O Cots

P Delpy

J Gergaud

D Goubinat
email: damien.goubinat@enseeiht.fr

In this article, the minimum time optimal control problem of an aircraft in its climbing phase is studied. First, a reduction of the initial dynamics into a three dimensional single-input system with a linear dependence with respect to the control is performed. This reduced system is then studied using geometric control techniques. In particular, the maximum principle leads to describe a multi-point boundary value problem which is solved by indirect methods. These methods are the implementation of the maximum principle and are initialized by direct methods. We check that the extremal solution of the boundary value problem satisfies necessary and sufficient conditions of optimality. From this reference case and considering small-time optimal trajectories, we give a local classification with respect to the initial mass and final velocity of BC-extremals for the climbing phase.

Introduction

An aircraft travels along several phases during a flight. These phases are the take-off, the climb, the cruise, the descent, the approach and the landing. In this article, we are interested in the time-optimal control problem of an aircraft in its climbing phase. This phase is determined by its own dynamics and governed by a system of ordinary differential equations of dimension 4. The aircraft is represented by the altitude h, the true air speed v, the mass m and the air slope γ. The performance model which describes the evolution of the thrust, the fuel flow and the drag coefficient is a Base of Aircraft Data (BADA) model from Eurocontrol. [START_REF] Poles | Base of Aircraft DAta (BADA) aircraft performance modelling report[END_REF] The atmospheric data as the pressure, the temperature and the air density are given by the International Standard Atmosphere (ISA) model. In our model, the lift coefficient is taken as the control variable.

The climbing phase has already been studied by the NASA. [START_REF] Ardema | Singular perturbations in flight mechanics[END_REF][START_REF] Nguyen | Singular arc time-optimal climb trajectory of aircraft in a two-dimensional wind field[END_REF] These works higlight a time scale separation between the different state variables. In his work, Ardema [START_REF] Ardema | Singular perturbations in flight mechanics[END_REF] classified the air slope γ as a very fast variable, the true air speed v and the altitude h as fast variables and the mass m as a slow variable. This time scale separation between the variables is tackled by a singular perturbation analysis. The method used here to deal with the singular perturbation consists in turning the system of ordinary differential equations into a system of differential algebraic equations by taking a quasi-steady approximation of the very fast variable γ. Numerically, this technique gives a very good approximation of the original time-to climb problem [START_REF] Cots | Time-optimal aircraft trajectories in climbing phase and singular perturbations (regular paper)[END_REF] and leads to the reduction of the dynamics into a singleinput dynamics of dimension 3 with a linear dependence with respect to the control. In this reduced problem, the new control variable is the air slope γ. This reduced problem is then studied through the Pontryagin's Maximum Principle [START_REF] Pontryagin | The mathematical theory of optimal processes[END_REF] (PMP) and second-order optimality conditions. The extremals solution of the maximum principle are classified using the work of Bonnard and Kupka. [START_REF] Bonnard | Théorie des singularités de l'application entrée/sortie et optimalité des trajectoires singuliéres dans le problème du temps minimal[END_REF] From this work and thanks to small-time analysis a local classification of timeoptimal trajectories is sketched.

The paper is organized as follows. The physical model with the optimal control problem are defined in section 2. In section 3, we analyze the optimal control problem with the application of the maximum principle and we classify the bang-bang extremals near the switching surface in the particular case of the dimension 3. Then, in section 4, a local time-optimal classification is sketched thanks to numerical methods and small-time considerations.

Physical model and Mayer optimal control problem

In this section, we present the dynamics of the climbing phase. We restrict the dynamics to the vertical motion of the aircraft. The aircraft is subjected to four forces, the drag On the minimum time optimal control problem of an aircraft in its climbing phase

ḣ = v sin(γ), (1) v
= T m - 1 2 ρS v 2 m C D -g 0 sin(γ), (2)
ṁ = -C s T, (3) γ
= 1 2 ρS v m C L - g v cos(γ), (4)
where the state variables h, v, m and γ represent respectively the altitude, the velocity, the mass and the air slope. In this model, the fuel flow C s , the thrust T and the drag coefficient C D are given by the BAse of Aircraft Data 19 model:

C s (v) C s,1 1 + v C s,2 , T (h) C t,1 1 - h C t,2 + h 2 C t,3 , C D C D,0 + C D,1 C 2 L ,
where the constants C s,i , C t,i and C D,i are specific to the aircraft. The International Standard Atmospheric model provides the expression of the pressure P, the temperature Θ and the air density ρ:

P(h) P 0 Θ(h) Θ 0 g 0 /(βR) , Θ(h) Θ 0 -βh, ρ(h) P(h) RΘ(h) .
The remaining data are positive constants: g 0 is the gravitational constant, S the wing area, R the specific constant of air, β the thermical gradient and P 0 , Θ 0 represent the pressure and temperature at the sea level.

Regarding the lift coefficient C L as the control variable looks like a natural idea in the considered dynamics. However, according to the work of Ardema 2 and Nguyen, [START_REF] Nguyen | Singular arc time-optimal climb trajectory of aircraft in a two-dimensional wind field[END_REF] the dynamics contains slow (the mass m), fast (the altitude h and the velocity v) and very fast variables (the air slope γ). The time scale separation between the slow and the very fast variable is handled by a singular perturbation analysis which consists here in taking the very fast variable as the new control variable and replacing its corresponding dynamics by a quasi-steady approximation:

0 = 1 2 ρS v m C L - g v ,
where we also consider that the air slope remains small (from 0 to γ max 0.262 radians) so we set cos(γ) ≈ 1 and sin(γ) ≈ γ. These assumptions lead to the following affine controlled dynamics:

ẋ(t) = f 0 (x(t)) + u(t) f 1 (x(t)), where x (h, v, m), u γ, f 0 (x)                   0 T (h) m - 1 2 ρS v 2 m C D,0 + 2mg 0 ρS v 2 C D,1 -C s T (h)                   and f 1 (x)             v -g 0 0             .
A previous work [START_REF] Cots | Time-optimal aircraft trajectories in climbing phase and singular perturbations (regular paper)[END_REF] has shown that this reduced system is a very good approximation of the original dynamics. Finally, the time-optimal climbing problem is defined by: min

u(•), t f t f , ẋ(t) = f 0 (x(t)) + u(t) f 1 (x(t)), |u(t)| u max , ∀t ∈ [0 , t f ] a.e., x(0) = x 0 , b(x(t f )) = 0, (P t f )
where u max γ max , x 0 (h 0 , v 0 , m 0 ) and x f (h f , v f , m f ) are totally determined and where b(x)

          h -h f v -v f m -m f           .
nonlinear point mass representation and we consider that all the forces apply on the center of gravity of the aircraft. Let assume that the thrust, the drag and the velocity vector -→

V are colinear. The application of the first dynamics principle, assuming that the earth system is galilean, provides the following four dimensional system:

3. Analysis of the time-optimal problem using the maximum principle

Necessary optimality conditions and singular trajectories

Let define the pseudo-Hamiltonian of the Mayer optimal control problem (P t f )

H : R n × (R n ) * × R → R (x, p, u) → H(x, p, u) p , f 0 (x) + u p , f 1 (x) .
If (ū(•), t f ) is a solution of (P t f ) with x(•) the associated trajectory, then, the maximum principle [START_REF] Pontryagin | The mathematical theory of optimal processes[END_REF] asserts that there exists a real number p 0 0 and an absolutely continuous function p(•) : [0 , t f ] → (R) * such that ( p(•), p 0 ) (0, 0). Besides, for t ∈ [0 , t f ] a.e., we have

ẋ(t) = ∂H ∂p ( x(t), p(t), ū(t)), ṗ(t) = - ∂H ∂x ( x(t), p(t), ū(t)), (5)
and the maximization condition holds:

H( x(t), p(t), ū(t)) = max |w| u max H( x(t), p(t), w). (6)
For our time-optimal problem, the following boundary conditions must be fulfilled:

b( x( t f )) = 0, (7)
H( x( t f ), p( t f ), ū( t f )) = -p 0 . (8)
Definition 1. We call an extremal a triplet (x(•), p(•), u(•)) which satisfies equations ( 5) and ( 6) and such that H > 0 along it. It is called a BC-extremal if it satisfies also (7) and (8).

Remark 1. We only consider extremals in the normal case (p 0 0), and by homogeneity we take p 0 = -1. Besides, for any extremal (x(•), p(•), u(•)), the adjoint vector p(•) never vanishes.

The singular extremals play an important role in this study. A complete study of these extremals can be found in the work of Bonnard and Chyba 5 but here, we focus our analysis on a single-input affine system in dimension 3. First, we give a propostion which characterizes the singular extremals. We use this proposition as a definition of the singular extremals.

Proposition 3.1. The control u(•) with its associated trajectory x(•) are singular on [0 , T ] if and only if there exists a non zero adjoint p(•) such that (x(•), p(•), u(•)) is solution a.e. on [0 , T ] of the following equations:

ẋ(t) = ∂H ∂p (x(t), p(t), u(t)), ṗ(t) = - ∂H ∂x (x(t), p(t), u(t)), 0 = ∂H ∂u (x(t), p(t), u(t)).
Definition 2. Let consider a smooth manifold M and a set of local coordinates x.

• Lie derivative. The Lie derivative of a smooth function f on M along the vector field F is defined by

f → F • f with (F • f )(x) d f (x)F(x) for all x in M.
We also say that the smooth vector field F acts on a smooth function f .

• Lie bracket. The Lie bracket of two smooth vector fields F 0 and F 1 defined on M is defined by

F 1 → [F 0 , F 1 ] F 0 • F 1 -F 1 • F 0 , i.e. [F 0 , F 1 ](x) = dF 0 (x)F 1 (x) -dF 1 (x)F 0 (x).
• Poisson bracket. Denoting H 0 and H 1 , the Hamiltonian lifts associated to F 0 and F 1 , we defined by In our problem, ∂ u H = H 1 = 0 has to be differentiated at least twice along an extremal to compute the control and one gets:

{H 0 , H 1 } - → H 0 • H 1 = H [F 0 ,F
H 1 = H 01 = H 001 + u H 101 = 0.
A singular extremal along which H 101 0 is called of minimal order and the corresponding control is given by

u s (z) - H H 001 101 ( ( z z ) ) ,
where z (x, p). Plugging such u s into H defines a t rue H amiltonian, d enoted h s , w hose s olutions i nitiating from H 1 = H 01 = 0 define the singular extremals of minimal order outside H 101 = 0 . We also have the following additional

(•) | s • | max singular extremal: ∂ ∂u ∂ ∂t 2 ∂H ∂u = H 101 0. ( 9
)
Introducing the determinants,

D 0 (x) det( f 1 (x), f 01 (x), f 0 (x)), D 001 (x) det( f 1 (x), f 01 (x), f 001 (x)), D 101 (x) det( f 1 (x), f 01 (x), f 101 (x)),
then the singular trajectories and control are given by the following proposition.

