D F Escande

F Doveil

Yves Elskens
email: yves.elskens@univ-amu.fr

N-body description of Debye shielding and Landau damping

Keywords: numbers : 52.20.-j Elementary processes in plasmas 52.35.Fp Plasma : electrostatic waves and oscillations 45.50.-j Dynamics and kinematics of a particle and a system of particles 05.20.Dd Kinetic theory basic plasma physics, Debye shielding, Landau damping, N -body dynamics

This paper brings further insight into the recently published N -body description of Debye shielding and Landau damping [Escande D F, Elskens Y and Doveil F

I. INTRODUCTION

In order to discuss microscopic plasma physics, most textbooks start with invoking the N -body description of plasmas, but they deem it too difficult to tackle, and they rapidly switch to kinetic equations. The kinetic approach has proved extremely powerful to compute a wealth of plasma effects, but it often hides some aspects of their mechanical nature ; for instance, after the Vlasovian derivation of Landau damping, most textbooks use simple mechanical models to provide physical interpretations of this damping.

However, the N -body description of plasmas has proved for more than six decades to be very powerful at describing analytically effects like Debye shielding, Landau damping, spontaneous emission of Langmuir waves, and at revealing the corresponding mechanical behavior of both particles and waves. This started in 1952 with a seminal paper by Pines and Bohm [START_REF] Pines | A collective description of electron interactions : II. Collective vs individual particle aspects of the interactions[END_REF]. From the nineties on, there was a revival of the N -body approach to describe analytically wave-particle interaction in plasmas [START_REF] Antoni | Explicit reduction of N -body dynamics to self-consistent particle-wave interaction[END_REF][START_REF] Elskens | Microscopic dynamics of plasmas and chaos[END_REF].

Last year, reference [START_REF] Escande | Direct path from microscopic mechanics to Debye shielding, Landau damping, and wave-particle interaction Plasma[END_REF] provided a direct and simultaneous derivation of Debye shielding and Landau damping by an N -body description of the plasma. This derivation works directly with Newton's second law and with a version of the electrostatic potential linearized from ballistic orbits of the particles ; it does not use probabilistic arguments or partial differential equations. However, in retrospect, the formulation of this paper now appears to be unnecessarily intricate, because it was still too close to the heuristic derivation of the theory ; it also required an erratum [START_REF] Escande | Corrigendum : Direct path from microscopic mechanics to Debye shielding, Landau damping, and wave-particle interaction Plasma[END_REF].

The heuristic derivation substituted the discrete summations over the particles, which are intrinsic to the N -body approach, with integrals over a continuous velocity distribution, and was presented at the beginning of reference [START_REF] Escande | Direct path from microscopic mechanics to Debye shielding, Landau damping, and wave-particle interaction Plasma[END_REF]. In retrospect, this substitution turns out to be an efficient shortcut to the simultaneous derivation of Debye shielding and Landau damping, but this non rigorous step can be avoided by considering the initial N -body distribution as close to a set of monokinetic arrays of particles. This other approach was presented in a second part of reference [START_REF] Escande | Direct path from microscopic mechanics to Debye shielding, Landau damping, and wave-particle interaction Plasma[END_REF]. In strong resonance with Dawson's seminal multi-beam fluid theory [START_REF] Dawson | Plasma oscillations of a large number of electron beams[END_REF], this second approach shows that Landau damping is due to phase mixing à la van Kampen [START_REF] Escande | Direct path from microscopic mechanics to Debye shielding, Landau damping, and wave-particle interaction Plasma[END_REF], while the first approach leads to the Vlasovian expression introduced by Landau that hides this fact.

Sections II and III (and the appendix) provide a shorter and more rigorous version of the approach of reference [START_REF] Escande | Direct path from microscopic mechanics to Debye shielding, Landau damping, and wave-particle interaction Plasma[END_REF] using a smooth distribution function. The calculation reduces to a mere exercise using Fourier series and Laplace transform. Section IV discusses the various physical consequences of the present N -body approach. Section V is devoted to the comparison of the present N -body approach with the 1952 one by Pines and Bohm [START_REF] Pines | A collective description of electron interactions : II. Collective vs individual particle aspects of the interactions[END_REF]. This latter work is often overlooked by plasma physics textbooks and was overlooked also by the present authors in reference [START_REF] Escande | Direct path from microscopic mechanics to Debye shielding, Landau damping, and wave-particle interaction Plasma[END_REF]. The main technical difference of the present theory with reference [START_REF] Pines | A collective description of electron interactions : II. Collective vs individual particle aspects of the interactions[END_REF] is the use of the Laplace transform. It turns out to have the same heuristic power as in the Vlasovian approach, and enables a compact and straightforward simultaneous derivation of Debye shielding and Landau damping without any a priori physical intuition.

II. SIMPLER DERIVATION OF THE FUNDAMENTAL EQUATION FOR THE ELECTROSTATIC POTENTIAL

As in reference [START_REF] Escande | Direct path from microscopic mechanics to Debye shielding, Landau damping, and wave-particle interaction Plasma[END_REF], we deal with the One Component Plasma (OCP) model [START_REF] Abe | Giant cluster expansion theory and its application to high temperature plasma Progr[END_REF][START_REF] Baus | Statistical mechanics of simple Coulomb systems[END_REF][START_REF] Salpeter | On Mayer's theory of cluster expansions Ann[END_REF], which considers the plasma as infinite with spatial periodicity L in three orthogonal directions with coordinates (x, y, z), and made up of N electrons in each elementary cube with volume L 3 . Ions are present only as a uniform neutralizing background, enabling periodic boundary conditions. We now provide a shorter and more rigorous version of the derivation of reference [START_REF] Escande | Direct path from microscopic mechanics to Debye shielding, Landau damping, and wave-particle interaction Plasma[END_REF] using a smooth distribution function. The actual calculation involves sums over the N particles and over all Fourier components of the potential. However, the principle of this calculation can be explained more simply by performing it formally for the unphysical case of a single electron acted upon by a single Fourier harmonic of its own field. This is done in the next subsection, while the complete derivation is in appendix.

A. Single particle acted upon by a single Fourier harmonic of its own field While the Fourier decomposition of the periodic delta distribution corresponding to the charge density of an electron entails an infinity of harmonics, in this subsection we keep just a single one with wavevector k which provides through Poisson equation the single harmonic of the electrostatic potential

φ(t) = - e ǫ 0 k 2 exp[-ik • r(t)], (1)
where k = k , ǫ 0 is the vacuum permittivity, and r(t) is the position at time t of the particle acting as a source with charge -e. Let r 0 and v be respectively the initial position and velocity of the particle, and let ∆r(t) = r(t)r 0vt.

