Tab. 4).

The volumetric heat capacity (Fig. 15) decreases during the first 24 hours of hydration. All the curves present the same decreasing evolution.

The high initial value of ρC P can be explained by the presence of non-chemically bound water within the material. The solid phase has indeed a much lower volumetric heat capacity than "free" water [START_REF] Bentz | [END_REF]. The progressive consumption of water by the hydration reactions causes an increase of the solid phase volume fraction and indirectly a decrease in the volumetric heat capacity of the material.

The thermal diffusivity, noted a, also characterizes the thermal behaviour of a medium. Table 4. Thermophysical properties of cement pastes after 24 hours of hydration at 294 ± 1 K. The evolution of the thermal diffusivity is plotted in Figure 16, for the studied cement pastes. The rise during the first hours of hydration can be explained by the increase of λ (Fig. 14) and the diminution of ρC P (Fig. 15).

W/C = 25% W/C = 30% W/C = 35% W/C = 40% λ [W m -1 K -1 ] 0.
The influence of the W/C ratio on the evolution of thermal diffusivity, in the range of W/C ratios investigated, is not very sensitive: the curves exhibit a similar evolution and reach, after 24 hours, similar thermal diffusivity values (see Tab. 4).

HFIC method -Drying sand

The sand sample is heated during 5 minutes through one of its larger faces (4 × 16 cm 2 ). The rise in temperature is measured by 6 thermocouples embedded at about 10 mm from the heating film (see Fig. 6). The precise thermocouple positions can be found in Table 5.

Parabolic functions are chosen to represent the space evolution of λ (x) and ρC P (x). The computation time is about 1 hour.

Space evolutions of the water content and of λ and ρC P are presented in Figure 17. The water content evolves linearly from 6.5% to 8.8% between the sample free surface and the opposite face. A quasi linear evolution is also observed for both λ and ρC P . From the drying face, their values increase and tend towards those found for wet sand (see Tab. 3).

These results show that drying is faster near the surface in contact with air. The replacement of liquid water by air/vapour mixture in the pores due to moisture exchange with the exterior provokes a diminution in the values of λ and ρC P . Liquid water indeed exhibits higher thermal conductivity and volumetric heat capacity than air and vapour.

Figure 18 shows the evolution of the thermal properties versus the water content. The results are compared to those previously determined by Mounanga et al. [1] and Delacre [2]. Although differences can be observed concerning the heat capacity for a water content of 0%, the values presented by Mounanga et al. [1] and Delacre [2] are in good agreement for both parameters. The difference at w = 0% can be related to the fact that the sand used by the authors is not of the same origin.

The results of the present paper are plotted on the same graph (see Fig. 18). The thermal conductivity values measured are consistent with those presented by Delacre [2]. The heat capacity values are situated slightly above those obtained by Delacre [2]. This can be, once again, attributed to the difference in composition of the two granular media.

Conclusions

In this paper, the "heated and non-heated wire" method (HnHW) and the "heating film with inverse calculation" method (HFIC) have been validated and applied to the determination of the thermophysical properties of evolutive materials. The results of these techniques have been compared with those obtained with the classical "heating film" method on a clayish material (HnHW method) and on hardened cement mortars and glass (HFIC method). A good agreement has been obtained between the different methods.

The HnHW method has then been used to study the thermal property evolution of hardening cement pastes as a function of their equivalent age. From tests performed at 294 ± 1 K, it is observed that, during the first day of hydration, the thermal conductivity increases slightly, the W/C ratio having little influence on this parameter. The evolution of the heat capacity shows a strong initial diminution, explained by the gradual decrease in the amount of interstitial "free" water during hydration. Indeed, water has a higher heat capacity than the solid phase of the material. These tests have been carried out under uniform temperature conditions. However, cement hydration is a thermo-activated process. It would thus be of interest to study the influence of the temperature on λ and ρC P in order to propose values adapted to "realistic" conditions (e.g. large concrete structures with non uniform temperature field). The present study focusing on the influence of W/C ratio, is a part of a larger research work. This latter includes the quantification of the temperature effect on the thermal properties [25].

The HFIC method has been applied to the study of drying sand. This technique together with the water content (w) measurement permitted the determination of the evolution of the thermal properties versus w. The results, showing the decreasing of both thermal conductivity and volumetric heat capacity with the diminution of w, are in agreement with previous papers.

Further works will aim at studying the thermal properties of more complex evolutive materials. In the field of Civil Engineering, it is the case for hardening mortars and cement pastes submitted to drying, for which the HFIC method is particularly suitable. following relation:

T c (r, t) = P l 4πλ n=+∞ n=-∞ E 1 x 2 + (y -2nh) 2 4at -E 1 x 2 + (y -(2n + 1) h) 2 4at - P l 4πλ n=+∞ n=-∞ E 1 x 2 + (y -2nh) 2 4a (t -t h ) -E 1 x 2 + (y -(2n + 1) h) 2 4a (t -t h ) (A.4)
Relations A.3 and A.4 are rigorous, but converge rapidly. In order to check for the convergence of these series, tests are carried out. The varying parameter is the number of sources and sinks whose temperature field is taken into account. The profile for a single source is given as a reminder. In these simulations, the medium width is 20 mm, the linear power is 20 W m -1 , the distance between the wires is 15 mm, λ is 1 W m -1 K -1 , a is equal to 10 -6 m 2 s -1 and t h is 120 s.

For the heated wire, Figure 19 shows that the number of sinks and sources has little influence on the temperature. The differences appearing at the end of the observed post-heating phase remain low. This shows the rapid convergence of the series given by equation A.4, near the source.

For the non-heated wire, no difference is observed during the heating phase (see Fig. 20). Only the post-heating phase exhibits a discrepancy between the profiles calculated with only few sources and sinks.

To calculate the temperature evolution of the heated and non-heated wires, it is possible to use a little number of sources and sinks to know the heating phase. For the post-heating phase, it is necessary to increase the number of reflections of the original source to get a better accuracy. For the determination of ρC P from equation ( 11) as explained in Section 3.1.2, only two sinks are used to correct the experimental data since the point of inflection is located in the heating phase during which the number of heat sources and sinks has little influence on the material temperature evolution.