Proposition 3.2. The singular trajectories satisfy

ẋ(t) = f 0 (x(t)) - D 001 (x(t)) D 101 (x(t)) f 1 (x(t)), outside x ∈ R 3 D 101 (x) = 0 .
Proof. Along a singular extremal, H 1 (z(•)) ≡ 0, so differentiating twice this quantity with respect to the time leads to

p , f 1 (x) = p , f 01 (x) = p , f 001 (x) + u f 101 (x) = 0
along the extremal. Since p(•) does not vanish, then we have det( f 1 (x), f 01 (x), f 001 (x) + u f 101 (x)) = 0 along the singular extremal and by linearity of the determinant, we obtain the expected result.

Remark 2. In this case, the singular control is given in feedback form and we note with the same notation:

u s (x) - D 001 (x) D 101 (x) . ( 10
)
Proposition 3.3. Along any singular extremal such that D 0 0 we have

H 101 0 ⇐⇒ D 0 D 101 0. Proof. Along any singular extremal (x(•), p(•), u(•)), if D 0 (x) 0 then span({ f 1 (x), f 01 (x)}
) is a plane of R 3 which separates this space into two half spaces and for which p is a normal vector (since p , f 1 (x) = p , f 01 (x) = 0 along the singular extremal. Besides, since H(x, p, u) = H 0 (x, p) = p , f 0 (x) > 0 along the singular extremal, then p and f 0 (x) are in the same half space. At the end, H 101 (x, p) = p , f 101 (x) ≥ 0 if and only if f 101 (x) and f 0 (x) are in the same half space, whence the result.

Corollary 3.4. The strict generalized Legendre-Clebsch condition (9) becomes D 0 D 101 > 0 along a singular arc.

Generic classification of the bang-bang extremal near the switching surface.

Along a singular extremal we have ∂ u H ≡ 0. In our particular case of a single-input affine system this condition becomes H 1 (z(t)) = 0. Let define the sets Σ 1 and Σ s , such that Σ 1 represents the switching surface, i.e.

Σ 1 {z (x, p) ∈ R n × (R n ) * | H 1 (z) = 0}, and where Σ s {z (x, p) ∈ R n × (R n ) * | H 1 (z) = H 01 (z) = 0}
contains all the singular extremals. We define the switching function Φ : t → Φ(t) H 1 (z(t)) and we note Φ + (resp. Φ -) if the control along the extremal is u max (resp. -u max ). The first and second derivatives of Φ ± are given by:

Φ± (t) = H 01 (z(t)), Φ± (t) = H 001 (z(t)) ± u max H 101 (z(t)).
Definition 3. An extremal is said bang-bang on [0 , t f ] if it is composed of a finite number of arcs of the form σ + and σ -where σ + (resp. σ -) represents a bang arc for which u(•) ≡ u max (resp. -u max ). In a similar way, σ s represents a singular trajectory and we denote by σ 1 σ 2 an arc σ 1 followed by an arc σ 2 .

Near the switching surface Σ 1 , the behavior of the extremals for a single-input affine system is detailled in the work of Kupka. [START_REF] Kupka | Geometric theory of extremals in optimal control problems. i. the fold and maxwell case[END_REF] We have the following:

Ordinary point. Let consider z 0 (x 0 , p 0 ) ∈ Σ 1 \ Σ s and t 0 its associated time such that z + (t 0 ) = z -(t 0 ) = z 0 , where z + (•) (resp. z -(•)) represents the extremal with u(•) ≡ u max (resp. u(•) ≡ -u max ). According to the maximum principle, near Σ 1 , the extremal is of the form σ -σ + if Φ(t 0 ) > 0 and σ + σ -if Φ(t 0 ) < 0.
Fold point. Let consider now a point z 0 ∈ Σ s and assume that Σ s is a smooth manifold of codimension two. Let denote t 0 the time associated to z 0 . At t 0 , the extremal has a contact of order at least two with Σ 1 . If the contact is exactly of order two, i.e. Φ± (t 0 ) 0, then locally z 0 is a fold point and three different behaviors can be encountered.

-Parabolic case: Φ+ (t 0 ) Φ-(t 0 ) > 0. The singular extremal at the switching time is not admissible and every extremal is of the form σ + σ -σ + or σ -σ + σ -with at most two switchings.

-Hyperbolic case: Φ+ (t 0 ) > 0 and Φ-(t 0 ) < 0. A connection with a singular extremal is possible and locally each extremal is of the form σ ± σ s σ ± (by convention each arc of the sequence can be empty).

-Elliptic case: Φ+ (t 0 ) < 0 and Φ-(t 0 ) > 0. In this case, a connection with a singular extremal is not possible. The elliptic trajectories are then bang-bang with no uniform bound on the number of switchings.

Let assume D 0 (x) 0, then the family ( f 0 (x), f 1 (x), f 01 (x)) forms a basis of R 3 and there exists (α, α 1 , α 01 ) ∈ R 3 and (β,

β 1 , β 01 ) ∈ R 3 such that f 001 (x) -u max f 101 (x) = α f 0 (x) + α 1 f 1 (x) + α 01 f 01 (x), f 001 (x) + u max f 101 (x) = β f 0 (x) + β 1 f 1 (x) + β 01 f 01 (x).
By linearity of the determinant, we have

D 001 (x) -u max D 101 (x) = α D 0 (x), D 001 (x) + u max D 101 (x) = β D 0 (x),
and since D 0 (x) 0 then we can compute α and β along the trajectory. Besides, along any singular extremal, these quantitites can be linked to the signs of Φ± (t):

Φ-(t) = p(t) , f 001 (x(t)) -u max f 101 (x(t)) = α(t) H 0 (z(t)), Φ+ (t) = p(t) , f 001 (x(t)) + u max f 101 (x(t)) = β(t) H 0 (z(t)),
with H 0 (z(t)) > 0. Denoting t 0 the time when the extremal has a contact with the switching surface Σ 1 , the behavior of the trajectory near z 0 is given by the signs of α(t 0 ) and β(t 0 ). Indeed, the trajectory is

-Hyperbolic if α(t 0 ) < 0 and β(t 0 ) > 0, -Elliptic if α(t 0 ) > 0 and β(t 0 ) < 0, -Parabolic if α(t 0 ) β(t 0 ) > 0.
-→

Second-order optimality conditions

This section relies on the work of Bonnard and Kupka 7 about second-order optimality conditions. Definition 4 . Let z(•) be a reference singular solution o f h s on [0 , t f ] and contained in Σ s . The variational equation

δż(t) = d - → h s (z(t)) • δz(t), dH 1 (z(t)) • δz(t) = dH 01 (z(t)) • δz(t) = 0, is called Jacobi equation. A Jacobi field J(•) (δx(•), δp(•)
) is a non-zero solution of the Jacobi equation. It is said semi-vertical at time t if δx(t) ∈ R f 1 (x(t)). The time 0 ≤ t 1 ≤ t f is said to be conjugate if there exists a Jacobi field J(•) semi-vertical at t = 0, t = t 1 and the points x 0 = x(0) and x 1 = x(t 1 ) are said to be conjugate. -The space K(t)

ad k f 0 • f 1 (x(t)) k = 0, . . . , n -2 has codimension one, where ad 0 f 0 • f 1 (x(t)) = f 1 (x(t)) and ad f 0 • f 1 (x(t)) = f 01 (x(t)). -h s (z(t)) = p(t) , f 0 (x(t)) + u s (z(t)) f 1 (x(t)) is not equal to zero along z(•).
Then we have the following theorem in the case of a single-input affine system. Theorem 3.6. Let consider t c the first conjugate time. Under assumptions 3.5, the reference singular trajectory x(•) is C 0 -locally time minimizing in the hyperbolic case and time maximizing in the elliptic case on [0 , t c ).

Remark 3. Moreover, the trajectory x(•) is not time optimal in L ∞ -topology on [0 , t], for every t > t c .

From now on, the sufficient conditions of optimality come down to the search of the first conjugate time t c . In the case of an affine system in dimension 3, the singular control is independent of the adjoint variable p and we can give the following characterization of a Jacobi field J(•) along the state variable x(•), assuming that x(•) is a singular trajectory contained in the set x ∈ R 3 D 101 (x) 0 on [0 , t f ]. Proposition 3.7. A Jacobi field J(•) along x(•) for a three dimensional system is a non trivial solution of the variational equation

δ ẋ(t) = ∂ ∂x ( f 0 + u s f 1 )(x(t)) • δx(t), δx(0) ∈ R f 1 (x(0)).
Then the first conjugate time is the first time t c > 0 such that det(J(t c ), f 0 (x(t c )), f 1 (x(t c ))) = 0, since by assumption f 0 and f 1 are linearly independent and J is colinear at f 1 only at the times 0 and t c .

Application to a medium-haul aircraft

4.1 Solving (P t f ) with direct and indirect numerical methods

Numerical methods

For small-time trajectories, near the switching surface the parabolic, elliptic and hyperbolic classification is valid. On the other hand, "long"-time optimal trajectories, i.e. solution of (P t f ), must minimize the travel time on each sub-interval no matter what the length of the interval is. From this fact, we consider that the time-optimal climbing trajectory is composed of a concatenation of hyperbolic, parabolic or elliptic arcs. Two types of numerical methods are used in order to solve the problem (P t f ): a direct method to determine the optimal structure and to get a good initial guess (see figure 1) in order to make the indirect method converge (see figure 3) which gives a refined solution and make possible a fine analysis of the problem (see figure 8). Firstly, we use the so-called direct approach wich consists in transforming the infinite dimensional optimal control problem (P t f ) into a finite dimensional optimization problem (NLP). This is done by a discretization in time applied to the state and control variables, as well as the dynamics equation. These methods are usually less precise than indirect methods based on Pontryagin's Maximum Principle, but more robust with respect to the initialization. Also, they are more straightforward to apply, hence their wide use in industrial applications. We refer the reader to for instance the books of Betts 3 and Gerdts [START_REF] Gerdts | Optimal control of ODEs and DAEs[END_REF] for more details on direct transcription methods and NLP algorithms. About the direct methods, all the tests were run using the Bocop 4 software and the discretized nonlinear optimization problems were solved by the well-known Ipopt 21 solver.