We now compute a perturbative solution to the dynamics. To this end, setting r(t) = r 0 + vt + ∆r(t) in equation ( 1), we replace φ with its expansion to first order in ∆r(t)

φlin (t) = - e ǫ 0 k 2 exp[-ik • (r 0 + vt)] [1 -ik • ∆r(t)]. (2)
We now use the time Laplace transform which maps a function g(t) to g(ω) = ∞ 0 g(t) exp(iωt)dt (with ω complex). Since the arguments of functions are spelled explicitly, from now on we omit diacritics for the Laplace (or Fourier) transformed quantities. The time Laplace transform of equation ( 2) is

ϕ lin (ω) = - e ǫ 0 k 2 exp[-ik • r 0 ] [ i ω -k • v -ik • ∆r(ω -k • v)], (3)
where the Doppler shift -k • v comes from the linear dependence on t of the exponent of equation (2).

To compute ∆r(ωk • v), we use Newton's equation, which writes r = e m e ∇ϕ lin (r)

for the particle with mass m e . The single harmonic of the electrostatic potential defined by equation (1) implies

ϕ lin (r) = 1 L 3 φlin exp(ik • r), (5)
where r now is the position of the particle as a massive object subjected to a force. In order to derive our perturbative solution, we consider k ∆r to be small in equation [START_REF] Antoni | Explicit reduction of N -body dynamics to self-consistent particle-wave interaction[END_REF], so that the linearized particle dynamics defined by equation (4) reduces to

∆r = ie L 3 m e k φlin (t) exp[ik • (r 0 + vt)]. (6)
Since ∆r(0) = 0 and ∆ ṙ(0) = 0, the time Laplace transform of equation ( 6) is

-ω 2 ∆r(ω) = ie L 3 m e k exp(ik • r 0 ) ϕ lin (ω + k • v), (7)
where the Doppler shift k • v comes from the linear dependence on t in the exponent of equation ( 6). Computing ∆r(ωk • v) in equation ( 3) from the expression of ∆r(ω) given by equation ( 7) yields

k 2 ϕ lin (ω) - e 2 m e ǫ 0 L 3 k • k ϕ lin (ω + k • v -k • v) (ω -k • v) 2 exp[i(k -k) • r 0 ] = k 2 ϕ (bal) lin (ω), (8)
where

ϕ (bal) lin (ω) = - ie ǫ 0 k 2 exp[-ik • r 0 ] ω -k • v , (9)
is the time Laplace transform of the ballistic potential obtained by setting ∆r(t) = 0 in equation ( 2). We keep on purpose the terms (kk) • r 0 and k • vk • v due to the two Doppler shifts, in order to exhibit the structure of the equation when the full set of wavevectors is taken into account.

B. Fundamental equation for the electrostatic potential

As derived in the appendix, when the full set of wavevectors and particles is taken into account, equation ( 8) becomes

k 2 m ϕ lin (m, ω)- ω 2 p N n, knb smooth ≤1 k m •k n N j=1 ϕ lin (n, ω + k n • v j -k m • v j ) (ω -k m • v j ) 2 exp[i(k n -k m )•r j0 ] = k 2 m ϕ (bal) lin (m, ω), (10)
where ω p = [e 2 n/(m e ǫ 0 )] 1/2 is the plasma frequency (with n = N/L 3 ), where ϕ lin (m, ω) is the time Laplace transform of the Fourier component of the electrostatic potential with wavevector k m = 2π L m, and m = (m x , m y , m z ) is a vector with three integer components (we comment below on the restriction k n b smooth ≤ 1 making k n run over a bounded domain). Note that g(m, ω) * = g(-m, -ω * ) for real m and complex ω if g(r, t) is real-valued. We also defined

ϕ (bal) lin (m, ω) = N j=1 ϕ (bal) lin,j (m, ω), (11)
with

ϕ (bal) lin,j (m, ω) = - ie ǫ 0 k 2 m exp[-ik m • r j0 ] ω -k m • v j , (12)
the ballistic potential related to particle j, i.e. the unshielded Coulomb potential of an electron with position r j (t) = r j0 + v j t. In this complete derivation, for any finite value of the ∆r 's due to the various particles, the linearization in the ∆r's is justified for finite values of the wavenumber moduli k m = k m only. Therefore, as in section 4.5 of reference [START_REF] Escande | Direct path from microscopic mechanics to Debye shielding, Landau damping, and wave-particle interaction Plasma[END_REF], we restrict the Fourier expansion of the Coulomb potential to

k m 's such that k m b smooth ≤ 1, where b smooth ≪ λ D = [ǫ 0 k B T /(ne 2 )] 1/2 ,
the Debye length, with k B the Boltzmann constant, and T the temperature ; let us denote by M smooth the set of such wavevectors. Then, we may consider finite ∆r 's, with ∆r ≪ b smooth . The above truncation of the Fourier expansion smoothes the Coulomb potential. We notice that a similar smoothing of the 1/r singularity of this potential (or some penalization of very close approaches [START_REF] Kiessling | The microscopic foundation of Vlasov theory for jellium-like newtonian N -body systems[END_REF]) is performed in the mean-field derivation of the Vlasov equation [START_REF] Braun | The Vlasov dynamics and its fluctuations in the 1/N limit of interacting classical particles[END_REF][START_REF] Dobrushin | Vlasov equations Funkts[END_REF][START_REF] Elskens | Vlasov equation and N -body dynamics -How central is particle dynamics to our understanding of plasmas ?[END_REF][START_REF] Neunzert | Die Approximation der Lösung von Integro-Differentialgleichungen durch endliche Punktmengen Numerische Behandlung nichtlinearer Integrodifferential-und Differentialgleichungen[END_REF][START_REF] Neunzert | An introduction to the nonlinear Boltzmann-Vlasov equation Kinetic theories and the Boltzmann equation[END_REF][START_REF] Spohn | Large scale dynamics of interacting particles[END_REF]. In any case, the Vlasov equation cannot describe scales smaller than the inter-particle distance. Since the self-field due to the smoothed Coulomb potential vanishes, it is not necessary to exclude self-interaction in the complete derivation [START_REF] Escande | Corrigendum : Direct path from microscopic mechanics to Debye shielding, Landau damping, and wave-particle interaction Plasma[END_REF].

Equation [START_REF] Elskens | Irreversible behaviours in Vlasov equation and many-body hamiltonian dynamics : Landau damping[END_REF] is the fundamental equation of this paper. This equation is of the type Eϕ lin = source term, where E is a linear operator, acting on the infinite dimensional array whose components are all the Doppler shifted ϕ lin (m, ω)'s.