Appendix B: Discretization of the problem

The heat conduction equation together with the boundary conditions (heat flux is null on the medium boundaries excepted on the lower face receiving the heat flux Φ, Fig. 21) and initial condition (temperature is null at time t = 0) can be used to determine the temperature field in the medium. In some cases, the thermophysical properties can evolve not only through space but also over time as the material internal structure changes. If this change is slow enough as compared with the experiment duration, the time evolution can be neglected. The system to be solved can thus be written as follow:

ρC P (x) ∂T ∂t = -∇. -λ (x) ∇T (B.1) ρC P (x) ∂T ∂t = λ (x) ∂ 2 T ∂x 2 + ∂ 2 T ∂z 2 + ∂λ ∂x ∂T ∂x (B.2) ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ -λ (x = 0) ∂T ∂x x=0;z = 0 -λ (x = L x ) ∂T ∂x x=Lx;z = 0 -λ (x) ∂T ∂z x;z=0 = Φ -λ (x) ∂T ∂z x,z=Lz = 0 T (x; z; t = 0) = 0. (B.3)
Functions λ (x) and ρC P (x) describe the spatial variations of the thermal conductivity and of the volumetric specific heat respectively. The problem is thus non linear and must be solved numerically.

The discretization can be done by the finite difference method. Özışık [START_REF] Özışık | Heat Conduction[END_REF] explains the different techniques that can be used to make the discretization. Central differences and the ADI method (Alternating Direction Implicit) are chosen to approximate the heat conduction equation. The ADI method consists in expressing the spatial derivatives with an implicit scheme in one direction and an explicit scheme in the other one. The directions of implicitness and explicitness are inverted at each time step. The advantage of this method is to obtain a linear system of equations that is unconditionally stable.

The following approximations are employed to express the spatial and temporal derivatives of the temperature field:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂T ∂x = T n i+1,j -T n i-1,j 2∆x ∂T ∂z = T n i,j+1 -T n i,j-1 2∆z ∂ 2 T ∂x 2 = T n i+1,j -2T n i,j +T n i-1,j ∆x 2 ∂ 2 T ∂z 2 = T n i,j+1 -2T n i,j +T n i,j-1 ∆z 2 ∂T ∂t = T n+1 i,j -T n i,j ∆t (B.4)
T n i,j is the temperature at time step n and at the node located by i along x and j along z. ∆x and ∆z are the space step. In the following calculi, ∆x = ∆z. ∆t is the time step.

If the implicitness direction is x and the explicitness direction is z, the heat conduction equation can be approximated as (general term):

[(b ia i ) T i-1,j + (1 + 2a i ) T i,j -(a i + b i ) T i+1,j ] n+1 = [a i T i,j-1 + (1 -2a i ) T i,j + a i T i,j+1 ] n (B.5) where parameters a j and b j are given by the following expressions: To calculate the temperature field, it is necessary to discretize the boundary conditions. These latter are treated through the method of the fictitious node. This allows to use central differences to express the spatial derivatives on the very boundaries. Figure 22 shows the principle of this method.

a i = λ i
For example, if the heat flux at the node i = 0 is null:

-λ ∂T ∂x = 0 ⇔ T -1 = T 1 (B.7)
In the expression of the general term at the node i = 0, T -1 can be thus replaced by T 1 and equation (B.6) becomes:

[-a 0 T 0,j-1 + (1 + 2a 0 ) T 0,ja 0 T 0,j+1 ] n+1 =

[(1 -2a 0 ) T 0,j + 2a 0 T 1,j ] n (B.8)

In the same way, the heat flux on the lower face of the medium is Φ, that is to say: The discretization yields a linear system of equations in the shape of two tridiagonal matrices for odd time steps and two others for even time steps:

-λ ∂T ∂z = Φ ⇔ T n i,-1 = T n i,1 -2 Φ λ ∆x (B.
-A z T 2n = B x T 2n-1 + C x implicit in z and explicit in x -A x T 2n+1 = B z T 2n + C z explicit in z and implicit in x A z , A x , B z , B x are tridiagonal matrices and C x and C z two vectors containing the flux boundary conditions. The resolution of this system requires the inversion of matrices A z andA x :

T 2n = A -1 z B x T 2n-1 + C x T 2n+1 = A -1 x B z T 2n + C z .
The initial temperature is null at each node of the medium:

T 0 = ⎡ ⎢ ⎣ 0 . . . 0 ⎤ ⎥ ⎦ .

Fig. 1 .

 1 Fig. 1. Schematic diagram of the "heated and non-heated wire" device (HnHW method).

Fig. 2 .

 2 Fig. 2. Diagram of the electric circuit for "the heated and non-heated wire" device.

Fig. 3 .

 3 Fig. 3. Temperature vs. non-dimensional time for the heated wire during the post heating phase.

Fig. 4 .

 4 Fig. 4. Temperature vs. time for the non-heated wire.

Fig. 5 .

 5 Fig. 5. Image method applied to the HnHW method.

Fig. 6 .

 6 Fig. 6. Schematic view of the experimental device for the HFIC method.

Fig. 7 .

 7 Fig. 7. Example of temperature time evolutions measured by thermocouples placed at different locations in a medium with non-uniform properties.

  [s], t max the observation time [s], N the number of temperature profiles [-], T exp n,t the experimental temperature profile [K], T num n,t the computed temperature profile [K].

Fig. 8 .

 8 Fig. 8. Principle of the inverse method.

Fig. 9 .

 9 Fig. 9. Calibration curve of the stainless-steel wire.

Fig. 10 .

 10 Fig. 10. Experimental electric potential field [V] at the surface of the heating film.

4. 4 . 1

 41 HnHW method Two tests on wet clay are carried out with the classical HF technique and three tests with the HnHW technique. Each test is achieved with a new sample.

Fig. 11 .

 11 Fig. 11. Mortar thermal conductivity vs. S/C ratio; comparison between the HF method and the HFIC method.

Figures 11

 11 Figures 11 and 12 show the evolutions of λ and ρC P as a function of mortar S/C ratio. The error bars are the standard deviations observed during the measurement by the HF method.A good agreement between the data measured by the HF method and the inverse method can be noted. The values of λ obtained by the inverse method are inside the zone of uncertainty of the HF method. The values of ρC P determined by the inverse method are slightly lower than the HF method results. However, the agreement is satisfactory. A better estimation of the parameters is possible by refining the grid of calculation. Indeed, although the discretization scheme is unconditionally stable, the solution can evolve according to the coarseness of the grid.