When the structure of the solution is known, we use indirect methods to refine the solution. More precisely, we use multiple shooting techniques which consists in solving a Multi-Point Boundary Value Problem (MPBVP) obtained from the application of the maximum principle. The MPBVP is then transformed into a system of nonlinear equations which is solved by a Newton-like algorithm. See the book of Bulirsch and Stoer 8 for details about indirect methods. About the indirect methods, all the tests were run using the HamPath 9 software and the system of nonlinear equations were solved by the hybrj 17 routine.

Numerical results

In this paper, the direct methods are used to determine the structure of the trajectory and to initialize the indirect methods. The figure 1 shows the resulting control for a middle-haul aircraft. This solution is computed using a gauss scheme of order 4 with 750 nodes of discretization. The final time is t f ≈ 654 s and the data which describes the aircraft and the atmosphere are presented in table 1. This aircraft starts from x 0 = (3480, 128.6, 69000) up to the beginning of the cruise defined by x f = (9144, 191, 68100).

Data

Value 1: Control law for a time-optimal trajectory using direct methods obtained with a gauss discretization scheme of order 4 using 750 nodes. The control seems to follow a structure of the form σ -σ s σ + .

The resulting control seems to be of the form σ -σ s σ + , this structure reminds the hyperbolic case where the structure of the control is given by σ ± σ s σ ± . Let check this structure with the indirect methods. The HamPath software computes from the maximized Hamiltonian the corresponding adjoint system thanks to automatic differentiation. Hence, we need to introduce the following true Hamiltonians h ± (x, p) H 0 (x, p) ± u max H 1 (x, p), h s (x, p) H 0 (x, p) + u s (x) H 1 (x, p), with u s given by eq. ( 10) and which composed the maximized Hamiltonian

H(x(t), p(t))            h -(x(t), p(t)), when 0 t < t 1 , h s (x(t), p(t)), when t 1 t < t 2 , h + (x(t), p(t)), when t 2 t t f ,
where t 1 and t 2 represent the switching times.

Remark 4. All the needed expressions of determinants and vector fields, for instance D 001 (x), D 101 (x) or f 01 (x), may be found in the work of Ref. [START_REF] Goubinat | Contrôle optimal géométrique et méthodes numériques : application à un problème de montée d'un avion[END_REF] Finally, we define the shooting function S :

R 5n+3 → R 5n+3 S (p 0 , t 1 , t 2 , t f , z 1 , z 2 )                          x f -π x (z(t f , z 2 , t 2 )) H(z(t f , z 2 , t 2 )) + p 0 H 1 (z 1 ) H 01 (z 1 ) z 2 -z(t 2 , z 1 , t 1 ) z 1 -z(t 1 , z 0 , 0)                          (11)
where z 0 (x 0 , p 0 ), π x (x, p) = x, z(t j , z i , t i ) is the solution1 at t j of the initial value problem ż(t) = -→ H(z(t)), z(t i ) = z i . The two first conditions represent the boundary conditions of the climbing problem. The third and the fourth conditions are related to the structure of the control and the two final conditions are matching conditions. The shooting method consists in finding a zero of S (y) with y (p 0 , t 1 , t 2 , t f , z 1 , z 2 ). To any zero of S is associated a unique BC-extremal of (P t f ). The HamPath software is used to solve the shooting equation S (y) = 0 and we find y * such that ||S (y * )|| ≈ 1.07 × 10 -10 . The solution y * is composed of p * 0 ≈ (4.09 × 10 -2 , 6.00 × 10 -1 , -1.92 × 10 -1 ), t * 1 ≈ 19 s, t * 2 ≈ 642 s and t * f ≈ 656 s. The values of z * 1 and z * 2 are omitted as they can be computed by numerical integration. Figures 2 and3 depict the evolution of the altitude, the velocity and the air slope (which is the control in our dynamics) associated to the σ -σ s σ + structure. At the boundaries of the trajectory, we observe that h and v have opposite variation, for example when the altitude is decreasing the true air speed is increasing. This behavior looks like an energy sharing strategy where, at the beginning of the trajectory, potential energy is transformed into kinetics energy whereas the opposite exchange happens at the end of the trajectory. In a second time, we check a posteriori that the trajectory is hyperbolic by computing the values of α(•) and β(•), defined on section 3.2, along the singular arc. The validity of the generalized Lendendre-Clebsch condition, see corollary 3.4, along the trajectory is also checked. At the same time, we compute the quantity Λ(•) det(J(•), f 0 (x(•)), f 1 (x(•))) in order to check the absence or presence of a conjugate time on

[t * 1 , t * 2 ]
. The results of these computations are presented in figures 4 and 5. The values of α(•) and β(•) on [t * 1 , t * 2 ] are consistent with the hyperbolic case. In addition, the hyperbolic trajectory satisfy the sufficient conditions of optimality since Λ(t) 0 on (t * 1 , t * 2 ]. In conclusion, this hyperbolic trajectory is locally a C 0 -optimal trajectory.

Small-time analysis at the beginning of the trajectory

From now on, we know that our hyperbolic trajectory is locally time-optimal. In this section and the following, we intend to study the deformation of this trajectory with respect to some parameters, focusing our analysis on the hyperbolic case. We aim to study the influence of the initial and final conditions by fixing the initial speed, the initial altitude and the final altitude but making the initial mass m 0 and the final velocity v f varying in predefined ranges (m 0 ∈ [48 000 , 72 000] kg and v f ∈ [190 , 250] m.s -1 ) which are admissible values for the chosen aircraft. Let emphasize that the final mass m f is free. The goal now is to give the optimal structure depending on the two parameters m 0 and v f . More precisely, we want to determine for a given couple (m 0 , v f ), the type of the bang arcs at the beginning and the end of the trajectory. We present the analysis made at the beginning of the trajectory. In a similar way, we can study the end of the trajectory.

(•) = det( f 0 (x(•)), f 1 (x(•)), J(•)) on [t * 1 , t * 2 ]
where J(•) represents the Jacobi field. As Λ(t) < 0 for all t in (t * 1 , t * 2 ], we observe the absence of conjugate points on [t * 1 , t * 2 ] and then the singular arc in σ -σ s σ + is locally time-optimal.

In the aircraft dynamics, the mass m is considered as a slow variable. In other words, the mass m evolves much more slowly than the altitude and the velocity. Let consider small-time variation, the evolution of the mass can be omitted and then the dynamics can be reduced to a planar dynamics, given by ẋ(t) = f0 ( x(t)) + u f1 ( x(t)), where x (h, v) and f0 ( x)

            0 T (h) m 0 - 1 2 ρ(h)S v 2 m 0 C D,0 -2 m 0 g 2 0 ρ(h)S v 2             , f1 ( x)       v -g 0       .
Let consider the beginning of the trajectory and let x0 (h 0 , v 0 ) denote the initial point of this dynamics associated to the initial constant mass m 0 . The singular set in dimension 2, which contains the singular extremals, is given by S { x ∈ R 2 | det( f1 ( x), f01 ( x)) = 0}. It depends on m 0 and locally if 0 is a regular value of the application x → Ψ( x) det( f1 ( x), f01 ( x)), then each connected component of Ψ = 0 is diffeomorphic to the circle or the real line. Let assume there exists only one connected component diffeomorphic to the real line and let consider a point x0 S . Then, two cases may occur: either x0 is below the line which represents S or it is above.

The figure 6 depicts the behavior of the negative bang and positive bang trajectories from x0 . Since the full trajectory is of the form σ ± σ s σ ± , the purpose of the bang arcs is to connect the singular arc. Then if x0 is below S , according to figure 6, an arc σ -must begin the trajectory. On the other hand if x0 is above S , an arc σ + must begin the trajectory. For a given mass m 0 and a given altitude h 0 , we denote by v S 0 the associated velocity such that the point (h 0 , v S 0 ) ∈ S . The figure 7 shows the evolution of v S 0 for m 0 ∈ [48 000 , 72 000]. In this figure we compare v S 0 and v 0 . Since v 0 is constantly equal to 128.6 m.s -1 we observe that v S 0 > v 0 for every m 0 in [48 000 , 72 000]. As a consequence, in any case the point x0 = (h 0 , v 0 ) is below the line which represents the singular set S and then we must start with a negative bang. The same argumentation may be used at the end of the trajectory to determine the nature of the final bang. In our situation, we can have both cases σ -σ s σ ± depending on the value of v f .

4.3

Local classification of the optimal structure with respect to m0 and v f Locally, we search BC-extremals of the form σ -σ s σ ± . Let consider the separating trajectory σ -σ s . We note again H the maximized Hamiltonian associated to this structure, and it is defined by The shooting function denoted S -s associated to this trajectory is defined by

H(z(t))      h -(z(t)), when 0 t < t 1 , h s (z(t)), when t 1 t t f . x0 • v h σ s σ - σ + x0 • v h σ s σ - σ +
S -s (p 0 , t 1 , t f , z 1 )                          h f -h(t f ; t 1 , z 1 ) H(z(t f ; t 1 , z 1 )) + p 0 p m (t f ; t 1 , z 1 ) H 1 (z 1 ) H 01 (z 1 ) z(t 1 ; 0, z 0 ) -z 1                         
where z 0 (x 0 , p 0 ), z(t j ; t i , z i ) is the solution at t j of the initial value problem ż(t) = -→ H(t), z(t i ) = z i and p m is the adjoint variable associated to the mass m. The HamPath package is used to solve the shooting equation S -s (y) = 0, y (p 0 , t 1 , t f , z 1 ), and we find y * (p * 0 , t * 1 , t * f , z * 1 ) such that ||S (y * )|| ≈ 6.1 × 10 -9 . The solution is composed of p * 0 ≈ (6.08 × 10 -2 , 1.12, -2.72 × 10 -2 ), t * 1 ≈ 78.7 s and t * f ≈ 889 s. The value z * 1 is omitted since it can be computed by numerical integration. The HamPath package may also be used to realize a differential continuation on a parameter. See the book of Allgower and Georg 1 for details about differential path-following methods and we refer to the presentation of the software [START_REF] Caillau | Differential continuation for regular optimal control problems[END_REF] for details about the implementation of these methods in the HamPath code. We use this functionality on the initial mass from m 0 = 72 000 kg up to m 0 = 48 000 kg. The figure 8 depicts locally the structure of the trajectories in relation with the initial mass and the final velocity. The blue curve, computed by numerical continuation, represents all the trajectories which end on the singular set. As the final altitude h f is fixed, the final velocity v f of each of these trajectories is adjusted to be on this set. We note v f,m 0 the final velocity associated to the initial mass m 0 for which the associated trajectory ends on the singular set. This particular trajectory is the boundary between σ -σ s σ - and σ -σ s σ + trajectories. The classification given in figure 8 shows that if v f is below v f,m 0 , the final bang is a positive one and if v f is above v f,m 0 it is a negative one.