III. SMOOTHED LINEAR OPERATOR OF THE POTENTIAL EQUATION

We now approximate the granular (or empirical, Kolmogorov) distribution F (r, v) = N j=1 δ(rr j0 ) δ(vv j ) with a position and velocity distribution function f (r, v) that is continuous in r, such that distribution f -F yields a negligible contribution when applied to space dependent functions which vary slowly on the scale b smooth . This means that we use a version of F coarse-grained on this scale. On replacing the discrete sums over particles with integrals over the smooth distribution function f (r, v) in its left hand side, equation [START_REF] Elskens | Irreversible behaviours in Vlasov equation and many-body hamiltonian dynamics : Landau damping[END_REF] becomes

k 2 m ϕ cg (m, ω) - ω 2 p N n k m • k n ϕ cg (n, ω + (k n -k m ) • v) (ω -k m • v) 2 f (k n -k m , v) d 3 v = k 2 m ϕ (bal) lin (m, ω), (13)
where ϕ cg is the smoothed version of ϕ lin , and f is the spatial Fourier transform of f . We consider almost uniform initial spatial distributions of particles. So, f is a spatially uniform distribution function

f 0 (v) ( f 0 (v) d 3 r d 3 v = L 3 f 0 (v) d 3 v = L 3 n = N ) plus a small perturbation f 1 (r, v)
of the order of ϕ cg . In linearizing equation ( 13) for ϕ cg , f (k, v) is substituted with f0 (k, v) which vanishes for k = 0. Therefore, operator E becomes diagonal with respect to both m and ω (a complex quantity), and equation ( 13) becomes

ǫ(m, ω) ϕ cg (m, ω) = ϕ (bal) lin (m, ω), (14)
with

ǫ(m, ω) = 1 - ω 2 p L 3 N f 0 (v) (ω -k m • v) 2 d 3 v. ( 15
)
The validity of the above coarse-graining was discussed in section 4.5 of reference [START_REF] Escande | Direct path from microscopic mechanics to Debye shielding, Landau damping, and wave-particle interaction Plasma[END_REF]. Here we only recall the main steps of the discussion. First, the presence of many particles in the Debye sphere justifies neglecting the nondiagonal terms in equation [START_REF] Elskens | Irreversible behaviours in Vlasov equation and many-body hamiltonian dynamics : Landau damping[END_REF]. Second, Dawson's work [START_REF] Dawson | Plasma oscillations of a large number of electron beams[END_REF] justifies the Vlasovian expression [START_REF] Escande | Direct path from microscopic mechanics to Debye shielding, Landau damping, and wave-particle interaction Plasma[END_REF] for ǫ(m, ω) for a one-dimensional plasma. The generalization of Dawson's calculation to the three-dimensional case, though tedious, does not a priori involve any new conceptual difficulty. These results back up the above coarse-graining.

However, we now show these results come with conditions on the parameters of our problem. Invoking Dawson's calculation means that, when interested in a wave with phase velocity v w (with v w = v w ), there are many particles with parallel velocities in the range

[v w -|γ L |/k, v w + |γ L |/k],
where γ L is the Landau growth or damping rate of the wave, and k the modulus of its wavenumber. Indeed, this range is the one where the phase mixing à la van Kampen is occurring, and where the synchronization of particles with the wave brings the change of particle momentum inducing Landau damping or growth of this wave [START_REF] Doveil | Experimental observation of nonlinear synchronization due to a single wave[END_REF][START_REF] Elskens | Irreversible behaviours in Vlasov equation and many-body hamiltonian dynamics : Landau damping[END_REF][START_REF] Elskens | Microscopic dynamics of plasmas and chaos[END_REF][START_REF] Escande | Wave-particle interaction in plasmas : A qualitative approach Long-range interacting systems[END_REF][START_REF] Escande | Intuitive and rigorous microscopic description of spontaneous emission and Landau damping of Langmuir waves through classical mechanics[END_REF]. If the initial distribution of particles is meant to approximate a smooth distribution f 0 (v), this means that the length L of the system is taken large enough to fulfill the condition "many particles with parallel velocities in the range [v w -|γ L |/k, v w + |γ L |/k]". However, whatever L, for v large enough, one should not use ǫ(m, ω), but its discrete counterpart computed with the N -body distribution function.

IV. DEBYE SHIELDING, LANGMUIR WAVES AND LANDAU DAMPING

To keep this paper self-contained, we now discuss how Debye shielding and Landau damping were derived from equations ( 14)- [START_REF] Escande | Direct path from microscopic mechanics to Debye shielding, Landau damping, and wave-particle interaction Plasma[END_REF] in reference [START_REF] Escande | Direct path from microscopic mechanics to Debye shielding, Landau damping, and wave-particle interaction Plasma[END_REF]. These equations are obtained by introducing the continuous velocity distribution after particle dynamics is taken into account, and not before, as occurs when kinetic equations are used. However, they look very similar to those including initial conditions in Landau contour calculations of Langmuir wave growth or damping, usually obtained by linearizing Vlasov equation and using Fourier-Laplace transform, as described in many textbooks. Indeed, the smoothed self-consistent potential ϕ cg is determined by the response function ǫ(m, ω), viz. the classical plasma dielectric function, whose Vlasovian expression involving ∂f 0 /∂v obtains by a mere integration by parts if f 0 is differentiable. For a cold plasma, ǫ(m, ω) = 1 -ω 2 p /ω 2 , as expected (however this case is ruled out since the present theory is correct only for a large number of particles in the Debye sphere, as recorded in the previous section).

In equation ( 14), ϕ

lin (m, ω) is a discrete sum over the N particles. As computed in [START_REF] Balescu | Statistical mechanics of charged particles[END_REF][START_REF] Gasiorowicz | Dynamics of ionized media[END_REF][START_REF] Rostoker | Superposition of dressed test particles[END_REF], in the inverse Laplace transform of ϕ cg (m, ω), the pole ω = k m • ṙj (0) of ϕ (bal) lin (m, ω) brings a contribution to ϕ cg (m, t) which is the shielded Coulomb potential of particle j. This potential is often computed in textbooks by adding a test particle to a Vlasovian plasma (see for instance chapter 9 of [START_REF] Nicholson | Introduction to plasma theory[END_REF]). Since Debye shielding is effective over distances larger than the Debye length, this discussion is relevant under the assumption that our box size is large in comparison, viz. L ≫ λ D .

As in the Vlasovian approach, the part of ϕ cg (m, t) due to dominant poles of 1/ǫ(m, ω) corresponds to one Langmuir wave propagating in the direction of k m and one propagating in the opposite direction, with the corresponding Landau growth or damping. Therefore, the full potential in the plasma turns out to be the sum of two contributions only : one due to Langmuir waves and one due to all the shielded Coulomb potentials of individual particles.

Equation ( 14) becomes the Vlasovian one on replacing in equation ( 12) the discrete sums over particles with integrals over the smooth distribution function f (r, v). This yields

ϕ (bal) cg (m, ω) = - ie ǫ 0 k 2 m f 1 (m, v) ω -k m • v d 3 v, (16)
the Vlasovian formula for the ballistic potential. The N -body description reveals that the contribution of ϕ (bal) cg (m, ω) to ϕ cg (m, t) in the Vlasovian approach is the continuous limit of the sum of all the Debye-shielded potential of the particles. In this approach, this contribution is only a transient [START_REF] Hazeltine | The framework of plasma physics[END_REF]. Though its physical meaning is unclear, it does not correspond to a robust supplementary contribution to the full electrostatic potential in the plasma which would add to the potential of Langmuir waves and to the shielded Coulomb potentials of particles.