Fig. 12 .

 12 Fig. 12. Mortar volumetric heat capacity vs. S/C ratio; comparison between the HF method and the HFIC method.

Fig. 13 .

 13 Fig. 13. Temperature of the cement paste samples during the tests.

Fig. 14 .

 14 Fig. 14. Thermal conductivity of cement pastes with different W/C ratios (test temperature: 294 ± 1 K).

Fig. 15 .

 15 Fig. 15. Volumetric heat capacity of cement pastes with different W/C ratios (test temperature: 294 ± 1 K).

Fig. 16 .

 16 Fig. 16. Thermal diffusivity of cement pastes with different W/C ratios (test temperature: 294 ± 1 K).

Table 5 .Fig. 17 .

 517 Fig. 17. Evolution of the thermal properties and water content of wet sand after 3 days of drying.

Fig. 18 .

 18 Fig. 18. Sand thermal properties vs. water content; comparison with previous studies.

Fig. 19 .

 19 Fig. 19. Temperature time evolution of the heated wire; influence of the number of sources and sinks.

Fig. 20 .

 20 Fig. 20. Temperature time evolution of the non-heated wire; influence of the number of sources and sinks (large cross: point of inflection).

Fig. 21 .

 21 Fig. 21. Scheme of the problem.

Fig. 22 .

 22 Fig. 22. Method of the fictitious node.

  ∆t (ρC P ) i ∆x 2 ; b i =∂λ ∂x i 1 2 (ρC P ) i ∆t ∆xInverting the directions of explicitness and implicitness yields the following general term:[-a i T i,j-1 + (1 + 2a i ) T i,ja i T i,j+1 ] n+1 = [(a ib i ) T i-1,j + (1 -2a i ) T i,j + (a i + b i ) T i+1,j ] n (B.6)Equations (B.5, B.6) are the general terms given by the discretization of equation (B.2) inside the medium.

9 )

 9 Replacing T i,-1 by this latter expression in equation (B.5) yields:[-a 0 T i-1,0 + (1 + 2a 0 ) T i,0a 0 T i+1,0 ] n+1 = [(1 -2a 0 ) T i,0 + 2a 0 T i,1 ] n + (a 0b 0 ) R 0 (B.10) be done for each face and for the explicit and implicit discretizations, 16 different equations describing the flux boundary conditions can be written: 2 equations for each face and 2 equations for each corner. These latter equations are obtained by taking into account the flux on each face determining the corner. They can be found by combining the equations written for each face. For example, for a corner located at the junction of an insulated face and a face receiving the flux Φ (i = 0 and j = 0), equation (B.5) becomes:[(1 + 2a 0 ) T 0,0 -2a 0 T 1,0 ] n+1 = [(1 -2a 0 ) T 0,0 + 2a 0 T 0,1 ] n + (a 0b 0 ) R 0

Table 1 .

 1 Chemical composition of the cement. CPA-CEM I 52.5 PM ES Blaine specific surface area: 3320 cm 2 g -1

	Composition	CaO	SiO2 Al2O3 Fe2O3 MgO SO3 NaO2 K2O CO2
	% (mass)	66.39 21.17	2.69	1.96	0.65	2.43	0.3	0.22	1.56

Table 2 .

 2 Comparison between the experimental results obtained with HF and HnHW techniques.

	Wet clay		HF method			HnHW method
	(water content: 123%) Test No. 1 Test No. 2 Mean value Test No. 1 Test No. 2 Test No. 3 Mean value
	λ [W m -1 K -1 ]	0.91	0.93	0.92	0.88	0.90	0.90	0.89
	ρCP [10 6 J m -3 K -1 ]	3.95	3.74	3.85	4.07	3.96	3.93	3.99

Table 3 .

 3 Comparison between the experimental results obtained by HF and HFIC methods.

			HF method		HFIC method	Discrepancy
	Mortars	λ	ρ C P	λ	ρ C P

The authors are grateful to Mrs. Véronique Baroghel-Bouny, head of the section "Microstructure and Durability of Concrete" at LCPC (Paris) as well as the technical staff of the section "Characterization and Study of Materials" for their assistance as regards the characterization of the cement used.

Appendix A: Image method

The HnHW method lays on theoretical developments explained hereafter. In this part, the equations are related to the temperature field around a continuous line source in an infinite medium. The resolution of the transient heat transfer equation in cylindrical coordinates yields the following relation:

where r is the distance to the source [m], P l the linear power [W], λ the thermal conductivity of the medium

is the exponential integral of x, that is to say:

Let us consider a finite medium at the boundaries of which homogeneous conditions of the first kind are imposed (Fig. 5a). The time evolution of the temperature field can be found using the "image method". The problem results in the superimposition of the temperature fields generated by heat sources and sinks disposed according to a precise pattern. This pattern is given in Figure 5b. The positions of the sources and sinks are given hereafter (the origin is located on the heated wire and n ∈ ] -∞;+∞[).

Position of the sources: y = 2nh Position of the sinks: y = (2n + 1) h.

The following relation gives the temperature field in any point of the medium during the heating phase T h (r, t):

It can also be used to calculate the temperature field after heating T c (r, t). In this case, heat sinks are superimposed to the heat sources, and inversely, which yields the


Mehdi Yedroudj
email: mehdi.yedroudj@lirmm.fr

Marc Chaumont
email: marc.chaumont@lirmm.fr

Frédéric Comby
email: frederic.comby@lirmm.fr

Fr Éd Éric Comby

How to augment a small learning set for improving the performances of a CNN-based steganalyzer?

, and explores the ways to improve the performances of CNN. It aims at studying the effects of base augmentation on the performance of steganalysis using a CNN. We present the results of this study using various experimental protocols and various databases to define the good practices in base augmentation for steganalysis.