Remark 5. It has not been proven that this classification gives time-optimal trajectories but we may emphasize that the reference trajectory, from section 4.1.2, satisfies necessary and sufficient conditions of local optimality and we mention that this classification has been obtained thanks to an analysis of small-time optimal trajectories.

Conclusion

In this paper, the time-optimal control problem of an aircraft in its climbing phase is modeled as a Mayer problem. Firstly, the four dimensional dynamics is reduced into a three dimensional dynamics to deal with the singular perturbation phenomenon. This reduced system is a single-input control system where the control appears linearly. The behavior of bang-bang extremals near the switching surface with the characterization of the singular trajectories are studied in the particular case of the dimension 3 throughout geometric analysis. Combining these results with numerical methods and small-time considerations leads to sketch a local classification of fairly time-optimal climbing trajectories depending on the initial mass and the cruise speed.

This study does not take into account states constraints which should be fulfilled during the climbing phase. For instance, the Computed Air Speed (CAS) and the number of mach (M) are limited by the Operation Maximal Speed (VMO) and by the Maximal Mach Operation (MMO). The figure 9 shows the evolution of the two velocity constraints c 1 (x) CAS(x) -VMO and c 2 (x) M(x) -MMO for a trajectory of the form σ -σ s σ + with m 0 = 69 000 kg and v f = 191 m.s -1 . In this example none of these constraints are saturated but a more exhaustic study is necessary to understand the behavior of the trajectories in presence of state constraints. A methodology closed to the unconstrained case can be applied except that the geometric study should be based on a maximum principle with state constraints which is presented, for example, in the work of Jacobson [START_REF] Jacobson | New necessary conditions of optimality for control problems with state-variable inequality constraints[END_REF] et al. and Maurer. [START_REF] Maurer | On optimal control problems with bounded state variables and control appearing linearly[END_REF] Besides, the small-time considerations could also be used and could give some interesting results on the structure of time-optimal trajectory with state constraints as it was done for the space shuttle reentry problem in the work of Bonnard et al.

T

  and its own weight -→ P. We use a DOI: 10.13009/EUCASS2017-463

Assumption 3 . 5 .

 35 Let z(•) = (x(•), p(•)) be a reference singular extremal curve on [0 , t f ] solution on Σ s and we assume the following:-f 0 (x) and f 1 (x) are linearly independant along x(•). H 101 (z(•)) is not equal to zero along z(•) and x(•) is injective.

Figure 2 :

 2 Figure2: Evolution of the altitude h and of the true air speed v associated to the σ -σ s σ + structure. At the boundaries of the trajectory, we observe that h and v have opposite variation, for example when the altitude is decreasing the true air speed is increasing. This behavior looks like an energy sharing strategy where, at the beginning of the trajectory, potential energy is transformed into kinetics energy whereas the opposite exchange happens at the end of the trajectory.

Figure 3 :

 3 Figure 3: Evolution of the control law associated to the σ -σ s σ + structure.

Figure 4 :

 4 Figure 4: Evolution of the quantities α(•), β(•) and D 0 (x(•)) D 101 (x(•)) along the singular arc. The generalized Legendre-Clebsch condition from corollary 3.4 is satisfied all along the singular arc as D 0 (x(•) D 101 (x(•)) > 0. The signs of α(•) and β(•) along the singular arc confirm that we have an hyperbolic trajectory.

Figure 5 :

 5 Figure 5: Evolution of Λ(•) = det( f 0 (x(•)), f 1 (x(•)), J(•)) on [t * 1 , t * 2 ]where J(•) represents the Jacobi field. As Λ(t) < 0 for all t in (t * 1 , t * 2 ], we observe the absence of conjugate points on [t * 1 , t * 2 ] and then the singular arc in σ -σ s σ + is locally time-optimal.

Figure 6 :Figure 7 :

 67 Figure6: Local representation of the negative bang (the green curve) and the positive bang (the red curve) trajectories from x0 . On the left (resp. right), x0 is located below (resp. above) the blue line which represents locally the singular set S . On the left figure, the only way to connect the singular trajectory σ s , which is time-optimal, is to follow an arc σ -whereas on the right figure the only way to connect σ s is to follow an arc σ + .

σ

  -σ s σ - σ -σ s σ + σ -σ s

Figure 8 :

 8 Figure 8: Local classification of the optimal structure with respect to the initial mass and the final velocity of the climbing phase. Note that this final velocity is also the cruise velocity.

6

 6

Figure 9 :

 9 Figure 9: Evolution of the speed constraints c 1 and c 2 along the time-optimal trajectory associated to the structure σ -σ s σ + . All along the trajectory none of these constraints are saturated.

  1 ] the Poisson bracket between H 0 and H 1 , where We note H 01 the Poisson bracket between H 0 and H 1 and H 001 the Poisson bracket between H 0 et H 01 .

	-→ H	(∂ p H, -∂ x H).

Table 1 :

 1 Constant data of a middle-haul aircraft and constant data of the atmospheric model during the climbing phase.

				Unit		Data	Values	Unit
	S C D,0 C D,1 C T,1 C T,2 C T,3 C S ,1 C S ,2	1.226 × 10 2 2.42 × 10 -2 4.690 × 10 -2 1.410 × 10 5 4.892 × 10 4 6.500 × 10 -11 6.333 × 10 -1 kg.min -1 .kN -1 m 2 N ft ft -2 8.590 × 10 -2 kts	Θ 0 β P 0 g 0 R γ air κ γ air /(γ air -1) 2.857 × 10 -1 2.882 × 10 2 6.500 × 10 -3 1.013 × 10 6 9.810 2.880 × 10 2 1.4	K K.m -1 Pa N.m -1 J.kg -1 .K -1
			0.3
			0.2		u max
			0.1
		u	0.0
			0.1
			0.0 0.3 0.2	0.2	0.4 -u max t/t f	0.6	0.8	1.0
	Figure

Numerically, z(t j , z i , t i ) is computed with a high-order Runge-Kutta scheme with adaptative step-size.


Mehdi Yedroudj
email: mehdi.yedroudj@lirmm.fr

Marc Chaumont
email: marc.chaumont@lirmm.fr

Frederic Comby
email: frederic.comby@lirmm.fr

Ahmed Oulad Amara
email: ahmed.oulad-amara@lirmm.fr

Patrick Bas
email: patrick.bas@centralelille.fr

Pixels-off: Data-augmentation Complementary Solution for Deep-learning Steganalysis

Keywords: CCS CONCEPTS, Applied computing → Computer forensics, • Computing methodologies → Image processing, • Computer systems organization → Neural networks steganalysis, deep-learning, CNN, data-augmentation, steganography, pixels-off

Figure 1: The "pixels-off" process illustrated on a 256×256 pixels image 1.pgm from BOSSBase [1]. Fig.a represents the cover and Fig.b its "pixels-off" version with 100 pixels switched-off. Fig.c (resp. Fig.d) is the embedding modification probabilities map for the cover (resp. "pixels-off" version) obtained from the S-UNIWARD model with a payload of 0.4bpp [12].

INTRODUCTION

Modern numerical steganography is the art of concealing a message in a digital cover [START_REF] Fridrich | Steganography in Digital Media[END_REF]. In other words, steganography is the set of techniques allowing secret communication between two protagonists, conventionally named Alice and Bob [START_REF] Simmons | The Subliminal Channel and Digital Signatures[END_REF].

When we study steganography, one has to check the security of the secret communication occurring between Alice and Bob. This security analysis is named steganalysis, and we conventionally attribute this role to another entity named Eve.

Since the use of deep-learning for steganalysis purposes [START_REF] Chaumont | Deep Learning in steganography and steganalysis[END_REF], many efficient models have been proposed, these are now seen as state-of-the-art references for spatial image steganalysis such as, Yedroudj-Net [START_REF] Yedroudj | Yedrouj-Net: An Efficient CNN for Spatial Steganalysis[END_REF], SRNet [START_REF] Boroumand | Deep Residual Network for Steganalysis of Digital Images[END_REF], CovPool-Net [START_REF] Deng | Fast and Effective Global Covariance Pooling Network for Image Steganalysis[END_REF], Zhu-Net [START_REF] Zhang | Depth-Wise Separable Convolutions and Multi-Level Pooling for an Efficient Spatial CNN-Based Steganalysis; (previously named "efficient feature learning and multi-size image steganalysis based on cnn[END_REF], Calpa-Net [START_REF] Tan | CALPA-NET: Channel-pruning-assisted Deep Residual Network for Steganalysis of Digital Images[END_REF].

For these networks, the number of images needed to reach the region of good performance (that is the performance of a Rich Model [START_REF] Fridrich | Rich Models for Steganalysis of Digital Images[END_REF] with an Ensemble Classifier [START_REF] Kodovský | Ensemble Classifiers for Steganalysis of Digital Media[END_REF]), is about 10,000 images (5,000 covers and 5,000 stegos) for the learning phase. This is the case when there is no cover-source mismatch, and the images' size is 256 × 256 pixels [START_REF] Yedroudj | How to Augment a Small Learning Set for Improving the Performances of a CNN-Based Steganalyzer?[END_REF].

However, this quantity of images is insufficient [START_REF] Yedroudj | How to Augment a Small Learning Set for Improving the Performances of a CNN-Based Steganalyzer?[END_REF] in the sense that performance can be increased simply by augmenting the size of the training set. In steganalysis, the so-called irreducible error region [START_REF] Hestness | Deep Learning Scaling is Predictable, Empirically[END_REF] probably requires many more images than those normally used today. In the case of [START_REF] Zeng | Large-Scale JPEG Image Steganalysis Using Hybrid Deep-Learning Framework[END_REF], it takes one million images, which corresponds to 100 times more images than those usually used for the learning phase.

Working with an extensive database is not necessarily the only solution to reach the irreducible error region faster. We can use transfer learning [START_REF] Qian | Learning and Transferring Representations for Image Steganalysis Using Convolutional Neural Network[END_REF] and/or curriculum learning [START_REF] Ye | Deep Learning Hierarchical Representations for Image Steganalysis[END_REF] to start learning from a network that has already been trained. We can also use a set of CNNs [START_REF] Xu | Ensemble of CNNs for Steganalysis: An Empirical Study[END_REF] or a network made of sub-networks [START_REF] Li | ReST-Net: Diverse Activation Modules and Parallel Subnets-Based CNN for Spatial Image Steganalysis[END_REF], which can save a few percentage points on accuracy.