To afford the description of nonlinear effects in wave-particle dynamics, section 6.1 of reference [START_REF] Escande | Direct path from microscopic mechanics to Debye shielding, Landau damping, and wave-particle interaction Plasma[END_REF] generalizes the linear analysis analogous to that in section II of the present paper by applying it to the particles which are not resonant with Langmuir waves, while keeping the exact contribution to the electrostatic potential of the remaining N ′ tail particles that may experience trapping or chaotic dynamics. This eventually leads to equations describing the self-consistent dynamics of M Langmuir waves and of N ′ tail particles (these equations could be also obtained with the formalism of the present paper). For a one-dimensional plasma, the linear theory of Langmuir waves can be formulated on this basis, with the bonus that the information on particle dynamics is provided in parallel with the waves' [START_REF] Elskens | Irreversible behaviours in Vlasov equation and many-body hamiltonian dynamics : Landau damping[END_REF][START_REF] Elskens | Microscopic dynamics of plasmas and chaos[END_REF][START_REF] Escande | Wave-particle interaction in plasmas : A qualitative approach Long-range interacting systems[END_REF][START_REF] Escande | Intuitive and rigorous microscopic description of spontaneous emission and Landau damping of Langmuir waves through classical mechanics[END_REF]. This approach also provides spontaneous emission [START_REF] Elskens | Microscopic dynamics of plasmas and chaos[END_REF][START_REF] Escande | Direct path from microscopic mechanics to Debye shielding, Landau damping, and wave-particle interaction Plasma[END_REF][START_REF] Escande | Intuitive and rigorous microscopic description of spontaneous emission and Landau damping of Langmuir waves through classical mechanics[END_REF], and some nonlinear effects like damping with trapping [START_REF] Firpo | Phase transition in the collisionless damping regime for wave-particle interaction[END_REF], and the decoupling of the waves from the strongly chaotic motion of the particles when there is a plateau in the particle velocity distribution function (see sec. 2.2 of [START_REF] Besse | Validity of quasilinear theory : refutations and new numerical confirmation Plasma[END_REF]).

V. COMPARISON WITH PINES AND BOHM 1952

In 1952, Pines and Bohm showed that an N -body approach can bring a lot of insight into basic plasma effects [START_REF] Pines | A collective description of electron interactions : II. Collective vs individual particle aspects of the interactions[END_REF]. We now summarize the part of their results overlapping with our theory. Pines and Bohm already considered a One Component Plasma in a periodic box and wrote the equations of motion of its N electrons acted upon by their electrostatic potential. Working with the physical time t, they got an equation (number [START_REF] Dobrushin | Vlasov equations Funkts[END_REF] in their paper) for the time evolution of a Fourier component ρ k of the charge density. This equation involves a contribution of all Fourier components whose structure is similar to equation ( 10), but without linearizing particle dynamics. In order to neglect the contribution of all other Fourier components, they introduced the now celebrated "random phase approximation" by showing that the particle positions may be considered as nearly random for all times if there are many particles in a Debye sphere of a plasma in thermal equilibrium.

As a first step, they considered the collective part of ρ k , and showed that, for small wavenumbers k , the charge density oscillates at the plasma frequency, and that plasma oscillations exist for k ≪ λ -1 D . Inductively, they derived a harmonic equation for non-vanishing k 's with an oscillation frequency ω assumed implicitly real, and satisfying a discrete version of the Vlasovian dispersion relation (their equation ( 23)). They recovered the Bohm-Gross dispersion relation for small k 's.

As a second step, in their section III they split ρ k into a collective and a granular part. The evaluation of the latter for thermal particles yielded correctly the Yukawa version of Debye shielding.

In contrast, as those presented in most textbooks, our approach to Debye shielding and Landau damping is a linear theory. Laplace transform can then be applied readily. As already demonstrated by Landau's theory, the Laplace transform is a powerful heuristic tool. In the N -body approach, it shows the full potential in the plasma to split naturally in two contributions related to two families of poles : one due to Langmuir waves subjected to Landau growth and damping, and one due to all the shielded Coulomb potentials of individual particles, whatever be their velocity. By considering the initial N -body distribution as close to a set of monokinetic arrays of particles, it was shown in Appendix B of reference [START_REF] Escande | Direct path from microscopic mechanics to Debye shielding, Landau damping, and wave-particle interaction Plasma[END_REF] that linearization provides naturally the cancellation of non-diagonal terms in equation [START_REF] Elskens | Irreversible behaviours in Vlasov equation and many-body hamiltonian dynamics : Landau damping[END_REF] ; it is not necessary to use the random phase approximation, or other statistical arguments.

Particle deflection and Gauss' theorem were used by Pines and Bohm to interpret their Yukawa version of Debye shielding. Section 5 of reference [START_REF] Escande | Direct path from microscopic mechanics to Debye shielding, Landau damping, and wave-particle interaction Plasma[END_REF] brings more mechanical insight to this interpretation. There, instead of applying the Laplace transform to linearized particle dynamics, one applies Picard iteration technique (a standard method to prove the existence and uniqueness of solutions to first-order equations with given initial conditions) to the equation of motion of a particle, say P , due to the Coulomb forces of all other ones without linearizing particle dynamics. This calculation shows that particle P acts on another particle P ′ , directly by its Coulombian force, and indirectly by deflecting the orbits of all other particles whose force on particle P ′ is modified by this deflection. The direct Coulombian force is reduced by the indirect action, since the Coulombian deflections due to particle P diminish the number of other particles in a sphere surrounding it, which diminishes the apparent charge of particle P according to Gauss' theorem. This shielding effect becomes dominant at a distance on the order of λ D .

When starting from random particle positions, the typical time-scale for shielding to set in is the time for a thermal particle to cross a Debye sphere, i.e. ω -1 p . This is also the time for the equilibrium pair-correlation function to settle at its time-asymptotic values. Therefore, shielding results from the accumulation of almost independent repulsive deflections with the same qualitative impact on the effective electric field of particle P . If point-like ions were present, the attractive deflection of charges with opposite signs would have the same effect. The Coulombian deflections induce a self-organization of the plasma : it becomes a dielectric. So, these deflections do not produce only the disorder leading to collisional transport, but also the order corresponding to shielding and to the behaviour as a dielectric. Paradoxically, "collisionless" Landau damping turns out to occur because of what is usually called "collisions". Since the global deflection of particles includes the contributions of many other ones, the density of the electrons does not change, at variance with what occurs for the shielding at work next to a Langmuir probe. Gauss' theorem was used in section 2.2.1 of [START_REF] Piel | An introduction to laboratory, space, and fusion plasmas[END_REF] to provide an intuitive explanation of the shielding of such a probe.

