Hussein Obeid

Salah Laghrouche

Leonid Fridman

Yacine Chitour

Mohamed Harmouche

Barrier function-Based Variable Gain Super-Twisting Controller

In this paper, a variable gain super-twisting algorithm based on a barrier function is proposed for a class of first order disturbed systems with uncertain control coefficient and whose disturbances derivatives are bounded but the upper bounds of those derivatives are unknown. The specific feature of this algorithm is that it can ensure the convergence of the output variable and maintain it in a predefined neighborhood of zero independent from the upper bound of the disturbances derivatives. Moreover, thanks to the structure of the barrier function, it forces the gain to decrease together with the output variable and the control signal follows the absolute value of the disturbances.

I. INTRODUCTION

For systems with matching disturbances, the sliding mode controllers have shown their high efficiency [START_REF] Utkin | Sliding modes in optimization and control problems[END_REF]. Indeed, they provide a closed-loop insensitivity with respect to bounded matched disturbances and guarantee the finite-time convergence to the sliding surface. However, the discontinuity of sliding mode controllers may cause a undesirable big level of chattering in the systems with fast actuators [START_REF] Pérez-Ventura | When is it reasonable to implement the discontinuous sliding-mode controllers instead of the continuous ones? frequency domain criteria[END_REF], [START_REF] Boiko | Discontinuous control systems: frequency-domain analysis and design[END_REF]. This major obstacle has been attenuated by some strategies. For systems with fast actuators, relative degree one and Lipschitz disturbances, the super-twisting controller [START_REF] Levant | Robust exact differentiation via sliding mode technique[END_REF] is one of the most popular strategies. It allows to achieve a second order sliding mode in finite-time by using a continuous control signal. However, the implementation of the super-twisting controller requires the knowledge of an upper bound of the disturbances derivatives, which is unknown or overestimated in practice. Moreover, even the disturbances derivatives are time varying, it will be desirable to follow their variation.

This problem motivates researchers to develop adaptive sliding mode controllers for the case where the boundaries of the disturbances exist but they are unknown. The general goal of these techniques is to ensure a dynamical adaptation of the control gains in order to be as small as possible while still sufficient to counteract the disturbances and ensure a sliding mode.

The adaptive sliding mode control approaches which exist in the literature can be broadly split into three classes ( [START_REF] Oliveira | Adaptive Sliding Mode Control Based on the Extended Equivalent Control Concept for Disturbances with Unknown Bounds[END_REF], [START_REF] Shtessel | Adaptive sliding mode control and observation[END_REF]):

This research was supported by Franche-Comté Regional Council (France) [project RECH-MOB15000008]; PAPIIT of UNAM grant IN115419 and CONACyT, grant 282013.

H. Obeid and S. Laghrouche are with Femto-ST UMR CNRS, Univ. Bourgogne Franche-Comté/UTBM, 90010, Belfort, France (e.mail: hussein.obeid@univ-fcomte.fr; salah.laghrouche@utbm.fr).

L. Fridman is with the Departement of Robotics and Control, Engineering Faculty, Universidad Nacional Autónoma de México (UNAM), D.F 04510. México (e.mail: leonid.fridman@unam.mx).

Y. Chitour is with the Laboratoire des Signaux et Systèmes, Université Paris-Sud, CentraleSupélec, Gif-sur-Yvette, Paris, France (e.mail:yacine.chitour@l2s.centralesupelec.fr).

M. Harmouche is with Actility, Paris, France (e.mail:mohamed.harmouche@actility.com).

(i) Approaches based on the usage of equivalent control value [START_REF] Oliveira | Adaptive Sliding Mode Control Based on the Extended Equivalent Control Concept for Disturbances with Unknown Bounds[END_REF], [START_REF] Utkin | Adaptive sliding mode control with application to super-twist algorithm: Equivalent control method[END_REF]- [START_REF] Oliveira | Adaptive sliding mode control for disturbances with unknown bounds[END_REF]. (ii) Approaches based on monotonically increasing the gains [START_REF] Negrete | Second-order sliding mode output feedback controller with adaptation[END_REF]- [START_REF] Shtessel | Super-twisting adaptive sliding mode control: A Lyapunov design[END_REF]. (iii) Approaches based on increasing and decreasing the gains [START_REF] Plestan | New methodologies for adaptive sliding mode control[END_REF]- [START_REF] Shtessel | Twisting sliding mode control with adaptation: Lyapunov design, methodology and application[END_REF].

Approaches in (i) propose to use the filtered value of the equivalent control as an estimation of the disturbance. The latter consist in increasing the gain to enforce the sliding mode to be reached. Once the sliding mode is achieved, the high frequency control signal is low-pass filtered and used as an information about disturbance in controller gain. The sliding mode controller gain consists in the sum of filtered signal and some constant to compensate possible error between real disturbance and its value estimated by filter. However, the algorithm in [START_REF] Utkin | Adaptive sliding mode control with application to super-twist algorithm: Equivalent control method[END_REF] requires the knowledge of the minimum and maximum allowed values of the adaptive gain, hence, it requires the information of the upper bound of disturbances derivatives. On the other hand, even the other algorithms [START_REF] Oliveira | Adaptive Sliding Mode Control Based on the Extended Equivalent Control Concept for Disturbances with Unknown Bounds[END_REF], [START_REF] Edwards | Adaptive dual layer super-twisting control and observation[END_REF]- [START_REF] Oliveira | Adaptive sliding mode control for disturbances with unknown bounds[END_REF] do not require theoretically the information of the disturbances derivatives, however, in practice, the usage of low-pass filter requires implicitly the information about this upper bound in order to adequately choose the filter time constant.

Strategies in (ii) consist in increasing the gain until the sliding mode is reached, then the gain is fixed at this value, ensuring an ideal sliding mode for some interval. When the disturbance grows, the sliding mode can be lost, so the gain increases and reach it again. This second strategy has two main disadvantages: (a) the gain does not decrease, i.e. it will not follow disturbance when it is decreasing; (b) one cannot be sure that the sliding mode will never lost because it is not ensured that the disturbance will not grow anymore.

To overcome the first of these disadvantages, approaches in (iii) have been developed. According to these approaches, the gain increases until the sliding mode is achieved and then decreases until the moment it is lost, i.e. the sliding mode is not reached any more. These approaches ensure the finite-time convergence of the sliding variable to some neighborhood of zero without big overestimation of the gain. The main drawback of these approaches is that the size of the above mentioned neighborhood and the time of convergence depend on the unknown upper bound of disturbance which are unknown apriori.

Recently, novel approaches based on the usage of a monitoring function [START_REF] Hsu | Adaptive Sliding Mode Control Using Monitoring Functions[END_REF] and a barrier function [START_REF] Obeid | Barrier function-based adaptive sliding mode control[END_REF] have been proposed to adapt the control gains. However, the first strategy has been only applied for the first order sliding mode controller. Whilst the second one has been applied for both first order sliding mode controller and the Levant's Differentiator [START_REF] Obeid | Adaptation of Levant's differentiator based on barrier function[END_REF].

Inspired by ( [START_REF] Obeid | Barrier function-based adaptive sliding mode control[END_REF], [START_REF] Obeid | Adaptation of Levant's differentiator based on barrier function[END_REF]), this paper proposes a variable gain super-twisting controller based on a new barrier function. This algorithm can drive the output variable and maintain it in a predefined neighborhood of zero, in the presence of Lipschitz disturbances with unknown boundaries. Compared to the earlier work [START_REF] Obeid | Adaptation of Levant's differentiator based on barrier function[END_REF], the class of systems considered in this paper contains an additional uncertainty namely the time-dependent uncertain control coefficient. Moreover, the convergence proof of this algorithm is quite different. The advantages of this suggested algorithm are based on the main features of the barrier functions:

• The output variable converges in a finite time to a predefined neighborhood of zero, independently of the bound of the disturbances derivatives, and cannot leave it. • The proposed algorithm does not require neither the upper bound of the disturbances derivatives nor the use of the low-pass filter.

• On the other hand, the barrier function strategy forces the super-twisting gains to decrease together with the output variable and the control signal follows the absolute value of the disturbances. • The usage of the barrier function strategy guarantees that once the derivatives of disturbances are growing the supertwisting gains are also growing ensuring that the output value belongs to the desired neighborhood. • When the super-twisting gains are big enough to ensure convergence to the origin, the barrier function strategy forces the super-twisting gains to decrease together with the output variable, i.e. barrier function methodology makes the control signal follows absolute value of the disturbances. This paper is organized as follows. In Section 2, the problem formulation is given. Section 3 presents the proposed variable gain super-twisting controller. Finally, some conclusions are drawn in section 4.

The notation x γ , for x, γ in R with γ ≥ 0, is used to represent |x| γ sign(x), where sign(x) is the set-valued function equal to the sign of x = 0 and [-1, 1] for x = 0 respectively.

II. PROBLEM FORMULATION

Consider the first order system described by

ṡ(t) = γ(t)u(t) + δ0(t), (1)
where s(t) ∈ R is the output variable, u(t) ∈ R is the supertwisting controller, and γ and δ0 are Lipschitz disturbances such that, if δ := δ0 γ , then one has for t ≥ 0,

g ≤ γ(t) ≤ G, | δ(t)| ≤ M, (2)
where the constant positive bounds g, G and the upper bound M are unknown.

In the presence of Lipschitz disturbances, the standard supertwisting controller [START_REF] Levant | Robust exact differentiation via sliding mode technique[END_REF] given by

u(t) = -k1 s(t) 1/2 + u2(t), u2(t) = -k2 s(t) 0 , (3)
drives both s(t) and ṡ(t) to zero in a finite time, i.e. it provides a second order sliding mode if the control gains k1 and k2 t - √ M and k2 = 1.1M . However, the implementation of this standard super-twisting controller requires the information of the unknwon upper bound M . Therefore, to overcome this problem, the following variable gain super-twisting controller is considered [START_REF] Negrete | Second-order sliding mode output feedback controller with adaptation[END_REF]

u(t) = -L(t, s0) s(t) 1/2 + u2(t), u2(t) = -L 2 (t, s0) s(t) 0 , (4)
where s0 := s(0) and L(t, s0) is the variable gain to be designed in the next section. Suppose that φ(t) = u2(t) + δ(t), then the dynamic of the first order system can be expressed as

ṡ(t) = γ(t) -L(t, s0) s(t) 1/2 + φ(t) , φ(t) = -L 2 (t, s0) s(t) 0 + δ(t). (5)
The idea behind the proposed algorithm is to first increase the variable gain L(t, s0) based on the strategy presented in [START_REF] Negrete | Second-order sliding mode output feedback controller with adaptation[END_REF] until the output variable reaches the neighborhood of zero |s| ≤ ε/2 at some time t. Then, for t > t, the variable gain L(t, s0) switches to the barrier function and the output variable belongs to the predefined neighborhood of zero |s(t)| < ε.