In this paper, we focus on the database enrichment in a clairvoyant scenario with no SCA. In this scenario, a CNN-steganalyzer is used inline to analyze test set images that have never been seen before (test phase). Prior to its inline deployment, during the learning phase, the CNN-steganalyzer has access to images whose distributions are similar to those of the test set (i.e. same sources, same development 1 , same sizes, same embedding process, and same payload).

The database enrichment is usually done with virtual augmentation [START_REF] Krizhevsky | ImageNet Classification with Deep Convolutional Neural Networks[END_REF], or by adding another similar dataset such as BOWS2 [START_REF] Ye | Deep Learning Hierarchical Representations for Image Steganalysis[END_REF], [START_REF] Yedroudj | How to Augment a Small Learning Set for Improving the Performances of a CNN-Based Steganalyzer?[END_REF]. We can make new acquisitions with the same devices used to produce the test database, and perform images developments similar to the ones used to generate the test database [START_REF] Yedroudj | How to Augment a Small Learning Set for Improving the Performances of a CNN-Based Steganalyzer?[END_REF].

Here, we propose to duplicate the learning set and to add a particular noise. Adding noise for database augmentation is a classical principle in machine learning and especially in image classification. Nevertheless, it did not give compelling results for CNN-based steganalysis. This can be explained by the fact that the added noise alters the initial surface statistical regularities [START_REF] Jo | Measuring the tendency of CNNs to Learn Surface Statistical Regularities[END_REF], leading to a mismatch between the learning set and the test one.

We thus propose the pixels-off noise principle. After explaining the reasoning behind this approach, its principle and its statistical properties in section 2, we present current state-of-the-art CNNs (section 3); then, we give the experimental methodology in section 4. We intensively evaluate its accuracy and usability in section 5. This evaluation is performed alongside two state-of-the-art networks (Yedroudj-Net [START_REF] Yedroudj | Yedrouj-Net: An Efficient CNN for Spatial Steganalysis[END_REF] and CovPool-Net [START_REF] Deng | Fast and Effective Global Covariance Pooling Network for Image Steganalysis[END_REF]) using two well-established embedding algorithms (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]) at 0.2 bpp and 0.4 bpp. We use two well-known test databases (BOSS [START_REF] Bas | Break Our Steganographic System': The Ins and Outs of Organizing BOSS[END_REF] and Alaska [START_REF] Cogranne | The ALASKA Steganalysis Challenge: A First Step Towards Steganalysis[END_REF]), and we compare the pixels-off technique with many other enrichment propositions. In section 6, we further analyze the property of this technique, and we discuss its usage.

PIXELS-OFF PROPOSITION 2.1 The enrichment approaches

Using an initial learning set and a fixed test set, database augmentation (enrichment) consists in increasing the number of images of the initial learning set. In this paper, we explore an enrichment technique which improves the CNN performance, without requiring access: 1) to images other than those of the initial learning database and 2) to the original cameras (or the original raw images) or any knowledge about the development. The approach we are proposing is, therefore, always feasible.

As mentioned previously, the virtual augmentation (VA) which flips and rotates the learning set without interpolation [START_REF] Krizhevsky | ImageNet Classification with Deep Convolutional Neural Networks[END_REF], or eventually adding another database [START_REF] Ye | Deep Learning Hierarchical Representations for Image Steganalysis[END_REF], can be used to enrich the initial learning set.

The steganalyzer, Eve, can also decide to duplicate the initial learning set and apply image processing on it with the purpose of enriching the initial learning set. For example, in [START_REF] Yedroudj | Yedrouj-Net: An Efficient CNN for Spatial Steganalysis[END_REF], the authors propose two processes. The first one consists in applying a sub-pixel image translation, of 0.5 pixel, on the padded images, followed by a cropping operation to obtain a 256×256 pixel image. The second one consists in upsampling with the Lanczos3 filter to obtain a 512×512 pixel image and then downsampling it with the same interpolation kernel to get an image of 256×256 pixels. Unfortunately, these processes applied directly to the images give an important accuracy reduction. Also note that the usual approaches consisting of adding noise, like in image classification, are also not as efficient (see section 5). These signal processing approaches are not preserving, or are too far from the initial learning set statistics. In short, a cover-source mismatch phenomenon [START_REF] Giboulot | Roots and Solutions of Cover-Source Mismatch in Image Steganalysis: a Comprehensive Study[END_REF] occurs because the added images do not share the same surface statistical regularities [START_REF] Jo | Measuring the tendency of CNNs to Learn Surface Statistical Regularities[END_REF], i.e. the same pixel distribution.

As experimentally observed in [START_REF] Jo | Measuring the tendency of CNNs to Learn Surface Statistical Regularities[END_REF], the networks tend to learn surface statistical regularities, instead of high-level abstraction. As appended comments, the authors of [START_REF] Jo | Measuring the tendency of CNNs to Learn Surface Statistical Regularities[END_REF] observe that:

(1) better results are obtained when tests are performed on an unfiltered database (unfiltered images) while training is performed on low-pass filtered images, compared to training and testing on unfiltered images. (2) when merging the filtered and unfiltered images databases, the generalization capabilities improve, but results are not surpassing the results obtained when the learning and testing databases are similar.

These first observations, even if they are related to an image classification context, suggest that well-chosen noises, applied to cover images, could lead to beneficial enrichment of an initial training set, i.e. an improvement in the steganalysis accuracy.

Our proposition consists in switching off a small proportion of pixels from the duplicated initial database of cover images. We name this filtering the "pixels-off" technique. We thus keep the idea of random masking, but rather than applying it to the Fourier domain; we apply it directly to the spatial domain so that we keep the average cover statistics. Pixel zeroing is similar to a noise addition, but to our knowledge, no other noise had previously worked to augment a database for steganalysis purposes. The pixels-off technique can be seen as a simulation of dead pixels, or faulty sensors, which frequently occurs in various application domains.

The pixels-off technique

The Algorithm 1 describes how the pixels-off technique is used for database enrichment. As shown in Algorithm 1, 2nd line, only the covers are retained out of the training set since they are the ones considered for pixelsoff.

The loop for (line 5) iterates through the entire cover list. For each iteration, a cover, 𝑥, is selected (line 6). A binary image, 𝑧, whose size is equal to the cover's one, is generated (line 7) by randomly spreading P zeros in an image made of ones. Note that the variable P defines the number of pixels to be switched to zero. In practice, the value of this variable should be small. Therefore, it is chosen so that only a small portion of the image (between 1.5‰and 1.5%) is switched off.

The cover pixels-off version, 𝑥 𝑜 𝑓 𝑓 , is obtained by applying an element-wise multiplication between the cover, 𝑥, and the binary image, 𝑧. The obtained image, 𝑥 𝑜 𝑓 𝑓 , is thus a new cover. The latter is subsequently used to generate a stego through the same embedding algorithm and the same payload as those used to obtain the trainingset-list stegos (line 9). These generated pairs (cover/stego) are then appended to the pixels-off-list (line 10), which will later be added to the training set.

The impact on the covers/stegos statistics

At first glance, the "pixels-off" noise could seem to be a noise of a high power relative to the original cover image. However, this is not the case. Take for example the noisy image in Figure 1.b, its PSNR is 31,932. The number of modified pixels is very small with respect to the image size. The proportion of the modified pixels is indeed between 1.5‰ and 1.5%. This can be visually observed in Figure 1. The cover is given in Figure 1.a, and the cover with 1.5 ‰ pixels-off, is given in Figure 1.b. For these two images, the difference between their average is 0.23, the difference between their standard deviation is also 0.23, and the difference between their Entropy is 0.0081.

The first and second moments and the probability distribution are therefore approximately equal. There are nevertheless, some local differences between the two images, such as the local variance computed on a window close to pixels-off.

That said, the majority of the pixels of a pixels-off noisy cover keep the same correlation/dependencies as the original cover. Those correlations/dependencies are what is defining the source (i.e. the surface statistical regularity). So, the "pixels-off" technique only implies a small perturbation of the statistics of the covers, but preserves the average statistics. This is exactly what we need to enrich the initial learning set with additional covers.

The impact on the stegos is described below. Figure 1.c shows the embedding modification probability, when using S-UNIWARD [START_REF] Holub | Universal Distortion Function for Steganography in an Arbitrary Domain[END_REF] embedding model, for each pixel of the cover image in Figure 1.a. Figure 1.d shows the embedding modification probability, for the cover with pixels-off given in Figure 1.b. The red pixels stand for the high probabilities and the blue ones for low probabilities. Figure 2 better illustrates the difference between these two images with a visualization by elevation. The "peaks" stand for pixel areas where there is a difference between the embedding modification probability for the cover image and its corresponding pixels-off one. The peaks only occur to the pixels-off positions, and they are all centred on the pixels-off positions.

One can observe that the probabilities between the two images are similar, except in areas close to a pixel-off. This occurs because a pixel-off area is producing a saliency in the image. It is captured by the filters computing the cost map, such as those of S-UNIWARD, and is identified as an area where the embedding is favorable, i.e. whose cost value is small, thus, whose modification probability is high. A stego image and its "pixels-off" noisy version will not have the exact same modified pixels when the same message is embedded. This property is also interesting because it produces noisy stegos different from initial stegos, but still statistically close.

PRESENTATION OF TWO STATE-OF-THE-ART CNNS FOR SPATIAL STEGANALYSIS

Our study on the pixels-off data augmentation technique, is conducted using two convolutional neural networks (CNNs) from existing state-of-the-art spatial steganalysis methods: Yedroudj-Net [START_REF] Yedroudj | Yedrouj-Net: An Efficient CNN for Spatial Steganalysis[END_REF] (section 3.1), and CovPool-Net [START_REF] Deng | Fast and Effective Global Covariance Pooling Network for Image Steganalysis[END_REF] (section 3.2), over two databases BOSS base [START_REF] Bas | Break Our Steganographic System': The Ins and Outs of Organizing BOSS[END_REF] and ALASKA [START_REF] Cogranne | The ALASKA Steganalysis Challenge: A First Step Towards Steganalysis[END_REF].