VI. CONCLUSION

This paper provided further insight into the unified N -body description of Debye shielding and Landau damping. First, by deriving the fundamental equation for the electrostatic potential of reference [START_REF] Escande | Direct path from microscopic mechanics to Debye shielding, Landau damping, and wave-particle interaction Plasma[END_REF] in a simpler and more rigorous way. Second, by stressing that the term from initial conditions in Landau's calculation of Langmuir waves does not bring new physics, but is a mere damped continuous version of the Debye shielded potentials of all particles whose physical meaning is unclear in the Vlasov picture. Third, by discussing more precisely the condition for approximating the N -body distribution by a smooth one in the dispersion relation. Fourth, by comparing the present approach with the 1952 one by Pines and Bohm [START_REF] Pines | A collective description of electron interactions : II. Collective vs individual particle aspects of the interactions[END_REF] which already brought a lot of insight into Debye shielding and Langmuir waves. In particular, we have shown how the Coulombian deflections of particles bring order to the plasma by producing Debye shielding and by making the plasma behave as a dielectric. Furthermore, the Laplace transform confirms to be a powerful heuristic tool providing a compact and straightforward simultaneous derivation of Debye shielding and Landau damping. Linearization was shown to provide naturally the cancellation of non-diagonal terms in the fundamental equation for the electrostatic potential. This avoids using the random phase approximation or statistical arguments.

The N -body approach provides a derivation of Debye shielding for a single mechanical realization of the plasma. This was used in an N -body calculation of collisional transport of the One Component Plasma (OCP) incorporating all impact parameters with no ad hoc cut-off [START_REF] Escande | Uniform derivation of Coulomb collisional transport thanks to Debye shielding[END_REF]. This calculation explains why a two-body calculation yields a correct estimate of collisional transport, while most of this transport is due to the simultaneous action of many particles with impact parameters between the inter-particle distance and the Debye length.

When generalizing this theory to the collisional transport of a plasma with granular electrons and ions, the singularity of head-on collisions of particles with opposite charges can be avoided by excluding the case of a vanishing impact parameter which has probability 0. Then, the deflection of particles with same sign is just the opposite of the deflection of particles with opposite signs, so that the classical Rutherford cross-section is the same for charges of like sign and opposite signs. However, one might wonder whether a quantum mechanical treatment would not modify the scattering cross-section. Yet, it does not. This result, traditionally derived with the Born approximation, was obtained more recently with a much simpler calculation using Fermi golden rule that makes intuitive why the classical and quantum results agree [START_REF] Gauthier | Simple quantum calculation of the Rutherford cross section[END_REF].

The theory of collisional transport of reference [START_REF] Escande | Uniform derivation of Coulomb collisional transport thanks to Debye shielding[END_REF] and the approach of the present paper show that N -body dynamics, which has always been the ultimate reference in plasma textbooks, is now also a practical tool. Furthermore, Laplace's dream was not a mere utopia, since classical mechanics can genuinely describe non trivial aspects of the macroscopic dynamics of a many-body system.

One of us (DFE) thanks the Program Committee of the 42st EPS Conference on Plasma Physics and G.L. Delzanno to have led him to look for a more pedagogical derivation of the results of reference [START_REF] Escande | Direct path from microscopic mechanics to Debye shielding, Landau damping, and wave-particle interaction Plasma[END_REF] in order to prepare an invited talk and a course ; this was the origin of section II of this paper. He also thanks J. Daligault for recording him the pioneering 1952 contribution of Pines and Bohm [START_REF] Pines | A collective description of electron interactions : II. Collective vs individual particle aspects of the interactions[END_REF] to the N -body approach to basic plasma physics, and J. Daligault and S. Khrapak for a critical reading of our manuscript. We thank the two anonymous referees for leading us to formulate our theory in a more rigorous way, and to write a clearer and more self-contained paper.

or of the order of λ D .

Appendix A: Full derivation

We now consider ϕ(r), the potential created by the N particles at any point where there is no particle. Its discrete Fourier transform is readily obtained from the Poisson equation, and is given by φ(0) = 0, and for m = 0 by

where r j (t) is the position at time t of particle j acting as a source, φ(m

Let r j0 and v j be respectively the initial position and velocity of particle j, and let ∆r j (t) = r j (t)r j0v j t. We now compute a perturbative solution to the full N -body dynamics and the resulting Fourier components of the potential for m running in the finite domain M smooth defined in subsection II B, such that k m b smooth ≤ 1.

To this end, setting r j = r j0 + v j t + ∆r j (t) in equation (A1), we replace φ with its expansion to first order in the ∆r

whose time Laplace transform is

where the Doppler shift -k m • v j comes from the linear dependence on t of the exponent of equation (A2).

To compute ∆r j (ωk m • v j ), we use Newton's equation for the particles rj = e m e ∇ϕ lin (r j ),

where m e is the electron mass. When m runs over the finite domain such that k m b smooth ≤ 1, the self-field due to ϕ vanishes, and it is not necessary to exclude self-interactions. Therefore, one may use the harmonics of the electrostatic potential due to all particles defined by equation (A1), which yields

where r j now is the position of particle j as a massive object subjected to a force. Using equation (A2), the linearized particles dynamics defined by equation (A4) is then given by

Since ∆r j = 0 and ∆ ṙj = 0 at t = 0, the time Laplace transform of equation (A6) is

where the Doppler shift k n • v j comes from the linear dependence on t in the exponent of equation (A6). Computing ∆r(ωk • v) in equation (A3) from the expression of ∆r(ω) given by equation (A7) yields the fundamental equation [START_REF] Elskens | Irreversible behaviours in Vlasov equation and many-body hamiltonian dynamics : Landau damping[END_REF] for the potential. Note that the assumption k m b smooth ≤ 1, with b smooth ≪ λ D , defining the set M smooth , excludes scales which are irrelevant to Debye shielding and Landau damping, since these phenomena involve scales larger than,


Luca Maggiani

Cédric Bourrasset

Jocelyn Sérot

Matteo Petracca

Paolo Pagano

Claudio Salvadori

Parallel Image Gradient Extraction Core For FPGA-based Smart Cameras

One of the biggest efforts in designing pervasive Smart Camera Networks (SCNs) is the implementation of complex and computationally intensive computer vision algorithms on resource constrained embedded devices. For low-level processing FPGA devices are excellent candidates because they support massive and fine grain data parallelism with high data throughput. However, if FPGAs offers a way to meet the stringent constraints of real-time execution, their exploitation often require significant algorithmic reformulations.

In this paper, we propose a reformulation of a kernel-based gradient computation module specially suited to FPGA implementations. This resulting algorithm operates on-the-fly, without the need of video buffers and delivers a constant throughput. It has been tested and used as the first stage of an application performing extraction of Histograms of Oriented Gradients (HOG). Evaluation shows that its performance and low memory requirement perfectly matches low cost and memory constrained embedded devices.

I. INTRODUCTION

With the rapidly increasing interest in decentralized approaches for Smart Camera Networks (SCN), embedded computer vision has become a research field per se. The deployment of complex and smart applications on SCNs now crucially depends on the the possibility to perform complex and time-consuming vision algorithms directly at the node level. This places stringent constraints on the processing elements embarked on these nodes. This is specially true for the so-called low-level processing stages, operating on the raw pixels coming from the sensor, for which the increasing sensor resolutions lead to data throughputs which cannot be sustained by embedded general purpose CPUs.