1.Introduction

Convolutional neural networks (CNN) became very popular to solve classification problems in the last five years. Several authors have proposed to use CNNs to solve steganalysis problems [START_REF] Qian | Deep Learning for Steganalysis via Convolutional Neural Networks[END_REF], [START_REF] Pibre | Deep learning is a good steganalysis tool when embedding key is reused for different images, even if there is a cover sourcemismatch[END_REF], [START_REF] Xu | Structural Design of Convolutional Neural Networks for Steganalysis[END_REF], [START_REF] Ye | Deep learning hierarchical representations for image steganalysis[END_REF]. These methods yield encouraging results but remained comparable to the state-of-the-art algorithms performances. Authors have explored many approaches to improve it such as using a phase split [START_REF] Chen | JPEG-Phase-Aware Convolutional Neural Network for Steganalysis of JPEG Images[END_REF], an ensemble of CNN [START_REF] Xu | Ensemble of CNNs for Steganalysis: An Empirical Study[END_REF], the transfer learning [START_REF] Qian | Learning and transferring representations for image steganalysis using convolutional neural network[END_REF] or the augmentation of the database [START_REF] Ye | Deep learning hierarchical representations for image steganalysis[END_REF], [START_REF] Zeng | Large-scale jpeg image steganalysis using hybrid deep-learning framework[END_REF].

Let us put aside the quest of the best deep learning network architecture for the steganalysis task. In this paper, our objective is to look at a "real-world" problem [START_REF] Ker | Moving Steganography and Steganalysis from the Laboratory into the Real World[END_REF], which is to learn with a small size database. This problem is also known as low regime learning. It is well-known that supervised approaches based on the use of CNNs need a lot of samples when used for steganalysis purposes. The seminal propositions of Qian et al. [START_REF] Qian | Deep Learning for Steganalysis via Convolutional Neural Networks[END_REF] and Pibre et al. [START_REF] Pibre | Deep learning is a good steganalysis tool when embedding key is reused for different images, even if there is a cover sourcemismatch[END_REF] used from 8 000 to 80 000 spatial images resized to 256×256 (BOSSBase [START_REF] Bas | Break Our Steganographic System': The Ins and Outs of Organizing BOSS[END_REF] or ImageNet [START_REF] Krizhevsky | Imagenet classification with deep convolutional neural networks[END_REF]). In 2017 the authors mainly use around 5 000 pairs of images [START_REF] Xu | Structural Design of Convolutional Neural Networks for Steganalysis[END_REF], [START_REF] Ye | Deep learning hierarchical representations for image steganalysis[END_REF], [START_REF] Chen | JPEG-Phase-Aware Convolutional Neural Network for Steganalysis of JPEG Images[END_REF], [START_REF] Xu | Deep Convolutional Neural Network to Detect J-UNIWARD[END_REF], which is probably insufficient. The number of images for the learning has even reached five millions of samples in [START_REF] Zeng | Large-scale jpeg image steganalysis using hybrid deep-learning framework[END_REF].

In an operational and realistic protocol, the number of available images for the learning task could be much smaller than what is used in "laboratory". Because all the CNN-based steganalysis are sensitive to the cover-source mismatch phenomenon [START_REF] Cancelli | A comparative study of +/-1 steganalyzers[END_REF][START_REF] Ker | A Mishmash of Methods for Mitigating the Model Mismatch Mess[END_REF][START_REF] Kodovskỳ | Study of cover source mismatch in steganalysis and ways to mitigate its impact[END_REF], each time the source distribution is modified, the learning process has to be restarted. The aim of this paper is thus to look at the impact of artificial data-augmentation, which is probably more realist than having access to a huge database of a given source distribution. In all cases, using data-augmentation is an automatic process which requires less human time consumption than searching for images of similar distributions.

Today, the classical scenario used to test an embedding algorithm efficiency is to use the BOSSBase [START_REF] Bas | Break Our Steganographic System': The Ins and Outs of Organizing BOSS[END_REF] for training and testing, assigning 5000 of the 10000 images to the learning database, while the rest used as testing database. A classical way to artificially increase the learning database without changing the labels is to flip and rotate the learning database without interpolation [START_REF] Krizhevsky | Imagenet classification with deep convolutional neural networks[END_REF].

Recently, Ye et al. [START_REF] Ye | Deep learning hierarchical representations for image steganalysis[END_REF] proposed to increase the size of the training database, by adding to the initial 50% of BOSSBase, the whole BOWS2 [START_REF] Bas | BOWS-2 Contest (Break Our Watermarking System)[END_REF] database (this gives a total of 15000 pairs of images for the training set), while the test set is unchanged and is made of the remaining 50% of BOSSBase. This process effectively improves the results in terms of error probability of detection. However, it could be considered as a very lucky measure because the improvement is essentially due to the fact that BOSSBase and BOWS2 share some identical camera models, and a similar "development" process 1 .

The question is thus still open: how should we process in order to enrich a learning database? Can we enrich even more the BOSS learning base in order to obtain a huge learning base, and thus improve the steganalysis results? In this paper we intend to experimentally explore efficient ways to increase the learning database of a CNN based steganalyzer. In Section 2, we recall the topology of the CNN used for the various experiments [START_REF] Yedroudj | Yedrouj-Net: An efficient CNN for spatial steganalysis[END_REF]. In Section 3, we describe the experimental protocol and briefly present all the setups. In Section 4, we experimentally explore the different augmentation methods and we draw conclusions on the practical question of the learning database augmentation.