Yedroudj-Net

Yedroudj-Net [START_REF] Yedroudj | Yedrouj-Net: An Efficient CNN for Spatial Steganalysis[END_REF] is a shallow network that converges fast. It can learn on small databases whilst obtaining good performances. It was published in 2018 to become, in a very short period of time, one of the most important CNNs for spatial steganalysis. We report in Figure 3 the global architecture of this network. We can observe that it is built in three parts, called modules: the pre-processing, the convolution and the classification module. The pre-processing module includes one convolutional layer whose weights are initialized with high-pass filter kernels derived from the SRM linear filters [START_REF] Fridrich | Rich Models for Steganalysis of Digital Images[END_REF]. Unlike the original paper, we have decided to add a truncation activation layer (TLU) on this module. This helps the network to converge even faster. The convolutional module is composed of five convolutional blocks; each block starts with a convolutional layer and ends with an average pooling layer. In between, we find a batch normalization layer and an activation layer (block Type2). The average pooling layer is omitted from the first block to prevent premature signal loss (block Type1). The average pooling layer is replaced with a global average pooling in the last block (block Type3).

As for the classification module, it consists of three fully connected layers and a softmax activation function. The softmax function is used to normalize the two scores provided by the network between [0,1].

For more details on Yedroudj-Net, the reader can either check the original paper [START_REF] Yedroudj | Yedrouj-Net: An Efficient CNN for Spatial Steganalysis[END_REF] or the online code at 2 . The pytorch version of yedroudj-net, which has been used in this paper, will soon be available at3 .

CovPool-Net

Global Covariance Pooling Network, referred to as CovPool-Net [START_REF] Deng | Fast and Effective Global Covariance Pooling Network for Image Steganalysis[END_REF], is a deep network that is as efficient as SRNet [START_REF] Boroumand | Deep Residual Network for Steganalysis of Digital Images[END_REF], yet it requires no curriculum learning. Moreover, according to [START_REF] Deng | Fast and Effective Global Covariance Pooling Network for Image Steganalysis[END_REF] authors, CovPool-Net converges faster than SRNet. CovPool-Net is a recent CNN that was published in July 2019. Thanks to its deep architecture design, this network has been able to achieve a good performance level and thus takes its place among the state-of-the-art spatial steganalysis models.

Similar to Yedroudj-Net, CovPool-Net is composed of a preprocessing module, a convolutional module and a classification module (see Figure 4). As for the pre-processing module, it is composed of two layers, a convolutional layer whose weights are initialized in the same way as in Yedroudj-Net, and a truncation activation layer (TLU) [START_REF] Ye | Deep Learning Hierarchical Representations for Image Steganalysis[END_REF].

The CNN convolutional module is made up of 10 convolutional blocks. All of these ten blocks include a convolutional layer, a batch normalization layer and an activation layer. The average pooling layer is present only on blocks 4, 6 and 8, while block 10 includes a global covariance pooling which, according to [START_REF] Deng | Fast and Effective Global Covariance Pooling Network for Image Steganalysis[END_REF] authors, improves the detection performance while preserving similar training speeds compared to a global average pooling. The blocks without an average pooling layer are noted Type1. Block Type2 refers to those with an average pooling layer. Last block, the one with the global covariance pooling layer, is noted as block Type3.

Unlike the Yedroudj-Net classification module, which is composed of three fully connected layers, the CovPool-Net classification module is composed of only one fully connected layer, which is followed by a softmax function that produces the probability distribution on the two-class labels.

EXPERIMENTAL METHODOLOGY

This section describes the common core of all the experiences reported in section 5.

Datasets

In this paper, to assess the reliability of our approach, we conduct various experiments on different databases. In the following paragraphs, we list those databases with a brief description. [START_REF] Bas | Break Our Steganographic System': The Ins and Outs of Organizing BOSS[END_REF]. This database was made in 2011 for a steganalysis competition. This database has become the most known and used database for steganalysis, but also for steganography. It consists of 10,000 grayscale images of 512 × 512 pixels, uncompressed, and coming from 7 different cameras, 4.1.2 the BOWS2 base: Break Our Watermarking System [START_REF] Bas | BOWS-2 Contest (Break Our Watermarking System)[END_REF]. This database was originally created for a watermarking competition held in 2008, but since 2017 it has been widely used as a complementary database to the BOSS base. In particular, it is used to augment the provided samples' number when training on the Boss base. This database consists of 10,000 grayscale images of 512 × 512 pixels, uncompressed, and whose distribution is close to that of BOSS base, 4.1.3 ALASKA-10K base: The ALASKA database was formerly created for the "in the wild" steganalysis competition [START_REF] Cogranne | The ALASKA Steganalysis Challenge: A First Step Towards Steganalysis[END_REF]. The main objective of this competition was to bring steganalysis from research laboratories to real-life conditions.

the BOSS base: Break Our Steganographic System

Compared to BOSS and BOWS2, ALASKA is a large-scale database. It consists of 80,000 raw images from 51 different cameras (including smartphones and tablets...). This allows us to reflect better the wide media diversity that can be found "in the real world"; this diversity may be particularly fruitful for machine learning models. Moreover, a large database with many samples is important for training Deep Learning models dedicated to steganalysis, especially given that these models are getting deeper and deeper.

For the objectives of this paper, and in order to be able to fairly compare the results with those obtained on the BOSS base, we decided to randomly select 10,000 (the same number of images as on the BOSS database) of the 80,000 RAW images. Next, we applied the scripts provided with the ALASKA database to obtain 10,000 uncompressed gray-scale images in "tiff" format of 1024×1024 pixels. We referee to that base as ALASKA-10K.

Due to computational and time limitations, we conduct all the experiments on images of 256×256 pixels. To this end, we re-sampled all the images to 256×256 pixels, using the imresize() Matlab function with the default parameters (bi-cubic interpolation with antialiasing).

Note that the sub-sampling step supposes to make the pixels' value distribution of ALSAKA-10k database closer to that of BOSS'. However, by examining the steganalysis results in section 5, it appears that these two databases are statistically different. ALSAKA-10k is still a more "secure" database relative to BOSS, i.e. more difficult to steganalyze.

Software platform

In our experiments, we use Matlab for image manipulation and preprocessing tasks such as image re-sampling, data embedding, pixel manipulation, etc. For data embedding, two well-known contentadaptive spatial steganography algorithms with two payloads of 0.2 and 0.4 bit per pixel (bpp) are used, namely S-UNIWARD [START_REF] Boroumand | Deep Residual Network for Steganalysis of Digital Images[END_REF] and WOW [START_REF] Bas | BOWS-2 Contest (Break Our Watermarking System)[END_REF].

Note that we use Matlab implementations (online codes 4 ) with the simulator for embedding and a random key for each embedding. In this paper, both CNN's, i.e. Yedroudj-Net and CovPool-Net, are implemented using PyTorch 5 , an open-source deep learning framework widely used by the scientific community. Also, note that the results presented in the original Yedroudj-Net paper were obtained using Caffe. This explains the slight variations in performance compared to those presented in this paper. All the experiments were run on an NVidia Titan X GPU card.

Descriptions of the different experimental scenarios

Below, we list all the scenarios that we have adopted in our tests with a brief description of how we create the training sets for each scenario. For a fair comparison, the test set remains the same on all scenarios.

Classical scenario. This represents the conventional scenario which is often used to obtain the baseline performance of a steganalysis model. Basically, this scenario consists of evaluating a steganalysis model on a given database. Therefore, we split the database of pairs (cover/stego) into three sets. 40% of the cover/stego pairs are reserved for the model's training, 10% are assigned to the validation set, while the remaining 50% are reserved for the test set. When referring to this scenario, we use the name of the used database, e.g. if we use Yedroudj-Net on ALASKA-10k, the results based on this scenario will be noted as ALASKA-10k.

Database enhancement scenario. In the field of steganalysis, the databases used are still relatively small. To overcome this limitation, researchers are studying database augmentation techniques and their impact on the overall performance of steganalysis models. One of these techniques is what we call database enhancement. In practical terms, database enhancement consists of providing a training set of the classical scenario with cover/stego pairs coming from a different database (a complementary database). To refer to this scenario, we put side-by-side the names of the original base and the complementary base. For example, BOSS+BOWS2 is when we enrich the BOSS training set with images from BOWS2. Virtual augmentation scenario. This is the most commonly used technique for database enrichment, whether for machine learning or deep learning tasks. In steganalysis, VA consists of performing label-preserving flips and rotations on both covers and stegos of the training set, thus increasing the size of the learning set by a factor of 8. In this paper, we refer to this scenario as Database name_VA. If we take the previous example BOSS+BOWS2, when applying VA, this scenario will be noted "BOSS+BOWS2_VA". Note that the test set of this configuration remains unchanged (no virtual augmentation). Pixels-off scenario. This scenario is similar to that of database enhancement, but rather than enriching the training set with cover/stego pairs from a complementary database, we use the pixels-off technique outlined in Algorithm 1 and discussed in section 2.2. First, we generate a pixels-off version(s) of the training set's covers. Once the covers are generated, we proceed to the generation of their corresponding stegos. These new resulting base of cover/stego pairs are then added to the training set. This increases the size of the learning set by a factor of 2. Note that it is possible to further increase the size of the training set by changing the 𝑃 parameter each time (see Algorithm 1) and thus generating more pixels-off versions. To refer to this scenario, we use the following notation "Database name+P 1 _P 2 .._P 𝑛 -off", where P 𝑖 ∈ {1,..,𝑛 } is the number of pixels that are switched to zero, and 𝑛 is the number of duplicated versions. For example, "BOSS+100_400-off" refers to the scenario in which we augment the Boss training set with two pixels-off versions, 𝑛 = 2 with 𝑃 1 = 100, 𝑃 2 = 400.

EXPERIMENTAL RESULTS

In the following section, we present the obtained results for different setups. Note that the maximum number of epoch is fixed to 200 (resp. 400) epochs for CovPool-Net (resp.Yedroudj.Net). Nevertheless, the two networks reach convergence before attaining this number. Therefore, to give a rough estimate, we provide convergence times for each setup.

Setup 1: No pixels-off, Yedroudj-Net, BOSS base

In Table 1, we report the error probability obtained using Yedroudj-Net on WOW and S-UNIWARD at 0.2 bpp and 0.4 bpp. Different scenarios are tested, the classical scenario, virtual augmentation scenario and the database enhancement scenario, plus the combination of database enhancement and virtual augmentation scenarios. Regardless of payload size and embedding algorithms, this virtual augmentation scenario (BOSS_VA) provides superior results to the classical scenario (BOSS), with a 2-3% performance improvement. Note that the training time (to reach convergence) of Yedroudj-Net under the virtual augmentation scenario is the same as under the classical scenario; this is due to the fact that we use online virtual augmentation. Thus, in theory, more images are available for network training (32,000 pairs), but in practice, for each epoch, the network will see as many images as in the classical scenario (4,000 pairs). In other words, for each epoch, we randomly select 4,000 pairs from the 3,2000 pairs. As for the database enhancement scenario, noted BOSS+BOWS, the CNN's performance is even better. As we can see, the steganalyzer efficiency has increased by 2 to 5%. However, more time is needed to train (14-15h versus 4-5h in the VA scenario). This is because, in contrast to the VA scenario, the number of images provided to the network at each epoch in the database enhancement scenario increases. The results are even better when we apply both augmentation techniques (database enhancement and virtual augmentation); the error probability is reduced by 5-7% compared to the classical scenario.