Because they naturally support massive and fine grain data parallelism, FPGAs have often been advocated as a solution to the aforementioned issue. However, in most cases, this is only true if the algorithm to be implemented can be formulated to operate on sequential streams of pixels, without the need to memorize and iterate on intermediate results. This reformulation, which essentially amounts to convert an imperative, stateful model to a dataflow, stateless one is often not trivial or even feasible in case of non-Turing-complete algorithms. Moreover, the efficiency of the final implementation directly depends on it. Yet, it is seldom explicitly described, if not mentioned, in papers dealing with embedded vision.

Let's take for example a classical operation in low-level vision : the computation of the gradient. This operation is used in many higher-level algorithms such as the Hough transform, the Canny edge detector or the Histogram of Oriented Gradient (HOG). It relies on regular, pixel-wise operations but some of them -the square root and the arc-tangent -are non linear [START_REF] Bhardwaj | A survey on various edge detector techniques[END_REF] and thus do not map easily on hardware computing devices such as FPGAs. To circumvent this problem, two solutions have typically been proposed : using pre-computed look-up tables as in [START_REF] Kadota | Hardware architecture for HOG feature extraction[END_REF], [START_REF] Hahnle | Fpgabased real-time pedestrian detection on high-resolution images[END_REF] or relying on run-time iterative algorithms such as CORDIC or derivatives [START_REF] Mizuno | Architectural study of HOG feature extraction processor for real-time object detection[END_REF]. The former approach requires large amount of memory, which can quickly be a problem with tightly space and/or cost-constrained devices, especially when the required precision increases. The second approach has a strong impact on performances since it is no longer possible, in the general case, to maintain a 1to-1 ratio between the frequency of the input and output pixels respectively. In certain cases, by taking into account the algorithmic context -i.e. the subsequent utilisation of the gradient information computed by the operation -, it is however possible to overcome the limitations of the above mentioned solutions. This is the case, in particular, for the gradient used in the HOG application. The HOG algorithm [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF] relies on two steps : first the computation of the image gradient followed by a local spatial aggregation of the histogram of the computed gradients. This algorithm is the basis of several well-known methods for detecting objects like pedestrians or vehicles in image sequences [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF]- [START_REF] Yang | Recognizing actions using depth motion maps-based histograms of oriented gradients[END_REF]. In this context, the resulting histograms are then normalised and compared to reference models by a SVM linear classifier in order to make decision about the presence/non-presence of the target objects.

In this paper, we describe an optimized implementation of the gradient extraction operation which can be used as the first step of the HOG algorithm. By taking into account the specific needs of the subsequent steps and reformulating the extraction in terms of purely dataflow basic operators, we have obtained a highly parallel, fully pipelined implementation, producing a valid result at every clock cycle and with a memory footprint considerably smaller compared to other state of the art solutions. The accuracy of the produced results also meets the precision requirements of the HOG algorithm. The rest of this paper is organized as follows. Sec. II is a brief overview of the existing solutions proposed for computing the gradient, especially for the HOG algorithm. Our solution, named HOG-Dot is described in Sec. III. It is evaluated, in terms of accuracy, in Sec. IV. The corresponding hardware implementation is exposed in Sec. V and the performances of the resulting core are assessed in Sec. VI.

II. RELATED WORK

The state of the art of the gradient implementation is mainly related to its use in the HOG algorithm [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF]. Since spatial gradients techniques are based on magnitude ∇I and angle orientation θ, gradient can be expressed in polar coordinates by:

∇I = G x 2 + G y 2 θ = arctan G y G x (1)
where G x and G y are respectively the gradient in x and y direction defined as:

G x = I * -1 0 1 and G y = I * -1 0 1 T ( 2
)
where I is the input image. These operations require strong computing capabilities for real-time processing. In particular, the arctan evaluation in a hardware implementation either requires many resources or produces non efficient results. In [START_REF] Barbaro | A 100 x 100 pixel silicon retina for gradient extraction with steering filter capabilities and temporal output coding[END_REF] a low resolution VLSI gradient module has been proposed. The gradient approximation is carried out by deploying a steering filter architecture within a custom CMOS image sensor. More recently, a fixed point implementation has been proposed by Kadota et al. [START_REF] Kadota | Hardware architecture for HOG feature extraction[END_REF]. The authors designed a memory-based core that extracts the gradient with precomputed Look-Up table (LUT) entries. The memory contains all the possible entries of G x , G y . Therefore the memory footprint would grow as G x × G y sizes increase, i.e. for luminance range extensions or to achieve better gradient resolution. Although this system reaches real time performance, it requires a huge memory footprint (n × 2 2n+1 bits where n is the G x representation) and a square root based on multi-cycles operations per pixel. Mizuno et al. [START_REF] Mizuno | Architectural study of HOG feature extraction processor for real-time object detection[END_REF] has improved the Kadota et al. proposal by using a CORDIC method [START_REF] Volder | The CORDIC trigonometric computing technique[END_REF] for the orientation evaluation. While the CORDIC method provides an acceptable precision for the orientation, it requires a variable number of iterations which depends on the initialisation value of the algorithm and the input operand.

The most used technique for the gradient extraction has been introduced by Bauer et al. in [START_REF] Bauer | FPGA implementation of a hog-based pedestrian recognition system[END_REF] and then reproduced in more recent implementations [START_REF] Hahnle | Fpgabased real-time pedestrian detection on high-resolution images[END_REF], [START_REF] Negi | Deep pipelined one-chip FPGA implementation of a real-time image-based human detection algorithm[END_REF]. Bauer et al. propose an orientation binning scheme without computing the orientation angle explicitly. The x and y derivatives directly index the histogram using a LUT, while the magnitude is extracted with an hardware square root module. Even though this technique improves the performance with respect to CORDIC-based solutions, the square root module increments the processing latency and limits the maximum operating frequency as well.

III. HOG-DOT ALGORITHM

One of the main issue in developing the HOG pipeline for FPGA-based SC is the polar coordinates computations through non linear floating-point operations. Even though floating point arithmetic could be exploited within FPGA, as shown in [START_REF] Ercegovac | Design and FPGA implementation of radix-10 algorithm for square root with limited precision primitives[END_REF], its implementation require a non-negligible amount of hardware resources and furthermore it will decrease the output throughput. Thus, we propose a highly parallel architecture capable to approximate the square-root and the arc-tangent. It is based on a set of linear operations that can be easily implemented within a pipelined hardware circuitry. The proposed methods considers a discrete gradient orientation space divided into N uniformly spaced samples, which are denoted by îk with k = 0, ..., N -1. In Fig. 1 The versor îk can be expressed by its components as follows:

îk = x cos θ k + ŷ sin θ k k ∈ 0, • • • , N -1 (3)
For each îk versor, we express the overlaying gradient component as the scalar projection of ∇I over îk . In Eq. 4 the scalar product is given.