2.Yedroudj-Net CNN

In this paper, our study on the data augmentation for spatial steganalysis is conducted only on the Yedroudj-Net [START_REF] Yedroudj | Yedrouj-Net: An efficient CNN for spatial steganalysis[END_REF]. This CNN has been created in 2017 and is a mix of the Xu-Net [START_REF] Xu | Structural Design of Convolutional Neural Networks for Steganalysis[END_REF] and Ye-Net [START_REF] Ye | Deep learning hierarchical representations for image steganalysis[END_REF], which are the two best CNNs created up to 2017 for steganalysis purposes. Yedroudj-Net gives better results than Xu-Net [START_REF] Xu | Structural Design of Convolutional Neural Networks for Steganalysis[END_REF] and Ye-Net [START_REF] Ye | Deep learning hierarchical representations for image steganalysis[END_REF] on WOW [START_REF] Holub | Designing Steganographic Distortion Using Directional Filters[END_REF] and S-UNIWARD [START_REF] Holub | Universal Distortion Function for Steganography in an Arbitrary Domain[END_REF], and also provides better results than an Ensemble Classifier [START_REF] Kodovský | Ensemble Classi-fiers for Steganalysis of Digital Media[END_REF] with a Rich Model [START_REF] Fridrich | Rich Models for Steganalysis of Digital Images[END_REF] when compared on a baseline where there is only one CNN, and no tricks such as the use of an ensemble or transfer learning. We have also conducted database augmentation experiments on Xu-Net [START_REF] Xu | Structural Design of Convolutional Neural Networks for Steganalysis[END_REF] and Ye-Net [START_REF] Ye | Deep learning hierarchical representations for image steganalysis[END_REF] and they follow the same trend as Yedroudj-Net. Yedroudj-Net is composed of a pre-processing block, five convolutional blocks, and a fully connected block made of three fully connected layers followed by a softmax. The network produces a probability distribution over the two class labels: stego or cover image. Fig. 1 illustrates the overall architecture of our CNN.

For more details on Yedroudj-Net, the reader can have a look at the paper [START_REF] Yedroudj | Yedrouj-Net: An efficient CNN for spatial steganalysis[END_REF] and the online code at http://www.lirmm. fr/ ~chaumont/DemoAndSources.html Note that the hyperparameters are kept identical.

3.Experimental methodology

3.1.Objectives and Dataset baseline

Our final objective is to increase the size of the learning database of a CNN based steganalysis through dataaugmentation in order to improve its performances. Indeed, increasing the number of learning samples is often beneficial for learning efficient features dedicated to a specific task. But, for steganalysis, the samples have to be selected carefully. The "new" samples have to share a "similar distribution" compared to the "original" samples. One thus tries to find distributionpreserving transformations which, when applied on an input cover or precover image, generate synthesized images that follow the same distribution. Those synthesized images could then be integrated into the learning database as additional images in order to increase the CNN classifier efficiency.

In this paper, first, we explore the factors that are influencing a cover distribution such as the camera model, or the development, and second, we propose distribution-preserving transformations that allow to enrich an initial database and to improve the CNN efficiency.

Our baseline setup will thus be working with the BOSSBase split into two sets. We assign 50% of the cover/stego pairs to the "original" training set, and the rest, to the testing set. For the training set, 4000 out of 5000 pairs are randomly selected for training and the remaining 1000 pairs are set aside for validation. Thus, the testing set is made of 5000 pairs. Regardless of the learning database enrichment, the test database will always contain images from and only from BOSSBase. For a fair comparison, we will use the same test base for all the experiments. To summarize, the learning set will always contain at least 4000 pairs of BOSSBase images, and the validation set will always contain 1000 pairs of BOSSBase images.

3.2.Software platform

We used S-UNIWARD [START_REF] Holub | Universal Distortion Function for Steganography in an Arbitrary Domain[END_REF], and WOW [START_REF] Holub | Designing Steganographic Distortion Using Directional Filters[END_REF], two well-known content-adaptive methods for the embedding in the spatial domain. Note that we used the Matlab implementations (online codes2 ) with the simulator for the embedding and a random key for each embedding. We thus avoid any wrong use of the C++ codes, i.e. a fixed and unique embedding key, as reported in [START_REF] Pibre | Deep learning is a good steganalysis tool when embedding key is reused for different images, even if there is a cover sourcemismatch[END_REF].

All experiments were performed with the publicly available Caffe toolbox [START_REF] Jia | Caffe: Convolutional architecture for fast feature embedding[END_REF] with necessary modifications, plus digits V5. All tests were run on an NVidia Titan X GPU card.

3.3.Datasets

Due to our GPU computing platform and time limitation, we conduct all the experiments on images of 256×256 pixels. To this end, we resampled all the 512×512 images to 256×256 images, using the imresize() Matlab function with the default parameters (bicubic interpolation with anti-aliasing).

For the various experimental setup, we are using the different databases listed below, and convert them to 256x256 images:

• the BOSSBase v1.01 [START_REF] Bas | Break Our Steganographic System': The Ins and Outs of Organizing BOSS[END_REF] consisting of 10 000 grey-level images of size 512 × 512, never compressed, and coming from 7 different cameras,

• the BOWS2 [START_REF] Bas | BOWS-2 Contest (Break Our Watermarking System)[END_REF] consisting of 10 000 grey-level images of size 512 × 512, never compressed, and whose distribution is close to BOSSBase,

• the LIRMMBase [START_REF] Pibre | Deep learning is a good steganalysis tool when embedding key is reused for different images, even if there is a cover sourcemismatch[END_REF] consisting of 9 388 grey-level images of size 512 × 512, never compressed, and coming from 7 different cameras. All the used cameras are different from those used in BOSSBase. This database is a variant of the LIRMMBase (http://www.lirmm.fr/ ~chaumont/ LIRMMBase.html) where images with no semantic content have been suppressed. Note that the development (of the RAW images) used in order to obtain the 256×256 images have been done reusing the same script than the one used for generating BOSS and BOWS2 (http://www.lirmm. fr/ ~chaumont/LIRMMBase/macroProductPGM.sh).

• the PLACES2 [START_REF] Zhou | Places: A 10 million image database for scene recognition[END_REF] containing more than one million of JPEG images coming from unknown cameras. For the experiments, those images are decompressed, converted in grey-level images, and then resized.

For some experiments, we re-run a development process and we will use the ImageMagick free and open-source software.

During the CNNs training, we regularly observe the Loss and Accuracy curves, computed on the validation test, to manually stop the training when an over-fitting phenomenon appears. This over-fitting occurs when the Loss curve continues to decrease on the training set but starts to increase on the validation set. For all the experiments, we report the error probability evaluated on the testing set.