The trajectory of the proposed super-twisting algorithm in the phase plane (s(t), φ(t)) is illustrated in Fig. 1. It can be shown that, at time t, the trajectory enters inside the blue vertical strip with a constant bandwidth ] -ε, ε[. The size of this bandwidth remains constant even if the disturbances are time varying. However, when the disturbances change, the size of the vicinity to which converges s(t) changes but the main feature is that it still has an upper bound ε that it cannot surpass.

A. Preliminaries 1) Barrier function : Definition 1: [START_REF] Tee | Barrier Lyapunov Functions for the control of output-constrained nonlinear systems[END_REF], [START_REF] Tee | Control of nonlinear systems with partial state constraints using a barrier Lyapunov function[END_REF] Suppose that some ε > 0 is given and fixed. For every positive real number b, a barrier function can be defined as an even continuous function [START_REF] Shtessel | A novel adaptive-gain supertwisting sliding mode controller: methodology and application[END_REF] In this paper, the following barrier function is considered

L b : x ∈ (-ε, ε) → L b (x) ∈ [b, ∞[ strictly increasing on [0, ε[. • lim |x|→ε L b (x) = +∞. • L b (x)
L b (x) = √ εb (ε -|x|) 1/2 , x ∈ (-ε, ε), ( 6
)
where b is a positive constant.

III. MAIN RESULTS

To implement the proposed new algorithm, the variable gain L(t, s0) is defined as follows: first consider the function

l(t) = L1t + L0, t ≥ 0, (7)
with L0, L1 are arbitrary positive constants. Assume first that

|s0| = |s(0)| ≤ ε/2. Then, L(t, s0) = L b (s(t))
, with b = L0, as long as the trajectory of ( 5) is defined.

If |s0| = |s(0)| > ε/2, then L(t, s0) = l(t)
as long as the trajectory s(•) of ( 5) is defined and |s(t)| > ε/2. If there exists t(s0) defined as the first time for which

|s(t)| ≤ ε/2, then L(t, s0) = L b (s(t)), b = √ 2l( t(s0)) ≥ √ 2L0
, for t ≥ t(s0) and as long as the of trajectory s(•) of ( 5) is defined. Hence, the variable gain L(t, s0) is defined, as long as the of trajectory s(•) of ( 5) is defined, by

L(t, s0) = l(t), if 0 ≤ t < t(s0), L b (s(t)), if t ≥ t(s0), (8)
with the convention that t(s0) = 0 if |s(0)| ≤ ε/2 and t(s0) = ∞ if s(•) is defined for all times and |s(•)| > ε/2. Since b = √ 2l( t(s0)), then t → L(t, s0) defines a continuous function and hence the control u(•) is also continuous as long as it is defined.

Theorem 1: Let M be the (unknown) upper bound on | δ|, g, G be the bounds on γ(•) and ε, L0, L1 > 0 defining the barrier function L b in [START_REF] Shtessel | Adaptive sliding mode control and observation[END_REF]. Consider System (5) with variable gain L defined in [START_REF] Edwards | Adaptive dual layer super-twisting control and observation[END_REF]. Then, for every s0 ∈ R (and initial value of δ), the trajectory of (5) starting at s0 is defined for all non negative times t and there exists a first time t(s0) for which |s(t)| ≤ ε/2. Then, for all t ≥ t(s0), one has |s(t)| < ε. Moreover, there exists ν(M, G, g) > 0 such that, for every trajectory of ( 5), one has that lim sup t→∞ |φ(t)| ≤ ν(M, G, g).

The proof of Theorem 1 is given in Appendix A.

A. Simulation results

The performance of the aforementioned algortihm is compared with the results obtained through the adaptive supertwisting controller presented in [START_REF] Shtessel | A novel adaptive-gain supertwisting sliding mode controller: methodology and application[END_REF]. [START_REF] Shtessel | A novel adaptive-gain supertwisting sliding mode controller: methodology and application[END_REF] In [START_REF] Shtessel | A novel adaptive-gain supertwisting sliding mode controller: methodology and application[END_REF], the adaptive super-twisting controller is implemented as

   u(t) = -α s(t) 1/2 + u2(t), u2(t) = - β 2 s(t) 0 , (9)
where β = 2µα, and the adaptive gain α is obtained through

α =    w1 γ1 2 sign(|s(t)| -ε), if α > αm ν, if α ≤ αm (10)
where µ, w1, γ1, ε, αm, and ν are positive constants to be selected.

In the simulations, s(0) = 5 and the parameter values of the proposed algorithm are set as ε = 0.1, L1 = 1, L0 = 0.1. On the other hand, the parameter values of the adaptive supertwisting algorithm ( 9)-( 10) are tuned according to [START_REF] Shtessel | A novel adaptive-gain supertwisting sliding mode controller: methodology and application[END_REF]. Hence, µ = 1, w1 = 200, γ1 = 2, ν = αm = 0.01 while the parameter value of ε is chosen as the proposed algorithm, i.e. ε = 0.1. The disturbances are given by γ(t) = 4 + 2sin(3t).

(11)

δ0(t) =      6 sin(5t), if t ≤ 2π s, 15 sin(3t), if 2π s < t ≤ 5π s, 30 sin(5t), if t > 5π s. ( 12
)
The plots of the output variable s(t) with the proposed algorithm and the algorithm presented in [START_REF] Shtessel | A novel adaptive-gain supertwisting sliding mode controller: methodology and application[END_REF] are compared in Figs. 2(a)-2(b). In Fig. 2(a) it can be observed that for the proposed algorithm, the output variable does not exceed the predefined neighborhood of zero ε = 0.1. On the other hand, it can be noticed in Fig. 2(b) that the size of the neighborhood of zero to which converges s(t) with the algorithm presented in [START_REF] Shtessel | A novel adaptive-gain supertwisting sliding mode controller: methodology and application[END_REF] is changing together with the amplitude of disturbances derivatives, and, consequently, it cannot be predefined. Moreover, the size of the neighborhood ensured by the algorithm presented in [START_REF] Shtessel | A novel adaptive-gain supertwisting sliding mode controller: methodology and application[END_REF] can be very large when the amplitude of disturbances derivatives is large (for t > 5π s, |s(t)| ≤ 5 = 50 ε). Fig. 3 illustrates the convergence of the state variable φ(t) with the proposed algorithm and the algorithm presented in [START_REF] Shtessel | A novel adaptive-gain supertwisting sliding mode controller: methodology and application[END_REF]. It can be clearly seen that φ(t) with the proposed algorithm converges to some vicinity of zero depending on M . Moreover, it can be noticed that the size of vicinity to which converges φ(t) with the proposed algorithm is less than the one with the algorithm presented in [START_REF] Shtessel | A novel adaptive-gain supertwisting sliding mode controller: methodology and application[END_REF] (φ(t) is less than 4 with [START_REF] Shtessel | A novel adaptive-gain supertwisting sliding mode controller: methodology and application[END_REF] the proposed algorithm, however, with the algorithm presented in [START_REF] Shtessel | A novel adaptive-gain supertwisting sliding mode controller: methodology and application[END_REF], φ(t) is more than 17).

The behaviours of the variable gains with the proposed algorithm and the algorithm presented in [START_REF] Shtessel | A novel adaptive-gain supertwisting sliding mode controller: methodology and application[END_REF] are depicted in Figs. 4(a)-4(b). It can be noticed that both variable gains can increase and decrease based on the output variable value.

Fig. 5 compares the evolution of the control signal with the disturbances. It can be seen that the control signal with the proposed algorithm closely follows the disturbances which is not the case with the algoritm presented in [START_REF] Shtessel | A novel adaptive-gain supertwisting sliding mode controller: methodology and application[END_REF].

IV. CONCLUSION

This paper presents a variable gain super-twisting controller for a class of first order disturbed system where the upper bound of the disturbances derivatives exist, but it is unknown. This algorithm ensures the convergence of the output variable and prevents its violation outside a predefined neighborhood of zero. The usage of the barrier function strategy ensures that, when the derivatives of disturbances are growing, the super-twisting gains are growing ensuring that the output value belongs to the desired neighborhood and then start to decrease. On the other hand, the barrier function strategy forces the supertwisting gains to decrease together with the output variable. Moreover, the value of the control signal is very close to the absolute value of the disturbances. respect to the state variable (s, φ). Let I(s0) be the interval where such a dynamics is defined and I(s0) is of the form [0, T0). Then, one must prove that T0 is finite, which would at once imply that T0 is the desired time t(s0).

Reasoning by contradiction, we assume that |s| > ε/2 on I(s0) and we can assume with no loss of generality that s is positive on I(s0). From the second equation of ( 5), one gets that -l 2 (t) -M ≤ φ(t) ≤ -l 2 (t) + M, on I(s0), which yields easily that I(s0) = [0, ∞). In particular, we deduce that φ(t) becomes negative in finite time and remains so.

The first equation of ( 5) yields that ṡ(t) ≤ -gl(t) s(t)

1 2 and hence convergence of s to zero in finite time. This contradicts the assumption that |s| > ε/2 on I(s0) and the argument for the first step of the proof of Theorem 1 is complete.

Second step: We now prove that for all t ≥ t(s0), one has |s(t)| < ε and there exists ν > 0 only depending on M, G, g such that lim sup t→∞ |φ(t)| ≤ ν. We use I(s0) to denote the interval of times t ≥ t(s0) for which the corresponding trajectory of ( 5) is defined. In the argument below, we only work for times t ∈ I(s0). In particular, |s(t)| < ε for t ∈ I(s0). It will also be clear for the argument below that I(s0) is infinite. We consider the following change of variables y = (y1, y2) given by y1 = L 2 b s(t), y2 = φ(t).