Setup 2: With pixels-off, Yedroudj-Net, BOSS base

In order to study the importance of the proposed technique, several tests have been carried out. First, as a reference test, we tested the classical scenario noted B (BOSS). Then, we tested the "pixels-off" scenario. For that, four pixels-off versions of the BOSS (B) training set were generated (100-off, 256-off, 400-off, 1024-off). These generated versions were then added, gradually, to the classic scenario training set B (BOSS); the training set size is thus increased by 4,000 each time. Accordingly, the final training set (B 4 ) contains 20,000 cover/stego pairs. The results are presented in Table 2. For the experiment with the training set noted B 1 (B+100-off), when the initial learning set is increased using the 100-off version, the Yedroudj-Net error probability is decreased -on average-by 2%. The performance is further improved when we increase the initial training set twice by using both the 100-off and the 256-off versions (B 2 ). The detection error probability is reduced by 2-4% for WOW (resp. 5-6% for S-UNIWARD) in comparison to the classical scenario. However, more time is needed to train (∼ 14h). The B 3 training set slightly reduces the steganalysis error probability, for both payloads, compared to the B 2 training set (0,4% for WOW at 0.2bpp, and 0.8% for S-UNIWARD at 0.4bpp). Furthermore, no significant improvement is observed from B 3 to B 4 . As shown in Table 3, whatever the value of the variable 𝑃 used for the enrichment (100-off, 256-off, 400-off, or 1024-off) the steganalysis performance is improved, with 𝑃 = 400 (400-off) gives the best results. When grouping all those enrichment, as observed in Table 2, with B 4 , the results are even better. Thus, the optimal parameters is roughly around 𝑃 = 400 pixels-off, but combining various enrichments with 𝑃 between 100 and 1024 allows us to improve even more the steganalysis efficiency. Note that this behavior is also observed in the other experiments detailed hereafter.

In the case of B 1 _VA (Table 2) where the enrichment is done by applying, in addition to the pixels-off technique, the virtual data augmentation, a significant improvement is observed. The detection error probability is reduced by 3-6% for WOW (and 4-5% for S-UNIWARD) compared to the classical scenario. These results show that virtual data augmentation can be freely used in parallel with pixels-off techniques. This conclusion is very interesting as it shows that the pixels-off technique can offer a good and straightforward solution to further improve the performance of CNN steganalysis, while continuing to apply classical enrichment methods.

Setup 3: With pixels-off, Yedroudj-Net, ALASKA base

To investigate the usability of the pixels-off technique to different databases, we conducted similar tests to those presented in Table 2, except that we used a different database, ALASKA-10k. The results are presented in Table 4. For the pixels-off scenario using A 1 (ALASKA+100-off) training set, the error probability is reduced by an average of 1% in comparison with the classical scenario A (ALASKA). The performance keeps improving as we further enrich the training set (A 2 ). Better results are obtained for A 3 , as we achieve an improvement in performance of 2-4% on WOW, and 2-3% on S-UNIWARD with respect to the no-enrichment scenario (ALASKA). The trend for scenario A 4 is similar to those obtained with BOSS base with scenario B4 (Table 2). The results between A 3 and A 4 are comparable or even a little bit improved with A 4 . Again, combining various enrichments by varying the parameter 𝑃 between 100 and 1024 improves the performances. From previous results, we can conclude that the pixels-off technique improves the performance of the model, regardless of the database. Before moving to the next section, where we will examine the performance of the pixels-off technique using another state-ofthe-art steganalyzer CovPool-Net, we would like to point out two observations in agreement with those completed in [START_REF] Giboulot | Roots and Solutions of Cover-Source Mismatch in Image Steganalysis: a Comprehensive Study[END_REF] and [START_REF] Sedighi | Toss that BOSSbase, Alice![END_REF].

Firstly, we can notice that the intrinsic security of ALASKA-10k database is better than that of BOSS database. As an example, the accuracy of S-UNIWARD 0.4 bpp, steganalyzed with Yedroudj-Net, is 32% on ALASKA (A), whereas it is 22% on BOSS (B). This can be explained by the fact that ALASKA-10k is made up of images that come from many different camera models, and that the raw images have been developed with a various number of developments, among them some are producing a high level of mismatch [START_REF] Giboulot | Roots and Solutions of Cover-Source Mismatch in Image Steganalysis: a Comprehensive Study[END_REF].

Secondly, the superiority, in terms of security, of S-UNIWARD versus WOW, when evaluated on BOSS, is less apparent when the evaluation is done on ALASKA-10K. At 0.4 bpp, S-UNIWARD is 7% more secure than WOW on BOSS, whereas the gap falls to 2% on ALASKA-10K. As reported in [START_REF] Sedighi | Toss that BOSSbase, Alice![END_REF], most of the adaptive algorithms have been tuned on the BOSS base, and the internal parameters are not optimized for other databases. This can explain the security gap reduction between WOW and S-UNIWARD.

Setup 4: With pixels-off, CovPool-Net, BOSS

In order to analyse the performance of the pixels-off technique when using different steganalyzer, additional tests were carried out.

In this section, we present the obtained results when pixels-off enrichment technique is used with another steganalysis model, CovPool-Net. For embedding, we use WOW and S-UNIWARD with BOSS base. As in section 5.2, we start with the classical scenario (no enrichment) and then gradually extend the training set size by applying the pixels-off algorithm, the results are presented in Table 5. For B 1 , we see an improvement of 2-3% for S-UNIWARD and less than 1% for WOW compared to the classical scenario. Better results are achieved for B 2 (3-5% for S-UNIWARD and 1-2% for WOW) For B 3 , the pixels-off algorithm increases the performance by 3-2% for S-UNIWARD and 2% for WOW. Like on Yedroudj-Net, CovPool-Net shows a considerable improvement when we combine both pixels-off and virtual augmentation (B 4 ). The error rate can be reduced by 10-7% for S-UNIWARD (resp. 8-6% for WOW ) compared to classical scenario (BOSS). Note that CovPool-Net takes one to two hours longer for training compared to Yedroudj-Net.

DISCUSSION AND ADVANCED ANALYSIS

In an attempt to better understand the underlying properties resulting from the use of the pixels-off technique, we first evaluate two other noises and compare their performance to the pixels-off enrichment. Second, we detail the class of problems addressed by the pixels-off technique.

Discussion on the noise power

We evaluate the impact of two classical additive noises when used for database enrichment. The first noise is the intensity spikes noise, known as the salt and pepper noise. We set the density to the default value, d = 0.05. The second noise is the Gaussian one. We use it with a zero mean and a variance of 0.01. We use BOSS base with WOW and S-UNIWARD at a payload of 0.4 bpp. For the steganalysis we use Yedroudj-Net.

The results are presented in Table 6. Note that for a fair comparison, the tests are always performed on the same test set. Also, note that the training sets have the exact same number of cover/stego pairs. One can observe that the results are worse when we perform a database enrichment using Gaussian noise (BOSS+Gaussian), compared to the no-enrichment scenario. An increase of 1% in detection error probabilities is reported for both WOW and S-UNIWARD. These results suggest that the addition of high-power noise, such as Gaussian noise, can lead to an important modification of the initial statistical models of both cover and stego contents. Thus, it provokes a mismatch between the training and the test set and so causes a drop in performance.

For enrichment with the salt and pepper noise and a density d = 0.05, the results are similar to those of the classic scenario (BOSS), and significantly inferior to those of the pixels-off scenarios (BOSS+100-off), regardless of the embedding algorithms used. The density d = 0.05 stands for the default density value (in Matlab) and the value commonly used in machine learning for data enrichment, however for steganalysis this value may be too high. Indeed, for an image of 256 × 256 pixels, with d= 0.05, about 3,300 pixels are affected by the salt and pepper noise, that is three times more modifications than when the maximum value of the 𝑃 parameter is used (in pixel-off technique 𝑃 𝑚𝑎𝑥 = 1024 pixels).

It appears that only low-power noise affecting a very small percentage of pixels (less than 1.5%) can be useful for data-enrichment. To investigate this assumption, we test the salt and pepper noise but this time with a lower noise density, d=0.0016. Therefore, for an image of 256 × 256 pixels, about 100 pixels will be affected by the salt and pepper noise. In this case, the performance is significantly improved compared to the "BOSS+salt&pepper (d = 0.05)" scenario, while it is quite similar to that of the the "BOSS+100-off" scenario. This is because the salt and pepper noise, with d=0.0016, is inherently close to the pixels-off noise, as it randomly scatters white or black pixels in the image.

In the past, the steganalysis community did not obtain successful results with salt and pepper noise, this is mainly due to the use of the same density value used in the machine learning literature, which was actually too high for steganalysis' purpose. However, this type of noise gives satisfactory results when the lower density value is used. Finally, note that we have also observed that some other weak noises can be used fruitfully. Instead of switching off 100 pixels (setting their values to 0), we tested adding a noise of "+/-1" to 100 pixels randomly picked. We obtained an improvement, almost equal to that of "BOSS+salt&pepper (d = 0.0016)".

Class of problems addressed by the pixels-off technique

A signal like the pixels-off noise slightly changes the statistical properties of both cover and stego images [START_REF] Jo | Measuring the tendency of CNNs to Learn Surface Statistical Regularities[END_REF]. As a consequence, the enrichment of a database with pixels-off noisy images should introduce a mismatch between the learning and test set, and thus imply a drop off in steganalysis performance. Nevertheless, the results that we reported previously, show that this type of enrichment improves CNN steganalysis accuracy. What can appear as a counter-intuitive behaviour is, in fact, specific to deep-learning based steganalysis. Indeed, feature-based steganalysis, for its part, is badly impacted by the pixel-off noise. This was checked by evaluating the impact of a pixels-off-100 enrichment on the BOSS base with WOW at 0.4 bpp using an Ensemble Classifier [START_REF] Kodovský | Ensemble Classifiers for Steganalysis of Digital Media[END_REF] with Rich Model SRM [START_REF] Fridrich | Rich Models for Steganalysis of Digital Images[END_REF]. Instead of improving the accuracy, the error probability increases from 25.5% to 28.3%. This confirms that the pixels-off technique does not produce images sharing the exact same source-model, and/or does not produce stegos that comes from the exact same stego-source-model. The pixels-off noise is breaking a sufficient number of pixel dependencies (which are due to the development processing) to results in an image realization that cannot come from the source that produced the initial database.