3.4.Description of the different experimental setups

Below, we briefly listed all the experimental setups with a small description explaining each choice:

• Setup 1: Classical enrichment. In this setup, the goal is to obtain the performance baseline. The enrichment of the original learning database (made of 4000 pairs) is obtained thanks to the virtual augmentation using the labelpreserving flipping and rotations [START_REF] Ye | Deep learning hierarchical representations for image steganalysis[END_REF], and the enrichment with BOWS2 images. This experiment is presented in Section 4.1,

• Setup 2: Enrichment with other cameras. In this setup, the goal is to evaluate the gain/loss of adding images from different cameras from the ones used in the original learning set. This experiment is presented in Section 4.2,

• Setup 3: Enrichment with strongly dissimilar sources and unbalance proportions. In this setup, the goal is to evaluate the gain/loss of adding a huge number of images generated using cameras and a development, totally different from those used in the original learning set. This experiment is presented in Section 4.3,

• Setup 4: Enrichment with the same RAW images but with a different development. In this setup, the idea is to evaluate the gain/loss of adding the same original RAW images whose development is different from the one used for the original learning set. This experiment is presented in Section 4.4,

• Setup 5: Enrichment with a re-development of the learning set. In this setup, the objective is to evaluate the gain/loss of adding the same original images which are redeveloped. This experiment is presented in Section 4.5,

4.Results and discussions

4.1.Setup 1: Classical enrichment

In Table Rich Model + the Ensemble Classifier (SRM + EC), for the embedding algorithm WOW [START_REF] Holub | Designing Steganographic Distortion Using Directional Filters[END_REF] and S-UNIWARD [START_REF] Holub | Universal Distortion Function for Steganography in an Arbitrary Domain[END_REF] at payload 0.2 bpp. Yedroudj-Net has an error probability 8% lower for WOW algorithm at 0.2 bpp, and a similar error probability for S-UNIWARD at 0.2 bpp compared to SRM+EC. As reported in [START_REF] Yedroudj | Yedrouj-Net: An efficient CNN for spatial steganalysis[END_REF], Yedroudj-Net obtains similar or better results compared to the state-of-the-art (including versus Xu In Table 2, we report the results with no enrichment (noted BOSS), the results with the Virtual Augmentation (VA) of the BOSS's training set (noted BOSS + VA; Virtual Augmentation consists in label-preserving flipping and rotations), the results with BOWS2 enrichment (noted BOSS + BOWS2), and the results with BOWS2 enrichment + the Virtual Augmentation (noted BOSS+BOWS2+VA). Some of these results have already been given in [START_REF] Yedroudj | Yedrouj-Net: An efficient CNN for spatial steganalysis[END_REF], are re-presented in order to have a selfcontaining paper. Note that for BOSS+BOWS2, the training set is made of 14 000 pairs (without counting the validation), 32 000 pairs for BOSS+VA (without counting the validation), and for BOSS+BOWS2+VA, the training set is made of 112 000 pairs (without counting the validation).

When the enrichment is obtained by only applying a virtual augmentation (BOSS+VA), a significant improvement is ob-served. The decrease of the error probability detection is 3% for WOW (resp. 2% for S-UNIWARD). This enrichment measure was initially proposed in [START_REF] Krizhevsky | Imagenet classification with deep convolutional neural networks[END_REF] and it is indeed very efficient. The reader should understand that the VA is an easy and low-cost measure in order to significantly improve the performances.

One can also observe better performance when using BOSS+BOWS2 compared to only using BOSSBase. The CNN decreases its detection error probability by 4% for WOW (resp. 2% for S-UNIWARD). As stated in the introduction, BOSSBase and BOWS2 share some identical camera models and a similar "development" process. As also observed in Section 4.4, in a close setup, this enrichment setup ("similar cameras" + "similar development") allows to increase the performances. We guess that in that case, the added images increase the generalization capability of the network.

When the enrichment is obtained with BOSS+BOWS2+VA, again a significant improvement is observed. The decrease of the error probability detection is 7% for WOW (resp. 5% for S-UNIWARD) compared to the no-enrichment setup. Note that the results given in the current Section will be the reference performances for the comparisons given in the next sections.

The observations given in this Section are confirming that if the database augmentation ensures a good diversity of the database, the CNN can improve its detection accuracy. The experiments described in the next sections are thus done in order to better understand the properties that have to be kept when adding images to the original database. In Table 3, we report the results with no enrichment (noted BOSS), the results with LIRMM enrichment (noted BOSS + LIRMM), the results with LIRMM and BOWS2 enrichment (noted BOSS + LIRMM + BOWS2), and the results with LIRMM and BOWS2 enrichment + the Virtual Augmentation (noted BOSS + LIRMM + BOWS2 + VA). Note that for BOSS+LIRMM, the training set is made of 14 000 pairs, for BOSS+LIRMM+BOWS2, the training set is made of 23 388 pairs (without counting the validation), and for the BOSS+LIRMM+BOWS2, the training set is made of 187 104 pairs (without counting the validation).

4.2.Setup

One can observe that results are worst when using BOSS+LIRMM, compared to only using BOSSBase. There is 2% increase of the detection error probabilities for both WOW and S-UNIWARD. For this setup, the enrichment of the learning set is not strongly unbalanced (1 BOSS pair for 2 LIRMM pairs), done with images acquired with different cameras but processed with the same development. It seems that for a beneficial enrichment, the additional images have to be acquired with the same cameras. Additional facts seem to confirm this hypothesis in Section 4.3 and Section 4.4.

When enriching the BOSSBase with BOSS + LIRMM + BOWS2, the results are as good (or a slightly better for WOW) as using the BOSSBase alone. Finally, the results become better when BOSS+BOWS2+LIRMM2+VA is used, but the increase in performance is only of 0.9% for S-UNIWARD (resp. 2% for WOW), while using the BOSS+BOWS2 (see Tab.2) give 2% increasing for S-UNIWARD (resp. 4% for WOW).