In theses variables, the system can be written as follows

   ẏ1 = 2 Lb L b y1 + L 2 b ṡ, ẏ2 = φ, (14)
and

|y1( t(s0))| ≤ b 2 ε. Since Lb = 1 2 b √ ε(ε -|s|) -1/2 ṡ s 0 (ε -|s|) , (15)
then the term Lb L b can be written as

Lb L b = 1 2 ṡ s 0 (ε -|s|) = C0L 2 b ṡ s 0 , (16)
with C0 = 1 2b 2 ε . Substituting ( 16) and ( 5) into ( 14), we obtain

   ẏ1 (1 + 2C0|y1|) = L 2 b γ(t)(-y1 1/2 + y2), ẏ2 = -L 2 b y1 0 + δ(t). (17)
We use a new time scale τ defined by τ ( t(s0)) = 0 and dτ = L 2 b dt. We use to denote the derivative with respect to τ . System (17) can be rewritten as

       y 1 (1 + 2C0|y1|) = γ(t)(-y1 1/2 + y2), y 2 = -y1 0 + δ(t) L 2 b . (18)
Note that

| δ(t)| L 2 b
Let us first prove by contradiction that, for every trajectory of ( 18), there exists an increasing sequence of times (τn) n≥0 tending to infinity such that

V1(τn) < V * 2 , n ≥ 0. (25)
Hence, if (25) does not hold true for some trajectory (s(•), φ(•)) of ( 18), there exists τ0 ≥ 0 such that one has, for τ ≥ τ0 V1(τ ) ≥ V * /2. In the strip |y1| ≤ yM , one has |y2| ≥ √ V * /2 and |y 2 + y1 0 | ≤ 1/8. Moreover, an obvious argument shows that if any trajectory of ( 18) is in the strip at some time τ * ≥ τ0 then it must go through it in time τcross verifying

τcross ≤ 2 ln(1 + 4CM ) g|y2(τ * )| ≤ ln(1 + 4CM ) CM . (26)
We claim that the trajectory (s(•), φ(•)) of ( 18) must enter into the strip. Indeed, otherwise

V 1 ≤ - 1 2 |y1| 1 2 - 1 8 |y2|, for τ large enough, i.e. (V 1/2 1
) ≤ -C1 for some positive constant. One would then have convergence in finite time to zero, which is impossible. Hence the trajectory (s(•), φ(•)) of ( 18) must reach a point (yM , y2(τ * )) with y2(τ ) ≥ √

V * /2 (the last point due to the symmetry with respect to the origin of trajectories of ( 18)). Then, a simple examination of the phase portrait of [START_REF] Obeid | Barrier function-based adaptive sliding mode control[END_REF] in the region y1 > yM shows that the trajectory (s(•), φ(•)) must enter the region 0 < 2y2 < y This is clearly impossible and hence (25) is proved. We now prove that (21) holds true and the argument goes by contradiction. Indeed, if it were not true, then there exists a trajectory (s(•), φ(•)) of ( 18) and an increasing sequence of times (τn) n≥0 tending to infinity such that

V1(τn) > 3V * 4 , n ≥ 0. (28)
Then, there exists a pair of times (still denoted) τ1 < τ2 such that V1(τ1) = V * /2, V1(τ2) = 3V * /4 and

V * 2 ≤ V1(τ ) ≤ 3V * 4 , τ ∈ [τ1, τ2]. (29)
We will contradict the existence of such a pair of times τ1 < τ2, which will conclude the proof of Theorem 1. Recall that V1 can increase only by going through the strip |y1| ≤ yM . Since the time τcross needed to cross that strip is given by (26), one deduces from (A) and (29) that the increase of V1 by crossing the strip |y1| ≤ yM is upper bounded by ln(1+4CM ).

Therefore the trajectory (s(•), φ(•)) of ( 18) must go through the strip |y1| ≤ yM at least twice. Combining the above fact with the phase portrait of (18) in the region y1 > yM , one deduces that there exists a pair of times τ 1 < τ 2 in [τ1, τ2] such that y 1 2

1 (τ 1 ) = 2y2(τ 1 ) and y2(τ 2 ) = 0. We then arrive at an equation similar to (27) and reach a contradiction.

Fig. 1 .

 1 Fig. 1. Phase plane of the proposed super-twisting algorithm

  has a unique minimum at zero, i.e. L b (0) = b > 0.

Fig. 2 .

 2 Plot of the output variable s(t) with: a) proposed algorithm, b) algorithm presented in

Fig. 3 .

 3 Plot of the state variable φ(t) with: a) proposed algorithm, b) algorithm presented in

Fig. 4 .Fig. 5 .

 45 Plot of the variable gain with: a) proposed algorithm, b) algorithm presented in[START_REF] Shtessel | A novel adaptive-gain supertwisting sliding mode controller: methodology and application[END_REF] Evolution of the control signal with the disturbances with: a) proposed algorithm, b) algorithm presented in

1 2 1

 12 and y1 > yM and exit it in finite time along the axis y2 = 0. Hence, there exists an interval of time [τ1, τ2] such that √ y1(τ1) = 2y2(τ1), y2(τ2) = 0.

APPENDIX A PROOF OF THEOREM 1.

The proof will be done in two steps. First step: It is first shown that there exists a finite time t(s0) for which the output variable s(t), which is part the solution of (5) under the variable gain [START_REF] Edwards | Adaptive dual layer super-twisting control and observation[END_REF], becomes |s(t)| ≤ ε/2. With no loss of generality, we can assume at once that |s(0)| > ε/2. From (8), the variable gain dynamic is given by l(t) = L1t + L0 as long as |s(t)| > ε/2 and the corresponding trajectory of ( 5) is defined as long as |s(t)| > ε/2 since, in this case, the growth of the right-hand side of ( 5) is sublinear with

Then, the second equation of [START_REF] Obeid | Barrier function-based adaptive sliding mode control[END_REF] shows that y 2 is bounded and hence y2 has at most a linear growth on I(s0). Then, from the first equation of ( 18), one gets that the time derivative of the positive function ln(1 + 2C0|y1|) is upper bounded by a function with linear growth on I(s0), yielding easily that I(s0) = [0, ∞). Now consider the following Lyapunov function

where σ(y2) is a saturation function defined as

and

The following inequality holds for every (y1, y2) ∈ R 2 ,

hence V1 is positive-definite and moreover of class C 1 if y1 = 0 and |y2| ≥ 2. The statement of the theorem will be a consequence of the following fact: for every trajectory of ( 18), one has lim sup

where V * is a positive constant only depending on M, G, g. The time derivative of the Lyapunov function ( 19) is given by

After easy computations, one gets that

Note that a non trivial trajectory of (18) crosses the line y1 = 0 in isolated points. Moreover, at a time τ where y(τ ) = (0, y2(τ )) with y2(τ ) > 0, V1 admits an isolated discontinuity jumping from gy 2 2 (τ ) to Gy 2 2 (τ ) and conversely if y2(τ ) < 0. Finally, note that, outside of the line y1 = 0 and for

We will assume without loss of generality that CM := M/b 2 is much larger than 1 and hence

) > 0.

Define


Anna R L Carter

Miriam Sturdee
email: m.sturdee@lancaster.ac.uk

Alan Dix

Dani Kalarikalayil Raju
email: danikraju@gmail.com

Martha Aldridge
email: m.aldridge@kainos.com

Eunice Sari
email: eunice@uxindo.comux

Wendy Mackay
email: mackay@lri.fr

InContext: Futuring User-Experience Design Tools

Keywords: Human-centered computing → Human computer interaction (HCI) UCD, UX, user experience, digital tools, future

Technology is changing, which means the design processes supporting it must also change. Digital tools for user experience and interaction design are vital in enabling designers to create appropriate, enjoyable and functional human-computer experiences, and so will necessarily evolve alongside our technological development. This workshop aims to support the futuring of user experience and user interfaces, and will engage with stakeholders, practicing designers, researchers, students and educators in order to understand better the needs for next-generation design tools. We will

BACKGROUND

Design tools are a key part of user experience design and interaction prototyping, but what does the future hold? This workshop will bring together UX researchers, educators, practitioners and tool creators, so that we can explore and envisage next generation user experience design tools.

Humans have always been tool users: tools to do things, but also tools to make tools, even our neolithic ancestors used hammerstones to flake flints. Tools have also been a central part of the development of human-computer interaction. Many of the core elements of modern interfaces found their genesis in programming environments such as Smalltalk and user interface development tools have been a central to understanding user interaction since the 1970s [START_REF] Edmonds | Adaptable Interfaces for Interactive System[END_REF]. By the 1980s early task analysis and workflow tools were emerging as well as systems such as HyperCard and Visual Basic embodied many of the features we still see in high-fidelity prototyping tools we see today. Alongside paper and the ubiquitous post-it note, tools for storyboarding and sketch-based interaction design, such as Denim [START_REF] Lin | DENIM: An Informal Tool for Early Stage Web Site Design[END_REF] have helped support early design.

Design tools help us to work more easily and effectively: aiding communication with stakeholders, enabling collaboration within design and development teams, and potentially blurring the boundaries between early design and low-fidelity prototyping. However, tools also shape the way we think; they may constrain or limit us to a fixed palette, or open our eyes to new viewpoints.

Are the tools we have sufficient and appropriate, or are there gaps or pain points restricting adoption, or worse restricting the creativity and effectiveness of designers? Could new or renewed tools improve what we create and how we create?

This question is of growing importance as the nature of technology and of user interaction is changing. Many tools focus on screenby screen interactions, but rich design needs to consider wider Figure 2: Future tools for existing interactions and tools for future interactions spatial and social context. Furthermore, more novel interactions often include physical devices as well as digital interactions [START_REF] Carter | Cocreation of the Lookout[END_REF][START_REF] Dix | TouchIT: Understanding Design in a Physical-Digital World[END_REF]. These new wider contexts and new modes of interaction demand new tools and possibly even new classes of tool support.

So, we have two directions to consider: future tools for existing interactions and tools for future interactions (Figure 2). These are not independent. On the one hand, new interaction technologies will help us see the limitations in existing design tools. On the other hand, pushing design tools away from a focus on screens towards broader context and better overall experience support, will make them more ready to embrace emerging technology and changing patterns of use.

Prior Research Activities

Several members of the organising team have previously hosted or participated in four previous exploratory workshops as part of the InContext project [START_REF] Dix | Where, Who, Why? Tools to Encourage Design In Context[END_REF]. These workshops examined user experience tools from the perspective of industry, educational professionals, researchers and students. Participants from these diverse groups joined into a discussion on the future of user experience tools, to support the development of software to better support the novel and multi-modal aspects of user experience. These were hosted through the lens of computer science, and Human Computer Interaction, and also were privileged to gain attendance from several individuals who work across disciplines within social sciences, design, education and more. The tools we imagined and propose could democratize the design of interactions across disciplines.

The InContext team co-presented a paper "Where, Who, Why? Tools to Encourage Design In Context" at the EduCHI symposium at CHI 2021 [START_REF] Dix | Where, Who, Why? Tools to Encourage Design In Context[END_REF] (see also Figure 3). The paper presentation was used as the starting point of one of the focus-group discussion sessions. The Miro boards for this are part of the public record of the symposium and thus available for further analysis upon request from the organisers.

ORGANIZERS

The following academics, practitioners and students will be organising the workshop and pre/post-workshop asynchronous activities. Figure 4 shows a montage of research completed by the authors.

Anna Carter is a PhD Researcher in the Centre for Doctoral Training at Swansea University where she is focusing on enhancing human interactions and collaborations with intelligence driven systems. Swansea City and County are stakeholders within the PhD project which involves deploying a range of technologies within the £1.3bn city regeneration project. Her research focuses on using human centred methods to co-create an immersive digital experience throughout the city centre using a range of next generation technologies created through participatory design. Designing inclusively is essential to all of her work and she is currently the accessibility chair for EduCHI 2022.