∂I ∂ îk = ∇I • îk (4)
Then by replacing Eq. 3 in Eq. 4, we obtain:

∂I ∂ îk = cos θ k ∂I ∂x + sin θ k ∂I ∂y (5)
Finally, by recalling Eq. 2, in Eq. 6 the spatial gradient component over a generic îk versor is shown.

∂I ∂ îk (x, y) = cos θ k [I(x + 1, y) -I(x -1, y)] + sin θ k [I(x, y + 1) -I(x, y -1)] (6)
As in Eq. 6, the k th projection of the spatial gradient ∇I can be straightforward computed by applying linear arithmetic operations to the image pixel. In particular in Eq. 7 the complete matrix expression is shown. Once the number of the orientation samples N has been defined, the co-sinusoidal values are then a-priori fixed for each θ k thus resulting a constant coefficients matrix.

In Eq. 7 a generic k th component can be computed as convolution product between the kernel matrix and the image pixels. By applying the same approach for each îk within the N orientation samples, all the gradient projections are computed. In Fig. 2, N = 8 is assumed. The eight co-sinusoidal convolutions are carried out over the same 3 × 3 mask pixels, thus computing the gradient projections. The spatial gradient is then evaluated by extracting the greater magnitude projection among the available set. The resulting projection represents the closest gradient approximation among the available orientation samples. Once the maximum magnitude has been computed, the argmax operator extracts the relative kth orientation as follows:

N ∂I ∂ îk (x, y) =   0 -sin θ k 0 -cos θ k 0 + cos θ k 0 + sin θ k 0   •   I(x -1, y -1) I(x, y -1) I(x + 1, y -1) I(x -1, y) I(x, y) I(x + 1, y) I(x -1, y + 1) I(x, y + 1) I(x + 1, y + 1)   (7) i 0 i 1 i 2 i 3 i 4 i 5 i 6 i 7 argmax ∇I Dot (x, y)
k = argmax ∂I ∂ îk (x, y) k ∈ 0, • • • , N -1 (8)
Finally, in Eq. 9 the resulting gradient approximation is shown.

∇I Dot (x, y) = ∂I ∂ îk (x, y) , θk (9)
The original HOG implementation defines instead the gradient ∇I HOG (x, y) as:

∇I HOG (x, y) = ∇I(x, y) , θk (10)
By comparing Eq. 9 and Eq. 10, the resulting vectors have the same orientation direction îk thus both belong to the same histogram bin. With to the magnitude differences, in Fig. 3 both approximations are reported along with the mathematical formulation. This expression, defined ∇I, represents the spatial gradient definition as in Eq. 1 which results without the HOG orientation binning. In particular, it is clearly shown that ∇I Dot always represents a round down approximation of the ∇I HOG . Since HOG-Dot involves a dot-product operation, the ∇I Dot approximation can be also expressed as follows:

∇I Dot (x, y) = ∇I(x, y) cos(θ -θk) (11)
All in all, the HOG-Dot technique allows gradient computation by deploying only linear operations rather than square root and arc-tangent operator. By controlling the number of the considered orientations N , the algorithm accuracy can be estimated and then predicted. The higher N means a finer granularity of the gradient projections but requires more computing iterations. Nevertheless the increased iterative operations, our method is suitable for a parallel exploitation thus achieving the highest performance.

θ -θ k ∇ IHO G ∇ I ∇ IDo t îk

IV. HOG-DOT ACCURACY EVALUATION

To evaluate the approximation error a mathematical expression has been derived, in terms of gradient angle and magnitude. As previously introduced, the angle approximation depends on the number of orientation samples N . Given a fixed N , the orientation accuracy is then equal to 2π/N radiant. As far as the magnitude approximation concerns, the Error Vector Magnitude (EVM) metric has been used. The EVM is usually deployed to quantify the difference between transmitted and received constellation points and can be expressed as:

EV M = P error P ref (12)
where the power of the error vector is expressed as P error while the ideal one as P ref .

In this context, the EVM can be rewritten as:

EV M Dot = e 2 ∇I HOG (x, y) 2 = e ∇I HOG (x, y) (13
) where e is ∇I HOG (x, y) -∇I Dot (x, y), the difference vector between the HOG and our version. As in Eq. 11 the difference vector between ∇I HOG (x, y) and ∇I Dot (x, y) is straightforward evaluated as follows: e = ∇I HOG (x, y) -∇I Dot (x, y)

= ∇I(x, y) 1 -cos(θ -θk) (14)
Thus, the EV M Dot is a deterministic function of θ -θk, and, consequently of N (θk = kπ/N ). The EV M Dot is then described by:

EV M Dot = 1 -cos θ - kπ N (15)
In Fig. 4 the EV M Dot is then shown. It is expressed within a single bin span as function of θ -θ k . Since the angle resolution directly depends on N , two EVM curves are reported for N equal to 8 and 16 respectively. The error goes to zero whether θ is equal to θ k , while it achieves the maximum value when ∇I Dot lies in the middle between θ k-1 and θ k+1 instead. Nevertheless, with N = 8 the maximum error is bounded to 2%. To complete the comparison, in Fig. 5 the maximum and average errors, defined as max(EV M Dot ) and avg(EV M Dot ), are reported as function of N . By increasing the orientation samples N , the difference between θ and monotonically decreases, thus the maximum EV M decreases as well. Given all values on its period, the average EV M is evaluated as follows:

avg(EV M Dot ) = 2 T T /2 0 EV M Dot dθ (16) = 1 - 2N π sin( π 2N )
where T is equivalent to the EV M Dot period π N . The average value is clearly bounded by Eq. 15 and decreases when N increases. Although we exploit the gradient extraction with a linear kernel based method, our results are comparable with those obtained by the HOG technique. As shown in Fig. 5, by fixing N equal to 8, the resulting average EVM is close to 0.5%. Whether an improved accuracy is requested, the sampling factor N can be further increased to match the output specifications.

V. HARDWARE IMPLEMENTATION

In Section III the proposed method has been first formally introduced. In this Section a FPGA hardware implementation will be shown. In particular, the proposed hardware design has been developed as the gradient extraction circuit for the HOG pipeline. Therefore the number of orientation samples N is then equivalent to the number of the HOG histogram bins, usually equal to 8.