Those results confirm again that performance is increased if there is an enrichment with images acquired with the same cameras and with the same development (BOWS-2 share similar cameras and a similar development). This tendency seems to contradict the idea that using millions of images, whose distribution is diverse, would be the best solution for increasing the steganalysis results [START_REF] Zeng | Large-scale jpeg image steganalysis using hybrid deep-learning framework[END_REF]. Indeed, the added images have to share a very similar "distribution" and images have probably to be acquired with the same cameras. In Section 4.3 we explore a little bit more this hypothesis. In Table 4, we report the results with no enrichment (noted BOSS), the results with 1% of PLACES2 enrichment (noted BOSS + PLACES2 1%), the results with 10% of PLACES2 enrichment (noted BOSS + PLACES2 10%), and 1% of PLACES2 enrichment (noted BOSS + PLACES2 100%). Note that for PLACES2 1%, the training set is made of 14 000 pairs (without counting the validation), for PLACES2 10%, the training set is made of 104 000 pairs (without counting the validation), and for the PLACES2 100%, the training set is made of 1 004 000 pairs (without counting the validation).

4.3.Setup 3: Enrichment with strongly dissimilar sources and unbalance proportions

Whatever the enrichment and whatever the embedding algorithm, the results are always worse than using the BOSSBase alone. For the setup where 1%, resp. 10%, resp. 100% of PLACES2 are added to the learning, the results get worse and worse, with respectively an increase of the detection error for S-UNIWARD (resp. WOW) of 5% (resp. 6%), 7% (resp. 12%), and then 9% (resp. 17%). Note that with an enrichment of 100% of PLACES2 (1 BOSS pair for 251 PLACES2 pairs), the detection is close to a random guessing.

Since the distribution of BOSS and PLACES2 are totally different (PLACES2 results from a JPEG dequantization, and a very diverse set of sources of cameras), the BOSS distribution is lost, and since no re-balancing measures are used during the learning, the BOSS distribution is considered as anecdotal and it is not really taken into account during the learning. Practically, the total loss computed for BOSS images is negligible compared to the total loss computed for PLACES2 images, and thus a minimization of the global loss will mainly concentrate on minimizing the loss associated to the PLACES2 images. Coming back to our previous statement, using millions of images is not sufficient [START_REF] Zeng | Large-scale jpeg image steganalysis using hybrid deep-learning framework[END_REF], the added images have to share a very similar "distribution" and images have probably to be acquired with the same cameras. In Table 4, we report the results with no enrichment (noted BOSS), and the results with 6 different versions of the BOSS-Base, each generated from the RAW images. There is an enrichment with a resizing with a bicubic interpolation (noted BOSS+DEV:Res-Bicub), the an enrichment with a resizing with a spline interpolation (noted BOSS+DEV:Res-Spline), the enrichment with a resizing without any interpolation (noted BOSS+DEV:Res-NoInt), the enrichment with no resizing and a central crop (noted BOSS+DEV:Crop), the enrichment with a resizing to a 768×768 images without any interpolation and then a central crop (noted BOSS+DEV:Res+Crop), and finally an enrichment with the use of Adobe Photoshop Lightroom 6 instead of ImageMagick, for generating the color images and then resizing to 256 × 256 the images while keeping the width/length ratio (noted BOSS-APL).

From Table 5, we can observe that the enrichment with a crop development (BOSS+DEV:Crop) lead to very bad results. The increase of the detection error of 7% for S-UNIWARD (resp. 7% for WOW). The enrichment with a resize to 768×768 followed by a crop (BOSS+DEV:Res+Crop), to a lesser extent, also give bad results with an increase of the detection error of 1% for S-UNIWARD (resp. 0.3% for WOW). Those bad results suggest that a resolution change during the development has a strong impact on the pixels distributions. When looking to the extreme case of the crop development (BOSS+DEV:Crop), we easily understand that the resulting images content change; there is almost no variations and no edges. Thus, an enrichment with a BOSS version whose development does not ensure the same final pixel resolution than BOSS Base will not enrich favourably the learning data-base.

In counterpart, using the same resize procedure with a slight variation on the interpolation (spline, no-interpolation, bicubic), or with the Adobe Photoshop Lightroom Process allows scrounging at most 1% for S-UNIWARD (resp. 2% for WOW). This confirms that additional samples very close to the target BOSS distribution can improve the learning capabilities. Looking back to the various experiment done previously, one can observe that in order to enrich favourably a target database, a favourable measure is to use images acquired with the same cameras than the target database, and to use a very close resizing process than the one used for the target database. In order to push the reflection a little bit more, we made an additional experiment where we regrouped diverse versions of BOSSBase (BOSS+DEV:Res+Bicub, BOSS+DEV:Res+Spline, BOSS+DEV:Res+NoInt, BOSS+DEV:Res+Crop) to the exception of BOSS+DEV:Crop. In Table 6, we report the results with this gathering of various development (noted BOSS+all-DEV), and the results with LIRMM and BOWS2 enrichment (noted LIRMM+BOWS2 and already reported in Section 4.1). Note that for BOSS+all-DEV, the training set is made of 44 000 pairs (without counting the validation), and for LIRMM+BOWS2 the training set is made of 14 000 pairs (without counting the validation).

For those two enrichments, there is a real improvement with a decrease of the error probability of detection of 2-3% for S-UNIWARD (and 4% for WOW). This last result is very interesting and shows that in order to enrich a database, in a practical scenario, there are at least those two options:

Given a target database:

• either Eve (the steganalyst) finds the same camera(s) (used for generating the target database), capture new images, and reproduce the same development than the target database, with a special caution to the resizing,

• either Eve has an access to the original RAW images and reproduce similar developments than the target database with the similar resizing,

The reader should also remember that the Virtual Augmentation is also a good cheap processing measure. Note that it is unclear which option would be better in a practical case. Additional experiments have to be done in the future. Anyway, those two enrichments show that a very caution process has to be taken for really improving the results. We believe that those enrichments reduce the over-fitting and also improve the generalization of the learner.