Miriam Sturdee is a Research Fellow and lecturer in the School of Computing and Communications, Lancaster University, where she investigates visual and creative methods for the advancement of computer science, with a particular interest in the design of novel interfaces. Her current work is examining the user experience of shape-changing interfaces, exploring the visual side of cybersecurity [START_REF] Sturdee | A Visual Exploration of Cybersecurity Concepts[END_REF], and the co-existence of the arts and HCI [START_REF] Sturdee | A plurality of practices: artistic narratives in HCI research[END_REF]. She also has an MFA in Visual Communication from Edinburgh College of Art, and an MA in Psychology. She has previously organised workshops on Sketching in HCI (DIS 2017), art and tangible interaction (CHI 2020), novel cycling interfaces and futuring peer review (CHI 2021).

Alan Dix is Director of the Computational Foundry at Swansea University, a £31m investment by the University, Welsh Government and European Union to create a place to nurture fundamental digital research that makes a difference to real lives. Previously he spent 10 years in a mix of academic and commercial roles. Alan has worked in human-computer interaction research since the mid 1980s and is the author of one of the major international textbooks on HCI as well as of over 500 research publications from formal methods to design creativity. His work has often looked over the technological horizon, for example in 1992 he wrote possibly the first paper to highlight the potential for gender and ethnic bias in black-box machine learning algorithms, and in 2000 led development of an early (albeit too early!) social networking system. Social impact is also crucial to his work and he is currently responsible for the algorithmic social justice theme of the UK Not-Equal network for Digital Economy and Social Justice.

Martha Aldridge is a Design Consultant of 14 years with a wealth of experience in multiple domains including government, early years and special needs education and medicine. She has been a consultant for multiple academic research groups from scientific instrument design to improving independent online learning for autistic children, and has contributed to the design of accessible government services since 2013. She has also led the design of large scale public health studies and the effective use of public health data for driving policy and local authority spending decisions. Martha currently works remotely as a digital transformation consultant for public and private sector clients.

Dani Kalarikalayil Raju is co-founder of Studio Hasi, a startup with a mission to diversify future making. He is an alumnus of IDC School of Design, IIT Bombay and has been working with resource constrained user groups (Emergent Users) in India for the past 5 at the Université Paris-Saclay, where she teaches theoretical and practical interaction design. She has been actively involved in the creation of methods and interactive tools to support creativity, both in industry and research, since the 1980s and has published numerous articles on design, particularly with respect to the use of video and interaction design.

Elizabeth Churchill Elizabeth Churchill is a Director of UX at Google. With a background in psychology, Artificial Intelligence and Cognitive Science, she draws on social, computer, engineering, and data sciences to create innovative end-user applications and services. She has built research teams at Google, eBay, Yahoo, PARC and FujiXerox. Her current focus is on the design of effective designer and developer tooling. Elizabeth holds a PhD from the University of Cambridge and honorary doctorates from the University of Sussex and the University of Stockholm. She is a member of the Association for Computer Machinery's (ACM) CHI Academy, is an ACM Fellow, Distinguished Scientist, and an ACM Distinguished Speaker. She served as the ACM's Vice President for 2 years, from 2018 -2020. In 2016 she received a Citris-Banatao Institute Award Athena Award for Women in Technology for her Executive Leadership. She has been named one of the top women leaders in UX over the last several years.

WEBSITE

The workshop web page will be hosted as part of HCIbook.net and is linked to here: hcibook.net/incontext/chi2022. Participant submissions will be made available after acceptance on the website, and the website maintained after the initial workshop event.

PRE-WORKSHOP PLANS

Our previous scoping workshops have created a network of interested researchers and practitioners who we can reach out to. Additionally, many of the organisers are on Twitter and can utilise their networks to advertise the call, as well as using more traditional methods such as the CHI mailing list. We will also direct people to the website (see previous section) and keep this regularly updated with useful information in the run up to the live conference dates. As the workshop is intended to be run in hybrid fashion, we anticipate being able to offer a range of accessibility options (e.g. wheelchair access, captioning as available via the conference platform), and will make a "kit-list" for remote participants who wish to actively take part in the hands on activities. Miro, or another collaborative online whiteboard will be used to enable upload of collaborative images both in person and remotely. This whiteboard environment will be designed and made available to participants prior to the event to enable community building and networking.

WORKSHOP FORMAT

As InContext will be a hybrid format workshop we anticipate a number of in-room/technical requirements:

(1) A large screen/projector for bringing remote participants into the room, presenting material, and monitoring the Miro board in real time; (2) Flip-chart and markers for physical notes;

(3) Large tables to facilitate activities such as paper-prototyping and making; (4) Appropriate recycling facilities for used workshop materials.

(5) For remote attendees, we would request that live captions are available (either automated or manual).

More accessibility requirements may emerge once participation is confirmed, and we will put an accessibility statement on the website to enable requests to be made as early in the process as possible. We expect to use Miro for our collaborative whiteboard, and this platform has been improving its accessibility recently. Should a more accessible alternative be identified, this may be used instead. The Miro board (or alternative) will also be available asynchronously, with an activity list for individual participation prior to and after the synchronous event. If circumstances change, the workshop will move entirely online.

WORKSHOP STRUCTURE

The workshop will combine positional presentations, practical, hands-on activities and discussion to support innovation and networking. The hybrid format ensures we can maximise attendance and engagement both prior to, during and after the conference. We aim to assign a in-person laptop member to each table to enable zoom members to be part of the in person groups with break out rooms assigned to each table.

Pre-Workshop Activities (Asynchronous)

Introduce Yourself: Participants will be invited to fill out a short workshop "persona" on the online whiteboard.

Read & Discuss: Accepted submissions will be linked from the website, and summaries posted on the Miro for participants to comment on and discuss prior to the synchronous event.

Synchronous Hybrid Event

Indicative format and activities, breaks and lunch will be confirmed as per the CHI2022 schedule:

Introduction & Welcome: Organiser introductions and scoping the problem. (30 minutes) Participant Pecha Kucha: Each participant will have between 3 minutes to present their position paper or pictorial (we are anticipating up to 15 participants in addition to the organisers). (Up to 45 minutes) Break: Tea/coffee break Participatory Design Activity: This activity will focus on brainstorming and the current state-of-the-art of UX design tools (with introduction by WEM). Using Video Clipper, a tool that supports colocated or remote collaborative video capture during brainstorming and scenario-based video prototyping sessions, participants will form groups and discuss, iterate and present their thoughts following the initial pecha-kucha. Video Clipper will lightly guide video capture during the creation process to completely avoid post-hoc editing and produce a helpful record of the process. Both single-user iPad versions and a collaborative, web-based version is available which will work with any mobile device with a camera, hence supporting in person and remote attendance. (20 minutes) Question Formulation & Democratic Idea Generation: Writing questions for the UX Research using adapted Question Formulation From top left to bottom right: 1) A participant interacts with a white-box prototype of a shape-changing interface [START_REF] Sturdee | Visual Methods for the Design of Shape-Changing Interfaces[END_REF]; 2) Foto-foto training events [START_REF]We empower leaders and organizations to accelerate digital transformation[END_REF]; 3) Participatory design for creating virtual environments [START_REF] Kalarikalayil | Participatory Design for Creating Virtual Environments[END_REF]; 4) Brainstorming activity based on the pros and cons of educational platforms [START_REF] St-Cyr | Developing a Community of Practice to Support Global HCI Education[END_REF]; 5) Sketch for EduCHI work [START_REF] Dix | Where, Who, Why? Tools to Encourage Design In Context[END_REF]; 6) A user interacting with a tangible embedded interface designed for the Covid Era [START_REF] Carter | Cocreation of the Lookout[END_REF]; 7) Designing with Sticky Notes [START_REF] Mackay | Designing with sticky notes[END_REF].

Technique and democratic idea generation using 4 Steps of Sketching (introduced by ES). (15 minutes) Day in life Activity: Prototyping activity that asks the participants to immerse into different place based contexts including: one of a slum resident in Mumbai, one of a resident of Swansea's digital regeneration hub, a beach visitor in Perth and a inhabitant of Alcatraz, San Francisco. The participants will be divided into groups with each group given a different area of context (introduction by DKR). The aim will be to think about a design intervention that would aid the residents of each area (20 minutes).

Break for Lunch: Takeaway/Delivery lunch to eat in room with remote participants (orders taken before morning break, from identified options to support dietary requirements).

Sketching Prototyping activity that asks the participants to sketch eight new design tools beginning with a quick fire discussion about the range of pain points that they experience with current prototyping tools (introduction by AC). Then moving on to sketching using the Crazy 8's design sprint method to create a series of eight designs that would improve or replace existing tools [START_REF]Crazy 8's Design Sprint[END_REF]. The users will be asked to sketch eight designs within eight minutes and then present those to the group (introduction by ES).

(20 minutes) Collaborative Sketching: Collaborative sketching exercise, bringing together the images generated from the initial sketching activity (introduction by EC). Participants will be asked to iterate, combine, doodle on their ideas, working in small groups to take inspiration from the solo activity. (30 minutes) Break: Tea/coffee break UX 'Playgroup': A hands-on activity involving paper-prototyping, physical prototyping, fuzzy-felt and making. Working with ideas and thoughts from the sketching exercises, this maker activity will be based around design fiction and speculative design (with introduction by MS), participants will be asked to think about the connections between novel interfaces (including tangible and virtual interactions) and, by making, exploring some of the possible interactions and issues that arise when designing for novel interfaces. (45 minutes -with break options) Show & Tell: Small group presentations of pain points, solutions and possible UI design for the future of UX tools (2 minutes each). This will draw on experience of tool use for future contexts during the workshop as well as prior experiences discussed during the various activities. (30 minutes)

Reflections & Wrap-Up: End of day thoughts and takeaways. (20 minutes)

Post-Workshop Activities (Asynchronous)

Reflections on the event: We will invite thoughts, ideas and discussion points to keep the conversation going. There will be an area of the online whiteboard to place these items.

POST-WORKSHOP PLANS

InContext is a stand alone workshop looking at the future of user experience design tools, although we will make available initial findings from our scoping workshops in this area, subsequently inviting dialogue from invited participants. We anticipate at least three outputs:

• Publication of a workshop report "manifesto" including sketches, discussion and takeaways for the future of UX design tools, and/or Interactions article (post-proceedings) • Working group formation • Full paper synthesising and analysing pre/post-workshop project data for CHI 2023

CALL FOR PARTICIPATION

Technology is changing, which means the design processes supporting it must also change. Digital tools for user experience and interaction design are vital in enabling designers to create appropriate, enjoyable and functional human-computer experiences, and so will necessarily evolve alongside our technological development.

We are hosting a workshop aiming to support the futuring of user experience and user interfaces, and will engage with stakeholders, practicing designers, researchers, students and educators in order to understand better the needs for next-generation design tools. The workshop will combine presentations, hands-on practical experiences (in person and online), and group discussions to both create a user experience design tools futures manifesto, and the formation of a working group to share best practice and research.

InContext will be a one day, hybrid workshop to be held at CHI 2022. To take part, please submit a 4 -6 page position paper based upon the theme of The Future of User Experience Design Tools, either in ACM paper or pictorial format (e.g. Creativity & Cognition cc.acm.org/2022/pictorials/). User Experience is not simply about text, so we invite you to use the formats creatively.