The hardware design is obtained following Eq. 7. The matrix convolution between the kernel coefficients and the 3 × 3 mask pixel is computed by leveraging on pipelined hardware module. Since these coefficients are constant for a given angle θ k , the k th gradient component can be evaluated by deploying two hardware multipliers and three adder/subtractor modules. In Fig. 6 the hardware circuitry for a single gradient projection is shown. More in detail, the input pixel is represented with 8bit while each coefficient value has been represented with 9bit two's complement. Due to the internal pipeline structure, the module shown in Fig. 6 is capable of processing new input data every clock cycle to achieve the maximum throughput. In order to extend the gradient computation to all the available projections, N convolution modules are instantiated in parallel. As previously suggested by Fig. 2, the resulting hardware structure generates N parallel gradient projections. Each clock cycle, those resulting projections are compared each other to extract the maximum magnitude value. In Fig. 7 the hardware argmax implementation is shown. This is implemented as log 2 (N ) comparative stages within a pipelined architecture. With the assumption of N = 8, the first stage is composed by four parallel comparators, which are fed by the computed gradient projections. Each comparator propagates to the next stage the temporary maximum projection and its orientation. In the presented example, the comparison ends at the third stage, where the closest gradient approximation is generated. The resulting couple ∂I ∂ îk and θk represents the closest spatial gradient approximation retrieved from the HOG-Dot method, as in Eq. 9.

VI. PERFORMANCE EVALUATION

In this Section the proposed gradient extraction module is evaluated in terms of memory footprint, data throughput, data latency and output accuracy. We adopted the Terasic DE0nano development board which embeds an Altera Cyclone IV FPGA with 22k Logic Elements (LE), 600 kbits on-board memory and 144 9x9 bits multiplier modules. Considering a VGA resolution and N equal to 8, the proposed hardware module occupies only 5.4% of the Logic Elements (LEs), 1.2 KByte of the on-chip RAM (0.2%) and uses 16 of 144 available multiplier module. These results are depending on the number of gradient components N and to the maximum image width size W . Higher N would increase the number of the used multiplier as 2 • N while an increased image size will increase only the memory footprint due to neighborhood extraction. In Tab. I a with the state of the art is shown. In particular, due to the limited implementation details in the state of the art and the different FPGA technologies involved, numerical LEs comparisons are impractical. We rather compare the on-chip memory occupancy footprint and the used DSP modules among the considered solutions. Although the proposed work uses a higher amount DSP modules with respect to Kadota et al. [START_REF] Kadota | Hardware architecture for HOG feature extraction[END_REF] and Negi et al. [START_REF] Negi | Deep pipelined one-chip FPGA implementation of a real-time image-based human detection algorithm[END_REF], HOG-Dot memory footprint is significantly lower than others. Moreover, with respect to the implementations proposed by Bauer et al. [START_REF] Bauer | FPGA implementation of a hog-based pedestrian recognition system[END_REF] and Mizuno et al. [START_REF] Mizuno | Architectural study of HOG feature extraction processor for real-time object detection[END_REF], our method reduces the DSP usage. This results of the HOG-Dot pipelined architecture, which optimizes the hardware resource occupancy. In Table II the processing latency and the maximum operating frequency are shown with respect to the other implementations. As far as throughput concerns, our implementation achieves the same results as the Negi et al. proposal, while is deployed within a low-end FPGA device. With respect to Bauer et al., the resulting processing latency is increased by removing frame buffering and external memory accesses. The maximum operating frequency is 78MHz, which corresponds to the higher throughput achievable. Hence, the theoretical image frequency of 250 fps can be reached at VGA resolution. Given the limited processing latency, our module is then suitable for real-time processing applications.

A. HOG performance comparison

In this Section, our HOG-Dot method is then compared with the state of the art HOG implementation. In order to compare the results, an entire hardware HOG pipeline has been deployed following the module presented in [START_REF] Maggiani | Reconfigurable architecture for computing histograms in real-time tailored to FPGA-based smart camera[END_REF]. The HOG-Dot hardware module is then followed by the histogram circuit which provides to the classifier an ordered flow of oriented gradient over 8 × 8 pixel cells. The original HOG and our modified approach have been trained both with a classification SVM provided by SVMLight [START_REF] Joachims | Svmlight: Support vector machine[END_REF]. The INRIA pedestrian [START_REF]Person dataset[END_REF] dataset has been used to train and test both systems. The reference HOG implementation has been provided by the OpenCV framework, which implements a double precision floating point arithmetic with arc-tangent and square root operations. The comparison results are summarized with the Receiver Operator Characteristics (ROC) metric and shown in Fig. 8. This curve shows in the x-axis the relationship between the False Positive Rate (FPR) and the True Positive Rate (TPR) on the y-axis. The closer to the upper left the ROC goes, the higher performance are achieved. The plots are made by changing the SVM decision threshold within ±5 range as proposed by [START_REF] Fawcett | An introduction to ROC analysis[END_REF]. As a result, the proposed fixed As far as the accuracy evaluation is concerned, the proposed approximated algorithm performs up to 2% of error on the gradient magnitude.

VII. CONCLUSIONS

In this paper the parallel gradient extraction HOG-Dot algorithm is presented. This algorithm is suitable for real-time image processing in FPGA-based low-cost Smart Cameras. The fixed point and linear HOG-Dot implementation has been evaluated in a HOG-SVM pedestrian detection system. It shows comparable detection performance with respect to the floating point, non linear reference implementation. Moreover, the hardware implementation shows benefits in terms of memory footprint and hardware resources occupancy with respect to state of the art FPGA solutions. The final implementation shows real-time processing performance, enabling a wide range of applications in the smart camera scenario.

Fig. 1 .

 1 Fig. 1. The versors beam used to compute the dot product.

Fig. 2 .

 2 Fig. 2. The proposed HOG-Dot parallel implementation.

Fig. 3 .

 3 Fig. 3. ∇I, ∇I HOG and ∇I Dot comparisons.

Fig. 4 .

 4 Fig. 4. The EV M HOG Dot evaluation as function of θ -θk.

Fig. 5 .

 5 Fig. 5. Maximum and average EV M Dot as function of N .

Fig. 6 .

 6 Fig. 6. The hardware implementation for the i k gradient component.

Fig. 7 .

 7 Fig. 7. Pipelined argmax extraction (N = 8).

Fig. 8 .

 8 Fig. 8. ROC curves with INRIA datasetpoint HOG-Dot algorithm shows TPR up to 91% with FPR at 2%. These performance have been obtained with a L2 histogram normalization as in the OpenCV implementation. The original OpenCV instead results around 93% TPR with 2% FPR. The difference between the reference and our proposed method is clearly due to the approximation error shown in Fig.4. Although our method only involves fixed-point linear operation, the detection performance shows comparable results with the floating point, non linear reference implementation. As far as the accuracy evaluation is concerned, the proposed approximated algorithm performs up to 2% of error on the gradient magnitude.

TABLE I HARDWARE

 I RESOURCE UTILIZATION FOR VGA RESOLUTION.

		Platform	RAM	DSPs
	Kadota [2]	Altera Stratix II	N/A	12
	Mizuno [4]	Altera Cyclone IV	64 KB	68
	Bauer [10]	Xilinx Spartan-3	225 KB	18
	Negi [11]	Xilinx Virtex-5	162 KB	0
	HOG-Dot	Altera Cyclone IV	1.2 KB	16

ACKNOWLEDGMENT

This work has been sponsored by the French government research programme "Investissements d'avenir" through the IMobS3 Laboratory of Excellence (ANR-10-LABX-16-01).