4.5.Setup 5: Enrichment with a re-development of the learning set

In all previous setups, given a target database (never compressed 8-bits grey-level 256×256 images), we were presuming either a prior knowledge of the cameras used for the images acquisitions or a direct access to the RAW versions of the original images. In real-world cases, those knowledges are most of the time not available. Moreover, retrieving the camera models is a very complicated task in a real scenario due to the huge number of cameras. WOW 0. In Table 7, we report the results with no enrichment (noted BOSS), and the results with 2 different redeveloped versions of the BOSSBase, each generated from the original 256×256 8-bits grey-level BOSSBase images. The first redevelopment (noted BOSS+DEV:Translation) consists in applying a sub-pixel image translation, of 0.5 pixel, on the padded (symmetric padding) images, and then applying a crop operation to re-obtain a 256×256 images. The second redevelopment (noted BOSS+DEV:Up-Down-Sampling) consists in applying a Lanczos3 filter for the up-sampling in order to obtain a 512×512 images, and then downsampling with the same interpolation Kernel to re-obtain images of 256×256 size. The results are catastrophic with an increase of the error probability of 6% to 11% for S-UNIWARD and 4% to 7% for WOW. The use of a redevelopment does not seem to be a good idea.

5.Conclusion

In this paper, we have explored ways to enrich a learning database when steganalysis is done with a CNN. The enrichment is a crucial task since, in the majority of the today's experiments, the required number of images have to be extremely high due to the huge number of parameters to be learned. Using an insufficient set of examples (images) leads to CNNs that have not "learned enough" and the average efficiency is thus reduced.

After recalling the state-of-the-art of 2017 for the spatial CNN steganalysis, and briefly recalling the state-of-the-art steganalysis approach named Yedroudj-Net, we have presented various results. Additionally to the classical data augmentation which consists to apply flips and rotations on the learning images [START_REF] Krizhevsky | Imagenet classification with deep convolutional neural networks[END_REF], we observed two others ways for favorably enriching the learning database. The trend is that, in a clairvoyant scenario (knowledge of the embedding algorithm, knowledge of the payload size, approximate knowledge of the of the images distribution), for a given target (test) database, in order to augment its learning database, the steganalyst (Eve) has two choices:

• Either she is able to guess the camera(s) used for generating the target database. She thus captures new images, and reproduce a similar development than the target database, with a special caution to the resizing,

• Either she has an access to the original RAW images and reproduces a similar development than the target database with the similar resizing.

Those two possible ways to enrich the database are very restrictive. As explained in the paper some complementary solutions can be used such as transfer learning [START_REF] Qian | Learning and transferring representations for image steganalysis using convolutional neural network[END_REF], or the use of ensembles [START_REF] Xu | Ensemble of CNNs for Steganalysis: An Empirical Study[END_REF], but the underlying questions of generalizations / cover-source mismatch have to be explored deeper in the future.

Figure 1 .

 1 Figure 1. Yedroudj-Net CNN architecture. Figure taken from [1].

Table 1 : Steganalysis error probability of Yedroudj-Net, and SRM+EC for two embedding algorithms WOW and S-UNIWARD at 0.2 bpp and 0.4 bpp.

 1

	BOSS 256×256

1, we report the error probability obtained when there is no enrichment which means there are 4000 pairs in the learning set (+ 1000 pairs in the validation set), and 5000 pairs in the test set. All images are from BOSSBase. For a cursory comparison, the performance is reported for the Yedroudj-Net, and the Spatial

Table 2 : Base Augmentation influence: error probability of Yedroudj-Net, on WOW and S-UNIWARD at 0.2 bpp with and without Data Augmentation.

 2

	-Net and Ye-Net) in a fair

2: Enrichment with other cameras

		WOW 0.2 bpp S-UNIWARD 0.2 bpp
	BOSS	27.8 %	36.7 %
	BOSS+LIRMM	29.9 %	38.6 %
	BOSS+LIRMM+BOWS2	26.8 %	36.9 %
	BOSS+LIRMM+BOWS2+VA 25.7 %	36.1 %

Table 3 : Base Augmentation influence: error probability of Yedroudj-Net, on WOW and S-UNIWARD at 0.2 bpp with a learn- ing base augmented with either LIRMM, LIRMM+BOWS2, or LIRMM+BOWS2+VA.

 3

Table 4 : Base Augmentation influence: error probability of Yedroudj-Net, on WOW and S-UNIWARD at 0.2 bpp with a learn- ing base augmented with different portions of PLACES2.

 4

		WOW 0.2 bpp S-UNIWARD 0.2 bpp
	BOSS	27.8 %	36.7 %
	BOSS+PLACES2 1%	34.2 %	41.6 %
	BOSS+PLACES2 10%	40.0 %	43.9 %
	BOSS+PLACES2 100% 44.6 %	45.3 %

4.4.Setup 4: Enrichment with the same RAW im- ages but with a different development

		WOW 0.2 bpp S-UNIWARD 0.2 bpp
	BOSS	27.8 %	36.7 %
	BOSS+DEV:Res-Bicub	25.7 %	37.5 %
	BOSS+DEV:Res-Spline 26 %	35.8 %
	BOSS+DEV:Res-NoInt	25.6 %	36.2 %
	BOSS+DEV:Crop	34.8 %	44.2 %
	BOSS+DEV:Res-Crop	28.1 %	37.9 %
	BOSS+BOSS-ALP	26.0 %	35.5%

Table 5 : Base Augmentation influence: error probability of Yedroudj-Net, on WOW and S-UNIWARD at 0.2 bpp with a learn- ing base augmented with different BOSSBase versions.

 5

Table 6 : Base Augmentation influence: error probability of Yedroudj-Net, on WOW and S-UNIWARD at 0.2 bpp with a learn- ing base augmented with different versions of BOSSBase.

 6

		WOW 0.2 bpp S-UNIWARD 0.2 bpp
	BOSS	27.8 %	36.7 %
	BOSS+all-DEV 23.0 %	33.2 %
	BOSS+BOWS2 23.7 %	34.4%

Table 7 : Base Augmentation influence: error probability of Yedroudj-Net, on WOW and S-UNIWARD at 0.2 bpp with a learn- ing base augmented with a re-development of BOSSBase.

 7

			2 bpp S-UNIWARD 0.2 bpp
	BOSS	27.8 %	36.7 %
	BOSS+DEV:Translation	34.7.0 %	47.8 %
	BOSS+DEV:Up-Down-Sampling 31.2 %	42.6 %

The "development" stands for the numerical processes transforming a color RAW image to a

256×256 8-bit grey-levels image

http://dde.binghamton.edu/download/

Acknowledgments

This work was supported by the University of Montpellier (LIRMM), and the Algerian Ministry of Higher Education / Scientific Research.

Author Biography