Position papers and pictorials will be judged on appropriateness to the call, novelty, and openness to possibility. All accepted position papers will be hosted and maintained on the InContext website. Following CHI Guidelines all participants must register for both the workshop and for at least one day of the conference. Paper/Pictorial submissions: Easychair Workshop website: hcibook.net/ incontext/ chi2022

Figure 3 :

 3 Figure 3: Universal UX tool dashboard ideas by participants during workshops 2 & 4

  years in collaboration with the Computational Foundry at Swansea University. The consistent theme in his work is that of exploring futures with users having constraints such as low literacy, lack of technology experience and low self efficacy, by engaging them in workshops and deployment studies to re-imagine forward looking technologies for the world. Working with emergent users in two of Mumbai's slums, his recent work explored the value and uses of photovoltaic (PV) self-powered deformable digital materials for interhome connections. He was awarded the Gary Marsden Travel Awards for presenting this work at CHI 2021. Eunice Sari is the CEO and Co-Founder of UX Indonesia and Co-Founder of Customer Experience Insight Pty Ltd with more than 18+ years of global experience working in the field of User Experience and Customer Experience Insight in academia and industries. She holds Ph.D. in Education with a specialization in Design Research and Educational Technology. Eunice has pioneered many forward-thinking and innovative projects to affect lives and improve businesses' bottom line in various vertical industries in the USA, Europe, Australia, and Asia. Eunice was the first Asian female Google Developer Expert in Product Design and Strategy and currently is the Product Design expert mentor for the Google Startups program. She has helped hundreds of international startups through Google Launchpad and Accelerator Programs to develop their products and services and improve their business. Eunice has been helping and inspiring many academics, especially in the Asia Pacific, to design their HCI, UX, and Design teaching and learning programs with out-of-the-box tools and approaches that she has developed throughout these years. Wendy E. Mackay is a Research Director, Classe Exceptionnelle, at Inria, the French National Research Center for Computer Science, and is an ACM Fellow, CHI Academy member and the 2022 Chair for Computer Science at the Collège de France. Wendy directs the ExSitu (Extreme Situated Interaction) research lab, which focuses on creative professionals and 'extreme users' who push the limits of technology, to develop novel human-computer partnerships. Her lab is joint with the Human-Computer Interaction department of LISN (Laboratoire Interdisciplinaire des Sciences du Numérique)

Figure 4 :

 4 Figure 4: A montage of contextual design work completed by the authors.From top left to bottom right: 1) A participant interacts with a white-box prototype of a shape-changing interface[START_REF] Sturdee | Visual Methods for the Design of Shape-Changing Interfaces[END_REF]; 2) Foto-foto training events[START_REF]We empower leaders and organizations to accelerate digital transformation[END_REF]; 3) Participatory design for creating virtual environments[START_REF] Kalarikalayil | Participatory Design for Creating Virtual Environments[END_REF]; 4) Brainstorming activity based on the pros and cons of educational platforms[START_REF] St-Cyr | Developing a Community of Practice to Support Global HCI Education[END_REF]; 5) Sketch for EduCHI work[START_REF] Dix | Where, Who, Why? Tools to Encourage Design In Context[END_REF]; 6) A user interacting with a tangible embedded interface designed for the Covid Era[START_REF] Carter | Cocreation of the Lookout[END_REF]; 7) Designing with Sticky Notes[START_REF] Mackay | Designing with sticky notes[END_REF].

ACKNOWLEDGMENTS

We acknowledge and thank This work has been partially funded by The CHERISH-DE Centre -Challenging Human Environments and Research Impact for a Sustainable and Healthy Digital Economy (grant number EP/M022722/1). Many thanks also to Swansea City and County Council and the EPSRC Centre for Doctoral Training in Enhancing Human Interactions and Collaborations with Data and Intelligence Driven Systems (grant number EP/S021892/1), who fund Anna Carter's doctoral research.


Florian Chabot
email: florian.chabot@cea.fr

Quoc-Cuong Pham

Mohamed Chaouch

LapNet : Automatic Balanced Loss and Optimal Assignment for Real-Time Dense Object Detection

Keywords:

). To this end, we introduce LapNet, an anchor based detector, trained end-to-end without any sampling strategy. Our approach aims to overcome two important problems encountered in training an anchor based detector: (1) ambiguity in the assignment of anchor to ground truth and (2) class and object size imbalance. To address the first limitation, we propose a soft positive/negative anchor assignment procedure based on a new overlapping function called Per-Object Normalized Overlap (PONO). This soft assignment can be self-corrected by the network itself to avoid ambiguity between close objects. To cope with the second limitation, we propose to learn additional weights, that are not used at inference, to efficiently manage sample imbalance. These two contributions make the detector learning more generic whatever the training dataset. Various experiments show the effectiveness of the proposed approach.

Introduction

Object detection has been widely studied and it still remains a research field of interest for scene understanding applications. It needs to face several challenges: small object detection, heavily occluded objects and real time processing. Modern detectors are based on deep convolutional neural networks (CNN) which have shown their effectiveness in several computer vision tasks (classification, segmentation, detection...). These object detectors can be divided into two main categories: two-stage and single-stage detectors. Two-stage detectors such as [START_REF] Girshick | Fast r-cnn[END_REF][START_REF] Ren | Faster r-cnn: Towards real-time object detection with region proposal networks[END_REF] first propose a set of regions of interest in which deep feature maps are cropped and classified. Although these methods achieve good performance, they are time consuming because of their cascaded pipeline. In contrast, single-stage detectors [START_REF] Redmon | You only look once: Unified, real-time object detection[END_REF][START_REF] Redmon | Yolo9000: Better, faster, stronger[END_REF][START_REF] Liu | Ssd: Single shot multibox detector[END_REF][START_REF] Lin | Focal loss for dense object detection[END_REF][START_REF] Tian | FCOS: Fully convolutional one-stage object detection[END_REF] directly predict detection boxes without any region LapNet-608-53

Fig. 1. Speed/accuracy trade-off for single-shot detectors on COCO test-dev 2017 using TITAN X GPU. We only plot detectors with a computing time less than 200 ms.

Fast detectors correspond to approaches with a processing time per image less than 100 ms. Methods are denoted by "name-resolution-backbone" (50: ResNet-50, 101: ResNet-101, 53: DarkNet-53). LapNet gives significantly better results than YoloV3 [START_REF] Redmon | Yolov3: An incremental improvement[END_REF], using the same backbone network while keeping the same computing time. With an input image resolution of 608x608, LapNet has similar accuracy than RetinaNet [START_REF] Lin | Focal loss for dense object detection[END_REF] but a smaller inference time.

extraction and refinement step. Some of them [START_REF] Redmon | You only look once: Unified, real-time object detection[END_REF][START_REF] Redmon | Yolo9000: Better, faster, stronger[END_REF][START_REF] Redmon | Yolov3: An incremental improvement[END_REF][START_REF] Liu | Ssd: Single shot multibox detector[END_REF] are fast and thus suitable for real-time detection, at the cost of a lower accuracy.

Recent object detection algorithms generally use a set of discrete pre-defined boxes called anchors. The network predicts if an anchor contains an object (positive anchor) or not (negative anchor) as well as offsets applied to anchors to fit the objects. The strategy for finding positive and negative anchors during training is based on the absolute overlap criterion using Intersection over Union function (IoU): an anchor is labeled as positive if a ground truth box overlaps it above a fixed threshold. Thresholding the overlap is a way to define what an object is and what it is not. However, this strategy could be discussed. For instance, due to the discrete property of the anchor set, there is some chance that no anchor overlaps a ground truth box above the threshold, especially for small objects. In addition, an anchor highly overlapping two objects is assigned to only one of them. This ambiguous case perturbs the model training.

Another general issue when training object detectors is class imbalance and object size imbalance. Best represented classes (large number of training samples) and large objects are generally best trained. Positive anchors are more easily associated to large objects and dominate the loss function. A standard way to address imbalance is positive/negative sampling strategy [START_REF] Ren | Faster r-cnn: Towards real-time object detection with region proposal networks[END_REF][START_REF] Liu | Ssd: Single shot multibox detector[END_REF][START_REF] Fu | Dssd: Deconvolutional single shot detector[END_REF] based on heuristics. Another way to solve it is to design a specific loss function such as the focal loss [START_REF] Lin | Focal loss for dense object detection[END_REF] which explicitly manages the balance. The objective of the focal loss is to minimize the influence of easy samples (i.e background samples) in the total loss. It balances foreground and background samples in an automatic hard data mining way, but it does not provide a global weighting function which explicitly takes into account class representativeness and object sizes.

In this paper, we introduce LapNet, a new single-stage object detector combining real time processing and efficient detection. LapNet is based on a CNN encoder-decoder architecture which takes as input an image and a set of predefined anchors. It outputs a dense set of bounding boxes with associated per-class probabilities. In this work, we present two main contributions.

First, we propose a new method to choose positive/negative anchors. As stated above, the assignment based on standard absolute overlap (AO) criterion greatly depends of the number of anchors. To overcome this limitation, we introduce Per-Object Normalized Overlap (PONO), presented in section 3.2. PONO is the overlap between an anchor and a given ground truth box, normalized by the maximum overlap between this ground truth and all its assigned anchors. In other words, each ground truth box has at least one anchor which has a PONO value equal to one. It has two advantages compared to the absolute overlap criterion. The first advantage is an increased robustness when applying a threshold to select positive and negative anchors. When a ground truth box has too small AO values, it is discarded. PONO solves this issue by design. The second advantage is that PONO normalizes the overlap score which makes its more insensitive to the anchor discretization. LapNet takes its name from this idea. In addition, a strategy allowing the network to self-correct the positive and negative anchor assignment during training is proposed. The intuition is that a positive anchor could highly overlap almost equally several objects. When such an ambiguity happens, the network will not be able to fit the right object. To solve this issue, we propose to consider an ambiguous anchor as negative using an online Ambiguity Management Strategy (AMS) described in section 3.3.

The second contribution is related to the loss function. Dense optimization for object detection is still difficult because of large imbalance of classes and object sizes. The authors of [START_REF] Kendall | Multi-task learning using uncertainty to weigh losses for scene geometry and semantics[END_REF] proposed the homoscedastic loss for automatic loss balancing in the context of multi-task learning. In our work, we generalize this concept to address general sample imbalance by automatically learn weights which balance classes and object sizes (explained in section 3.3). It is a generic alternative to multi-scale detection used in recent litterature [START_REF] Lin | Feature pyramid networks for object detection[END_REF][START_REF] Lin | Focal loss for dense object detection[END_REF][START_REF] Tian | FCOS: Fully convolutional one-stage object detection[END_REF][START_REF] Redmon | Yolov3: An incremental improvement[END_REF]. These methods use Feature Pyramid Network (FPN [START_REF] Lin | Feature pyramid networks for object detection[END_REF]), which consists in predicting boxes for several feature levels (multi-scale prediction). During training, FPN based approaches balance object sizes in the network architecture itself: large objects are optimized in the lower pyramid level and small objects in the higher pyramid level. It forces to define which anchors correspond to each level. Differently, LapNet's generic weight learning avoids the use of multi-scale detection.