One should conclude that pixels-off enrichment incites the learning to be more sensitive to areas around switched-off pixels since these areas are now associated with strong embedding probabilities. Indeed, an area with a pixel switched off is creating a local abrupt signal change, for the covers and the stegos, and we think that it encourages the convolutional kernels to learn local weights more sensitive to those very singular areas. Knowing that modern adaptive embedding algorithms are mainly modifying the pixels from the textured areas, pixels-off may also encourage the network to focus more attention on these areas.

In order to challenge this rationale, we have forced the pixels that are switched-off to occur in the pixels of higher embedding probability. We conducted two additional experiments to check if the enrichment of the database can be guided by the adaptability of the embedding algorithm. In order to do so, we modify the pixels-off Algorithm 1 such that pixels switched-off are chosen by taking into account the Side Channel Information, which is in our case the embedding modification probabilities map. More precisely, instead of randomly switching off 100 pixels of the whole image, we switch off the same number of pixels while considering only the 10% pixels with the highest embedding probability (experiment noted BOSS+100_off-highP), then we test while considering only the 10% pixels with the lowest embedding probability (experiment noted BOSS+100_off-lowP). These results are presented in Table 7.

We can observe that the detection accuracy is reduced by 1% for the Boss+100_off-lowP scenario compared to the classical pixels-off scenario BOSS+100_off . On the other hand, the results are improved for the +100_off-highP scenario compared to the Boss+100_off scenario.

These results are interesting because they could lead to another way of doing SCA steganalysis, by generating SCA training sets. The usual approaches, such as Ye-Net [START_REF] Ye | Deep Learning Hierarchical Representations for Image Steganalysis[END_REF], SRNet [START_REF] Boroumand | Deep Residual Network for Steganalysis of Digital Images[END_REF], are using the probability map as an input of the network. On the contrary, with the adaptive pixels-off method, the side information is only used to enrich the database. In order to better understand how noises, like the pixels-off one, could be used as another way to do SCA learning, additional experiments can be carried out, such as selecting the pixels-off by drawing them from a scaled version of the probability map, or compare the results with a CNN-based informed steganalysis. We postpone the numerous additional experiments for future works.

CONCLUSION

Today, CNN-based steganalyzer architectures are getting deeper and deeper. Therefore, larger databases are needed in order to reach the region of good performance (irreducible error region). However, having a larger database in the steganalysis field can be a challenging option. This is where database enrichment techniques can help. This consists in increasing the number of images of the initial training set. Among existing techniques, we find the virtual data augmentation technique and the database enhancement technique.

In this paper, we propose and explore a novel technique for database enrichment for CNN-based steganalyzer. We call this enrichment technique "pixels-off". This technique is similar, in concept, to a noise addition, but it is performed in such a manner that it ensures that the pixel distribution of the resulting image remains as close as possible to the original one. Thus it avoids triggering a Cover Source Mismatch phenomenon (CSM).

The proposed technique is simple to implement, with low complexity and, most importantly, it improves the detection accuracy of the CNN-based steganalyzer. In addition, our technique is compliant with other techniques, so there is no restriction when it is performed in conjunction with other techniques such as virtual augmentation. Furthermore, the combination of the pixels-off technique with other data enrichment techniques leads to an even better performance.

We expect this paper to lead to fruitful research avenues that we plan to pursue. In future work, it would be interesting to study thoroughly how data enrichment impacts CNN-based steganalyzer performances. For this, we could examin the confusion matrix to understand the origin of the performance improvements. We could examine the learned filters of CNN, before and after data enrichment. We shall also study other techniques, and evaluate their scope for improving CNN-based SCA or not-SCA steganalysis performance.

Algorithm 1 : 6 x 7 z 8 𝑥 9 𝑦

 16789 Pixels-off database generation. Data: training-set-list, P; // P: numbers of pixels to nullify Result: pixels-off-list 1 pixels-off-list = []; 2 cover-list = get-covers (training-set-list); 3 N = length (cover-list); 4 (h,w) = size (cover-list(0)); 5 for 𝑖 = 1, 𝑖 ≤ 𝑁 , 𝑖 + + do = cover-list(i); // x: a cover = dispatch-P-zeros (h,w, P); // randomly spread P zeros, in an image made of 1, of size ℎ × 𝑤 𝑜 𝑓 𝑓 = x ⊙ z; // 𝑥 𝑜 𝑓 𝑓 : a cover with 𝑃 pixels-off 𝑜 𝑓 𝑓 = embedding (𝑥 𝑜 𝑓 𝑓 , 𝑝𝑎𝑦𝑙𝑜𝑎𝑑); 10 pixels-off-list.append([𝑥 𝑜 𝑓 𝑓 , 𝑦 𝑜 𝑓 𝑓 ]) 11 end 12 return pixels-off-list;

Figure 2 :

 2 Figure 2: Visualization by elevation for the differences between the embedding modification probability maps of the cover (Figure 1.c) and its pixels-off version (Figure 1.d).

Figure 3 :

 3 Figure 3: Yedroudj-Net CNN architecture.

Figure 4 :

 4 Figure 4: CovPool-Net CNN architecture.

Table 1

 1

	: Steganalysis error probability of Yedroudj-Net on
	BOSS base with two embedding algorithms WOW and S-
	UNIWARD at 0.2 bpp and 0.4 bpp under different scenarios
		WOW	SUNIWARD	examples	conv
		0.2bpp 0.4bpp 0.2bpp 0.4bpp	(pairs)	time
	BOSS	27.71	15.27	35.42	22.7	4,000	4-5h
	BOSS_VA	24.2	13.01	34.2	18.4	32,000	4-5h
	BOSS+BOWS2	23.12	12.84	33.06	17.8	14,000	14-15h
	BOSS+BOWS2_VA 20.85	10.13	29.12	16.37	112,000	14-15h

Table 2 :

 2 Steganalysis error probability of Yedroudj-Net on BOSS base with two embedding algorithms WOW and S-UNIWARD at 0.2 bpp and 0.4 bpp for pixels-off scenario

		WOW	SUNIWARD	examples	conv
		0.2bpp 0.4bpp 0.2bpp 0.4bpp	(pairs)	time
	B = BOSS	27.71	15.27	35.42	22.70	4,000	4-5h
	B 1 = B+100-off	25.31	14.3	33.1	19.4	8,000	9-10h
	B 2 = B 1 +256-off	23.95	13.41	29.8	17.8	12,000	13-14h
	B 3 = B 2 +400-off	23.5	13.44	29.3	16.95	16,000	17-18
	B 4 = B 3 +1024-off	23.8	13.65	29.2	16.98	20,000	21-22
	B 5 = B 1 _VA	21.5	12.4	31.4	14.8	64,000	9-10h

Table 3 :

 3 Steganalysis error probability of Yedroudj-Net on BOSS base with two embedding algorithms WOW and S-UNIWARD at 0.4 bpp with different numbers 𝑃 of pixels-off.

		WOW 0.4 bpp SUNIWARD 0.4 bpp
	BOSS	15.27	22.70
	BOSS+100-off	14.3	19.4
	BOSS+256-off	14.6	19.3
	BOSS+400-off	13.9	18.6
	BOSS+1024-off 14.8	19.9

Table 4 :

 4 Steganalysis error probability of Yedroudj-Net on ALASKA-10K base with two embedding algorithms WOW and S-UNIWARD at 0.2 bpp and 0.4 bpp for pixels-off scenario

		WOW	SUNIWARD	examples	conv
		0.2bpp 0.4bpp 0.2bpp 0.4bpp	(pairs)	time
	A = Alaska	40.11	30.4	42.85	32.45	4,000	4-5h
	A 1 = A+100-off	38.78	28.92	41.87	31.69	8,000	9-10h
	A 2 = A 1 +256-off	38.15	28.09	40.55	30.24	12,000	13-14h
	A 3 = A 2 +400-off	38.05	27.91	40.05	29.54	16,000	17-18h
	A 4 = A 3 +1024-off	37.5	28.02	40.01	28.94	20,000	21-22h

Table 5 :

 5 Steganalysis error probability of CovPool-Net on BOSS base with two embedding algorithms WOW and S-UNIWARD at 0.2 bpp and 0.4 bpp for pixels-off scenario

		WOW	SUNIWARD	examples	conv
		0.2bpp 0.4bpp 0.2bpp 0.4bpp	(pairs)	time
	B = BOSS	26.08	15.60	31.89	18.32	4,000	5-6h
	B 1 = B+100-off	25.33	14.63	28.54	16.25	8,000	10-11h
	B 2 = B 1 +256-off 24.88	13.11	26.61	15.00	12,000	14-15h
	B 3 = B 2 +400-off 23.34	13.02	26.64	15.44	16,000	19-20h
	B 4 = B 1 _VA	17.5	9.23	21.58	10.54	64,000	10-11h

Table 6 :

 6 Steganalysis error probability of Yedroudj-Net on BOSS base with two embedding algorithms WOW and S-UNIWARD at 0.4 bpp for different types of noise enrichment

	WOW0.4 SUNIWARD0.4

Table 7 :

 7 Steganalysis error probability of Yedroudj-Net on BOSS base with two embedding algorithms WOW and S-UNIWARD at 0.4 bpp for different scenarios of pixels-off

		WOW0.4 SUNIWARD0.4
	BOSS	15.27	22.70
	BOSS+100_off	14.3	19.4
	BOSS+100_off-lowP 15.17	20.85
	BOSS+100_off-highP 13.65	18.15

In this paper, the "development" stands for the numerical processes transforming a RAW color image to a

256×256 8-bits grey-levels image

online code: Caffe version

online code: Pytorch version

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

https://pytorch.org/

ACKNOWLEDGMENTS

The authors would like to thank the University of Montpellier (LIRMM) and the University of Nîmes. We would like to thank the French Direction Générale de l'Armement (DGA) for its support through the Alaska project ANR (ANR-18-ASTR-0009), and the French ANR DEFALS program (ANR-16-DEFA-0003). We would also like to thank the Algerian Ministry of Higher Education / Scientific Research, for its scholarship support.