The two contributions of this work lead LapNet to be comparable to best performing object detectors with a significant reduction of the processing time. This is detailed in section 4.

Related work

Given an input image, an object detector returns a list of bounding boxes and associated confidence score for each class. An object detector classifies image regions to predict those containing an object. A standard way to analyse all regions in the image is the sliding window scheme which has been widely used by reference work [START_REF] Viola | Rapid object detection using a boosted cascade of simple features[END_REF][START_REF] Felzenszwalb | Object detection with discriminatively trained part based models[END_REF][START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF]. With the advent of Deep Learning in computer vision community, sliding window and hand-crafted features for object detection have been surpassed by CNN based methods.

Anchor based detectors. Most CNN based methods [START_REF] Ren | Faster r-cnn: Towards real-time object detection with region proposal networks[END_REF][START_REF] Liu | Ssd: Single shot multibox detector[END_REF][START_REF] Redmon | Yolo9000: Better, faster, stronger[END_REF][START_REF] Lin | Focal loss for dense object detection[END_REF] use anchors to detect objects. Anchors are pre-defined boxes that are supposed to be representative of all objects in terms of scale and ratio. Each 2D position in the CNN output map corresponds to one anchor at this position. The objective is to predict if an anchor contains an object or not, as well as the offsets to apply to the anchor to fit the object. During training, positive and negative labels are assigned to anchors by using the absolute overlap criterion. Classification loss (cross entropy, focal loss) and regression loss (smoothL1) functions are used to train the model. One limitation of these methods is related to the positive/negative anchor assignment. Because of the absolute overlap criterion, some ground truth boxes can be left out. In this paper, we propose to solve this issue using a soft overlap criterion and an Ambiguity Management Strategy allowing to be less sensitivity to the number of pre-defined anchors.

Two-stage vs single-stage detectors. The first two-stage based detector was proposed by the authors of [START_REF] Girshick | Rich feature hierarchies for accurate object detection and semantic segmentation[END_REF]. They introduced the object proposal concept: a multi-scale segmentation algorithm [START_REF] Uijlings | Selective search for object recognition[END_REF] is executed to propose several candidate boxes. These regions are then extracted in the image and forwarded into a CNN that predicts class scores and regression offsets. This work was extended in [START_REF] Girshick | Fast r-cnn[END_REF] where deep feature maps are extracted in regions of interest (using ROI Pooling) to be more efficient and faster. In 2015, Faster R-CNN [START_REF] Ren | Faster r-cnn: Towards real-time object detection with region proposal networks[END_REF] was introduced: it combines both anchor based object proposal generation and region classification in the same deep network. Since, Faster R-CNN has been well studied and improved [START_REF] Lin | Feature pyramid networks for object detection[END_REF][START_REF] Dai | R-fcn: Object detection via region-based fully convolutional networks[END_REF][START_REF] Chabot | Deep manta: A coarse-to-fine many-task network for joint 2d and 3d vehicle analysis from monocular image[END_REF] making it the reference method for two-stage detection. However, cascaded methods such as Faster R-CNN are time consuming in inference phase. For this reason, single-stage methods were first introduced with the SSD algorithm [START_REF] Liu | Ssd: Single shot multibox detector[END_REF]. The network directly outputs a dense set of boxes and associated per class scores. Since, many single-shot approaches have been developed [START_REF] Redmon | You only look once: Unified, real-time object detection[END_REF][START_REF] Redmon | Yolo9000: Better, faster, stronger[END_REF][START_REF] Fu | Dssd: Deconvolutional single shot detector[END_REF][START_REF] Lin | Focal loss for dense object detection[END_REF][START_REF] Tian | FCOS: Fully convolutional one-stage object detection[END_REF]. The main advantage of these methods is shorter processing time: with a reasonable input resolution and a light backbone network, some of these detectors give access to real time processing [START_REF] Liu | Ssd: Single shot multibox detector[END_REF][START_REF] Redmon | You only look once: Unified, real-time object detection[END_REF][START_REF] Redmon | Yolo9000: Better, faster, stronger[END_REF][START_REF] Redmon | Yolov3: An incremental improvement[END_REF]. However, there still exists an important performance gap between fast single-shot detector and heavier ones such as RetinaNet [START_REF] Lin | Focal loss for dense object detection[END_REF] or the recent FCOS [START_REF] Tian | FCOS: Fully convolutional one-stage object detection[END_REF]. With LapNet, we provide high mean Average Precision (mAP) while keeping real time processing. Sample imbalance. Class imbalance is a well known issue which is not specific to object detection. For instance, Dice loss [START_REF] Sudre | Generalised dice overlap as a deep learning loss function for highly unbalanced segmentations[END_REF] is often used in medical image segmentation to address large class imbalance. In object detection, object scales and class representativeness are two important things to take into account. To not focus on large objects, some approaches use sampling strategy [START_REF] Ren | Faster r-cnn: Towards real-time object detection with region proposal networks[END_REF][START_REF] Liu | Ssd: Single shot multibox detector[END_REF][START_REF] Fu | Dssd: Deconvolutional single shot detector[END_REF] or bootstrapping [START_REF] Shrivastava | Training region-based object detectors with online hard example mining[END_REF]. In [START_REF] Lin | Focal loss for dense object detection[END_REF][START_REF] Tian | FCOS: Fully convolutional one-stage object detection[END_REF], the focal loss automatically balances foreground and background samples by giving more weight to hard samples. In our work, a generic method for both classes and object scales imbalance is proposed. To our knowledge, no method modeling sample balance with trainable parameters exists. Our approach generalizes the homoscedastic loss proposed by [START_REF] Kendall | Multi-task learning using uncertainty to weigh losses for scene geometry and semantics[END_REF] in the context on multi-task learning.

LapNet

Overview

Given an input image, LapNet predicts a dense set of bounding boxes for each class and the associated confidence scores. These boxes are then filtered using non-maximum suppression to provide final detections. As in [START_REF] Ren | Faster r-cnn: Towards real-time object detection with region proposal networks[END_REF][START_REF] Redmon | Yolo9000: Better, faster, stronger[END_REF][START_REF] Lin | Feature pyramid networks for object detection[END_REF][START_REF] Liu | Ssd: Single shot multibox detector[END_REF], LapNet uses anchors corresponding to pre-defined boxes. More specifically, the network returns for each anchor (1) the probability that it contains an object of a given class and (2) the offsets to apply to fit the object.

Per-class anchors. Anchors are computed on the training dataset with the K-means clustering method proposed in [START_REF] Redmon | Yolo9000: Better, faster, stronger[END_REF]. Contrary to related work, anchors are computed for each class independently. In this way, anchors are more specific and better fit the distribution of size and aspect ratio of objects for a given class. This is especially true in several contexts as in traffic scene analysis where the classes "car" and "person" do not share the same box distribution. We define N C as the number of classes and N A the number of anchors for each class. To simplify, N A is the same for each class. With this formulation, LapNet returns N C × N A score maps and N C × N A × 4 box offsets (corresponding to the four offsets values dx, dy, dw, dh to apply to an anchor to fit its associated object as in [START_REF] Girshick | Fast r-cnn[END_REF][START_REF] Ren | Faster r-cnn: Towards real-time object detection with region proposal networks[END_REF][START_REF] Lin | Focal loss for dense object detection[END_REF]).

Architecture. LapNet is a fully convolutional network based on an encoderdecoder architecture with skip connections. This architecture allows to keep information of objects at multiple resolutions and contextual information. Unlike FPN based approaches [START_REF] Lin | Feature pyramid networks for object detection[END_REF][START_REF] Lin | Focal loss for dense object detection[END_REF][START_REF] Redmon | Yolov3: An incremental improvement[END_REF], detection is not performed on each pyramid level. Instead, each pyramid feature map is passed through four convolutional layers and resized to the size of the highest resolution level. These feature maps are then concatenated and forwarded into a classification head and an offset regression head to predict the network outputs. In addition to its simplicity, the advantage of this architecture resides in its genericity compared to multi-scale detection based methods. Indeed, it does not require an explicit assignment between anchors and pyramid levels during training. Figure 2 illustrates the LapNet architecture.

Decoder Conv + UpSample Concatenate Classification N C x N A Localization N C x N A x 4
Backbone Fig. 2. LapNet encoder-decoder architecture overview. The input image is passed through a backbone network. A decoder with skip connections is then used to compute feature maps at several resolutions. Each of these feature maps is then processed by four 3x3 convolutional layers and resized to the size of the highest resolution feature map. After multi-scale feature concatenation, we use two convolutional heads to provide the two outputs. Each head is composed by four 3x3 convolutional layers and one final convolutional layer returning the output.

Dense association between anchors and ground truth boxes

LapNet requires to define labels on the entire anchor set. In previous work [START_REF] Ren | Faster r-cnn: Towards real-time object detection with region proposal networks[END_REF][START_REF] Lin | Focal loss for dense object detection[END_REF][START_REF] Liu | Ssd: Single shot multibox detector[END_REF], anchors with positive or negative label are defined according to a threshold applied to the absolute overlap (AO) criterion: if the overlap value of an anchor with a ground truth box is above the threshold, the anchor is considered as positive. The main drawback of absolute overlap measure is that some ground truth boxes may be missed by the label assignment because of the discrete property of anchor set. This usually happens in the case of small and strongly occluded objects.

To solve this issue, we introduce a new overlap measure called Per-Object Normalized Overlap function (PONO). It is a relative measure that increases the value of AO, especially since the AO value is low. Additionally, it always gives an overlap value equal to one for at least one anchor associated to a given object. Consequently, no object in the ground truth is missed.

Anchor to ground truth assignment. A grid of anchors is computed using pre-defined anchors (Figure 3b). It consists in a grid of size

H f ×W f ×N C ×N A ×4
where H f and W f are the height and width of the output feature map. N A and N C are respectively the number of anchors per class and the number of classes. Each element of the anchor grid is assigned to a ground truth box using the AO criterion. This assignment forms clusters where each anchor of the cluster corresponds to the same object in the ground truth (Figure 3c).

Per-Object Normalized Overlap. Assignment clusters are then used to compute the PONO map O. For each cluster, the highest overlap is selected. Each element of the cluster is divided by its corresponding maximum. Let A c,a,i,j be the anchor of class c, of index a at position (i, j) in the feature map. For a given ground truth box B n , we denote C Bn the set of anchors associated to B n . The PONO function is then defined by:

O(A c,a,i,j , B n ) = IoU (A c,a,i,j , B n ) max A c,a ,i ,j ∈C Bn IoU (A c,a ,i ,j , B n ) with A c,a,i,j ∈ C Bn (1)
In the following, we consider the resulting PONO map O = {O c,a,i,j = O(A c,a,i,j , B n )}. By definition, O has at least one element of value 1 for each object (Figure 3d). It will be used as input in the training process described in the next part.

LapNet training process

To train LapNet, two loss functions are used: a localization loss L loc , and a classification loss L cls . They use the pre-computed PONO map O described in the previous section. The localization loss is used to regress offsets applied to an anchor to fit its associated ground truth box. The classification loss is used to predict if an anchor contains an object of a specific class. In this work, we propose a weighted variant of these losses to avoid class and object size imbalance. Manually finding weights or standard hyper-parameters search to balance samples or losses for better training is time consuming. To deal with this issue, we propose an efficient optimization strategy to find these weights automatically in the training loop, without extra cost in computing time. The overall training process is illustrated in Figure 4.

Pixel-wise localization loss. In LapNet, the localization loss uses the Intersection over Union function to train the box offset regression task. Contrary to previous anchor based approaches [START_REF] Ren | Faster r-cnn: Towards real-time object detection with region proposal networks[END_REF][START_REF] Redmon | Yolo9000: Better, faster, stronger[END_REF][START_REF] Redmon | You only look once: Unified, real-time object detection[END_REF][START_REF] Lin | Focal loss for dense object detection[END_REF] which directly learn to regress box offsets using SmoothL1 loss, we propose to train the model to predict box offsets in a latent way, by optimizing the IoU function. It is a bounded function that is more stable when predicted offsets have high values. This is particularly true when using PONO because it recovers positive anchors associated to objects that are difficult to detect. First, by applying predicted offsets to the anchors, predicted boxes are generated (Figure 4a). Then, the IoU Ôc,a,i,j between a predicted box and its assigned ground truth box is computed. Ô is the resulting IoU map (Figure 4b). Finally, using the PONO map O, the localization loss function is defined as follows:

L loc (c, a, i, j) = 1 -Ôc,a,i,j ) 2 , if O c,a,i,j > 0.5 0, otherwise (2)
This loss function allows to learn latent offsets so that the overlap beetween a predicted box and its associated ground truth box is equal to one. The IoU loss has already been used for pixel-to-box regression [START_REF] Tian | FCOS: Fully convolutional one-stage object detection[END_REF][START_REF] Yu | Unitbox: An advanced object detection network[END_REF] but not for anchor based detector.

Pixel-wise classification loss with label ambiguity management. A standard way to define labels for classification loss would be thresholding the PONO map as it is done in the localization loss. Although PONO copes with the problem of unassigned ground truth objects, there still exists the issue of ambiguity between objects that are spatially close in the image. Indeed, an anchor that overlaps almost equally several objects is likely to fit any of these objects. Such an anchor is ambiguous and should not be considered as a positive anchor during training because it is generally unable to fit its assigned object correctly. Figure 5 illustrates the ambiguity situation.

To handle such cases, we propose a strategy for determining anchor labels by using the product of the PONO map O by the predicted geometric overlap Ô (Figure 4c). Using this strategy, positive and negative labels are defined as follows:

P c,a,i,j = 1, if O c,a,i,j × Ôc,a,i,j > 0.5 0, otherwise (3)
In case of ambiguity, if the network cannot predict offsets correctly for a given anchor, the value of Ôc,a,i,j will be low and therefore the product will return a low score potentially below the threshold. Such an anchor will then be considered as a negative one. Finally, the pixel-wise classification loss is defined by:

L cls (c, a, i, j) = CE(P c,a,i,j , Pc,a,i,j ) ( 4
)
where CE is the standard binary cross-entropy and Pc,a,i,j is the probability that the anchor A c,a,i,j contains an object of the class c. The sigmoid activation function is applied on logits to get P .

Balanced losses. One important challenge in training multi-class models for object detection or semantic segmentation is class and object scale imbalance. Underrepresented classes are much less well learned than well represented classes. Additionally, large objects dominate the loss function leading to poor performance on small objects. To solve it, several methods use sampling strategies [START_REF] Ren | Faster r-cnn: Towards real-time object detection with region proposal networks[END_REF][START_REF] Lin | Feature pyramid networks for object detection[END_REF], multi-scale detection head [START_REF] Lin | Feature pyramid networks for object detection[END_REF][START_REF] Lin | Focal loss for dense object detection[END_REF][START_REF] Tian | FCOS: Fully convolutional one-stage object detection[END_REF][START_REF] Redmon | Yolov3: An incremental improvement[END_REF], hand-crafted weights [START_REF] Zhao | Pyramid scene parsing network[END_REF][START_REF] Badrinarayanan | Segnet: A deep convolutional encoder-decoder architecture for image segmentation[END_REF] or specific loss [START_REF] Lin | Focal loss for dense object detection[END_REF].

In our work, we propose to automatically learn balance weights. Inspired by [START_REF] Kendall | Multi-task learning using uncertainty to weigh losses for scene geometry and semantics[END_REF], that addresses loss balancing for multi-task training, we learn two kinds of weights: loss weights and anchor weights. Loss weights are used to balance the different loss functions in the total loss. Our main contribution here, is the anchor weights intended for balancing classes and object sizes. More formally, we propose the following weighted losses:

L loc = λ loc 1 N + c a λ c,a loc i,j L loc (c, a, i, j) (5)
L cls = λ cls 1 N c a λ c,a cls i,j L cls (c, a, i, j) (6)
where λ cls , λ loc are the loss weights and λ c,a cls , λ c,a loc are the anchor weights, defined for a given anchor a and a class c. N + is the number of positive anchors for the localization task and N is the total number of pixels. In this formulation, all λ weights are trainable variables. A regularization loss L reg is added to avoid λ = 0:

L reg = log( 1 λ cls ) + log( 1 λ loc ) + 1 N C N A c a log( 1 λ c,a cls ) + log( 1 λ c,a loc ) (7)
In practice, instead of directly optimizing λ, we use s = log( 1 λ ) as trainable variables for better numerical stability and to avoid weights to be negative. In addition, if an anchor grid does not have a least one positive label, corresponding anchor weights λ c,a cls and λ c,a loc are not updated. This is important to avoid huge values of weights which could make the network diverge. In fact, the objective of anchor weights is to balance the loss of each anchor grid. If an anchor grid has only negative samples, it is an easy case for the network and thus its loss will be very small. In this case, the anchor weight increases too much compared to the other weights of anchor grids containing positive samples. Finally, the total loss for training LapNet is the sum of the localization, classification and regularization losses.

Experiments

In this section, the effectiveness of LapNet is demonstrated through various experiments. In particular, we provide results on two challenging datasets (PAS-CAL VOC [START_REF] Everingham | The pascal visual object classes (voc) challenge[END_REF], MSCOCO [START_REF] Lin | Microsoft coco: Common objects in context[END_REF]). Ablation studies are conducted to show the relevance of each contribution.

Training details. In all experiments, SGD with momentum of 0.9, initial learning rate of 0.005 and polynomial decay policy with power of 0.9 are used. For data augmentation, random scale and horizontal random flipping are applied.

We arbitrarily fix the number of anchors per object to N A = 10. The trainable variables s = log(λ) are all initialized to 1 (we observed that the initial value does not influence final performances). Two pre-trained backbone networks, DarkNet-53 [START_REF] Redmon | Yolov3: An incremental improvement[END_REF] and Inception-Resnet-V2 [START_REF] Szegedy | Inception-v4, inception-resnet and the impact of residual connections on learning[END_REF] are used, both pre-trained on ImageNet [START_REF] Deng | Imagenet: A large-scale hierarchical image database[END_REF]. The training is performed on 4 GPUs with a global batch size of 28 for DarkNet-53 and 16 for Inception-Resnet-V2. For DarkNet-53, the decoder structure proposed in YoloV3 [START_REF] Redmon | Yolov3: An incremental improvement[END_REF] is used and for Inception-Resnet-v2, the decoder structure proposed by [START_REF] Lin | Focal loss for dense object detection[END_REF] is plugged into the backbone. In both cases, the spatial output size of the network is eight times smaller than the input image (feat stride = 8).

PASCAL VOC

This dataset contains 20 classes of objects. LapNet is trained using the union of 2007 and 2012 trainval, and tested on 2007 test. 120K iterations are runned to get the final result shown in Table 1. We can see that LapNet improves detection results compared to previous work for both backbone networks. In particular, LapNet, trained end-to-end, offers better performance than the best previous method DSSD [START_REF] Fu | Dssd: Deconvolutional single shot detector[END_REF] which uses multiple training stages to get the final detector. In the following, an ablation study based on PASCAL VOC is presented.

PONO and ambiguity management strategy. In this part, we evaluate the relevance of Per-Object Normalized Overlap and ambiguity management strategy. The analysis is presented in Table 2a. All models are trained using loss weights, anchor weights and the DarkNet-53 backbone [START_REF] Redmon | Yolov3: An incremental improvement[END_REF]. In the first column of Table 2a, we compare the performance of LapNet models trained with PONO and the standard absolute overlap (AO) label assignement criteria. We observe better detection results are obtained with PONO than with absolute overlap (81.1 vs 80.3), which demonstrates that PONO is a better criterion for positive/negative label assignment. The last column of Table 2a provides results when ambiguity managment strategy, described in 3.3, is added to choose positive and negative anchors. Combining ambiguity management with AO decreases the mAP (78.4 vs 80.3). This could be explained by the fact that AO already misses some ground truth objects. Ambiguity management further deteriorates the result because it makes the overlap value lower. Combining PONO and ambiguity management gives the best result (81.7 vs 81.1).

Automatic balancing. We also study the influence of automatic weight learning for loss and anchor balance. Table 2b summarizes obtained results. All models are trained using the PONO map and the ambiguity management strategy. The first row is a model where weights are constant and equal to one. We observe that the model completely failed as it was expected. The main reason comes from the normalizer N in the equation of the classification loss [START_REF] Deng | Imagenet: A large-scale hierarchical image database[END_REF] which is an important parameter. As a matter of fact, N is simply the number of pixels (anchors) in the outputs. In other words, we just take the average over all pixels without any 5) and [START_REF] Everingham | The pascal visual object classes (voc) challenge[END_REF]. We provide results for different weighing strategies (hand-crafted fixed weights or learned weights). R = N N + represents the normalization used for classication loss in RetinaNet [START_REF] Lin | Focal loss for dense object detection[END_REF], see text for detail. CE is the cross entropy and FL the focal loss. The second line is considered as the baseline. The last line corresponds to the proposed full automatic weighting strategy.

focus on positive anchors unlike RetinaNet [START_REF] Lin | Focal loss for dense object detection[END_REF]. The resulting classification loss value is therefore very low because of the large number of easy negative samples leading the network to not be trained correctly. In the second row, weights are also constant but we use the normalization method used in [START_REF] Lin | Focal loss for dense object detection[END_REF] where the classification loss is divided by the number of positive anchors N + . With this formulation and equation ( 5), we give the constant value λ cls = R = N N + . This experiment is the baseline for our study. In the third line, we repeat the same experiment using focal loss [START_REF] Lin | Focal loss for dense object detection[END_REF] (with default parameters α = 0.25 and γ = 2) instead of cross entropy. Suprisingly, the focal loss decreases results significantly. It may be necessary to tune focal loss hyper-parameters to achieve better performance but it is a time consuming process which is specific to each training dataset. The fourth line of Table 2b shows that when using only automatic loss weights the model outperforms the baseline model of the second line. That proves that during training, even if classification loss values are low because of dividing by N , automatic loss weights naturally counterbalance that phenomenon. Finally the best result is obtained (Table 2b, last line) when using our proposed full automatic loss and anchor balancing. Figure 6 plots the values of some of the learned anchor weights. This plot shows that the larger is the area of an anchor, the smaller is its learned weight. Besides, the more the class is underreprensented, the larger are the weights. This is the expected behavior which would be hardly designed manually. Thanks to its automatic balancing method, LapNet provides state-of-the-art results on PASCAL VOC.

MSCOCO

We train LapNet on the 80 object categories using the 2017 trainval split (115K images) and test on 2017 test-dev (20K images) by uploading results on the evaluation server. Results presented in Table 3 are obtained using 500K training iterations. The performance of LapNet is compared to several other state-of-theart object detectors among which fast single-shot detectors (i.e running in less than 100 ms) [START_REF] Redmon | Yolov3: An incremental improvement[END_REF][START_REF] Redmon | Yolo9000: Better, faster, stronger[END_REF][START_REF] Liu | Ssd: Single shot multibox detector[END_REF], slower single-shot detectors [START_REF] Lin | Focal loss for dense object detection[END_REF][START_REF] Tian | FCOS: Fully convolutional one-stage object detection[END_REF][START_REF] Fu | Dssd: Deconvolutional single shot detector[END_REF][START_REF] Law | Cornernet: Detecting objects as paired keypoints[END_REF] and two-stage detectors [START_REF] Shrivastava | Beyond Skip Connections: Top-Down Modulation for Object Detection[END_REF][START_REF] Lin | Feature pyramid networks for object detection[END_REF][START_REF] Huang | Speed/accuracy trade-offs for modern convolutional object detectors[END_REF]. The main result is that LapNet widely outperforms the best of fast detectors, YoloV3 [START_REF] Redmon | Yolov3: An incremental improvement[END_REF], using the same backbone network and the same decoder (37.6 of mAP vs 33.0). This important improvement comes from our proposed learning strategy showing that with almost the same computing time (Figure 1), LapNet gives better detection results.

Compared to stronger but more time consuming methods such as RetinaNet [START_REF] Lin | Focal loss for dense object detection[END_REF] and FCOS [START_REF] Tian | FCOS: Fully convolutional one-stage object detection[END_REF] taking input images of 800 pixels for the smaller dimension (denoted RetinaNet-800 and FCOS-800), and using ResNet-101-FPN backbone, LapNet is less accurate (-0.9 to -2.8 in mAP) but is 2.5x faster (Figure 1). In previous work, the best results of RetinaNet and FCOS were obtained with the heavier ResNeXt-101 [START_REF] Xie | Aggregated residual transformations for deep neural networks[END_REF] backbone, outperforming LapNet by 2.6 and 3.9 points in mAP respectively. They are mentioned in Table 3 3. LapNet results on MSCOCO test-dev2017 with two different input resolutions (512x512 and 608x608). They are compared to results from several previous methods. LapNet clearly outperforms previous fast single-stage detectors. Figure 1 presents these results in the form of speed/accuracy graph.

we do not consider them in the speed/accuracy trade-off analysis because they are not suitable for real-time applications. For the same reason, we did not implement LapNet with ResNeXt-101 backbone. ResNet-101 was not implemented either because it was shown in [START_REF] Redmon | Yolov3: An incremental improvement[END_REF] that DarkNet-53 has equivalent performance.

Input image resolution is also an important parameter. LapNet is implemented with the same input image resolution as YoloV3 (512x512 or 608x608) to be fast. Compared to the version of RetinaNet using similar input image resolution (500 pixels for the smallest dimension) denoted RetinaNet-500, LapNet clearly achieves better performance (+3.2 in mAP at least, see Figure 1 and Table 3).

Conclusion

We have introduced LapNet, a real time single-shot detector which gives a very good trade-off between speed and accuracy. The training process is based on a new positive/negative label assignment method using Per-Object Normalized Overlap and Ambiguity Management Strategy. An automatic weighting method is also proposed to balance losses, classes and object sizes for efficient learning without any extra cost in computation time. Experiments on two reference datasets demonstrate the relevance of our contributions and position LapNet as the most accurate detector among real-time methods. We believe that the proposed generic automatic balancing could be applied to many other computer vision tasks.

Fig. 3 .

 3 Fig. 3. Per-Object Normalized Overlap (PONO) computation. (a) An input image with two ground truth objects (top) and three pre-defined anchors (bottom). (b) Dense anchor grid representation. (c) Anchor to ground truth assignment using absolute overlap, cluster colors (green and orange) correspond to the ground truth object assigned to each anchor of the grid. (d) Per-Object Normalization Overlap. The two ground truth boxes have at least one anchor with a PONO value equal to one (in red).

Fig. 4 .

 4 Fig. 4. LapNet training process. Given an image, the network outputs scores and offsets. (a) Offsets are applied to the anchor grid to get predicted boxes. (b) IoU map Ô is then computed using predicted boxes and associated ground truth boxes. Ô is used to compute the localization loss L loc . (c) The thresholded product between PONO map O and predicted IoU map Ô determines classification labels to optimize the classification loss L cls . Back-propagation is performed on the inverse path of flows colored in red.

Fig. 5 .

 5 Fig. 5. Illustration of ambiguity management (best viewed in color). (a) An image with four overlapping ground truth bounding boxes. (b) Selected positive anchors by thresholding PONO values. For readability, anchors are represented by red points. (c) Selected positive anchors (green points) by thresholding PONO map O weighted by geometric overlap Ô. We observe that green points form four clusters corresponding to the four objects. Remaining red points correspond to ambiguous anchors (on object borders) labeled as negative anchors.

Fig. 6 .

 6 Fig. 6. Analysis of weights automatically learned on VOC2007-2012 trainval. Anchor weights of two classes (car and person) are represented (NA = 10). The proportion of samples over the entire training dataset is reported in the top-right corner. This plot shows that weight values decrease with the anchor size and decrease with the number of samples of the class.

Table 1 .

 1 Results on the VOC 2007 test set. Two-stage detectors are on the top of the table and single-stage at the end. Compared to both detector families, LapNet gives the best results.

	method	backbone	mAP areo bike bird boat bottle bus car cat chair cow table dog horse mbike person plant sheep sofa train tv
	Faster[22]	VGG-16	73.2 76.5 79.0 70.9 65.5 52.1 83.1 84.7 86.4 52.0 81.9 65.7 84.8 84.6 77.5 76.7 38.8 73.6 73.9 83.0 72.6
	Faster[22]	ResNet-101 76.4 79.8 80.7 76.2 68.3 55.9 85.1 85.3 89.8 56.7 87.8 69.4 88.3 88.9 80.9 78.4 41.7 78.6 79.8 85.3 72.0
	MR-CNN[9] VGG-16	78.2 80.3 84.1 78.5 70.8 68.5 88.0 85.9 87.8 60.3 85.2 73.7 87.2 86.5 85.0 76.4 48.5 76.3 75.5 85.0 81.0
	R-FCN[3]	ResNet-101 80.5 79.9 87.2 81.5 72.0 69.8 86.8 88.5 89.8 67.0 88.1 74.5 89.8 90.6 79.9 81.2 53.7 81.8 81.5 85.9 79.9
	YOLOv2[20] DarkNet-19 78.6	-	-	-	-	-	-	-	-	-	-	-	-	-	-	-	-	-	-	-	-
	SSD300[18] VGG-16	77.5 79.5 83.9 76.0 69.6 50.5 87.0 85.7 88.1 60.3 81.5 77.0 86.1 87.5 83.97 79.4 52.3 77.9 79.5 87.6 76.8
	SSD512[18] VGG-16	79.5 84.8 85.1 81.5 73.0 57.8 87.8 88.3 87.4 63.5 85.4 73.2 86.2 86.7 83.9 82.5 55.6 81.7 79.0 86.6 80.0
	SSD513[8]	ResNet-101 80.6 84.3 87.6 82.6 71.6 59.0 88.2 88.1 89.3 64.4 85.6 76.2 88.5 88.9 87.5 83.0 53.6 83.9 82.2 87.2 81.3
	DSSD513[8] ResNet-101 81.5 86.6 86.2 82.6 74.9 62.5 89.0 88.7 88.8 65.2 87.0 78.7 88.2 89.0 87.5 83.7 51.1 86.3 81.6 85.7 83.7
	LapNet512 DarkNet-53 81.7 88.1 88.7 82.0 75.0 65.8 88.1 91.7 90.0 65.1 85.9 77.4 88.5 90.8 86.1 86.2 51.5 82.7 81.8 89.2 79.4
	LapNet512 IncResV2	83.2 89.8 89.8 83.8 76.1 65.2 89.8 90.7 92.0 64.6 89.8 79.0 91.8 91.8 89.5 84.9 53.5 86.3 82.4 89.6 84.7
										λ loc			λ cls		λ c,a loc		λ c,a cls	Loss mAP
	AMS AO PONO	no 80.3 81.1		yes 78.4 81.7			1 1 1 learned learned 1 R R	1 1 1 1		1 1 1 1		CE 16.4 CE 79.0 FL 71.4 CE 81.3
										learned learned learned learned CE 81.7
			(a)												(b)

Table 2 .

 2 (a) Influence of PONO and ambiguity management strategy (AMS). AO is the standard absolute overlap. The bottom right result corresponds to the complete LapNet model which outperforms other kind of positive/negative assignment. (b) Influence of automatic weights learning in the loss equation (

  for information. However, CNN w/ TDM[START_REF] Shrivastava | Beyond Skip Connections: Top-Down Modulation for Object Detection[END_REF] Inception-ResNet-v2-TDM 36.8 57.7 39.2 16.2 39.8 52.1 Faster R-CNN w/ FPN [16] ResNet-101-FPN 36.2 59.1 39.0 18.2 39.0 48.2 Faster R-CNN by G-RMI [12] Inception-ResNet-v2 34.7 55.5 36.7 13.5 38.1 52.0 Faster R-CNN w/ TDM [24] Inception-ResNet-v2-TDM 36.8 57.7 39.2 16.2 39.8 52.1

	method	backbone	AP AP50 AP75 APS APM APL
	Two-stage Faster R-Slow
	Single-stage
	Table