Hermine Bierm

Antoine Lerbet
email: antoine.lerbet@math.univ-poitiers.fr

STUDY OF THE PERIMETER OF A SHOT NOISE RANDOM FIELD BY AN ELEMENTARY APPROACH

Keywords: April 1, 2022. 2010 Mathematics Subject Classification. Primary: 28A75, 60G60, 60D05; Secondary: 60G10, 60G15, 62H30 Perimeter, excursion sets, stationary random fields, Gaussian random fields, Shot noise random fields

The study of the geometry of excursion sets of 2D random fields, especially the perimeter or length of level lines, has a growing interest from both the theoretical developments and the statistical applications. In this paper we are interested in the relationship between the perimeter of the excursion sets of a shot noise random field compared to the well known Gaussian framework. Our approach follows the weak framework of special bounded variation functions in which we consider the functions that map the level of the excursion set to the perimeter of the excursion set. In this unified framework we exhibit two different regimes with respect to the intensity of the shot noise random field. The first one is the classical Gaussian regime in high intensity, while the second new one, in low intensity, is related to an elementary approximation. In the explicit case of Gaussian correlation functions, we show the pertinence of such approximation for statistical evaluation. At least, this enables us to propose a classification procedure to discriminate Gaussian or shot noise fields.

Introduction and notation

Random fields are a popular branch of probabilities that allow to model a lot of domains, including texture synthesis [START_REF] Van Wijk | Spot noise texture synthesis for data visualization[END_REF][START_REF] Galerne | Random phase textures: Theory and synthesis[END_REF], image analysis with applications in the medical field [START_REF] Cao | The geometry of correlation fields with an application to functional connectivity of the brain[END_REF][START_REF] Bernardino | Statistics for Gaussian random fields with unknown location and scale using Lipschitz-Killing curvatures[END_REF] and extensive spatial modelling. The most popular random field models are certainly the Gaussian fields, but in this paper we will consider shot noise models defined on R 2 by [START_REF] Abaach | Testing marginal symmetry of digital noise images through the perimeter of excursion sets[END_REF] ∀x ∈ R 2 , X(x) = i m i g(x -x i ),

where g : R 2 → R is the kernel, the x i are the points of a homogeneous Poisson point process of intensity λ in R 2 and the m i are "marks", independent of the Poisson point process such that {(x i , m i )} i∈I is a Poisson point process on R 2 ×R of intensity λL⊗F with F a probability measure on R, defined on (Ω, A, P) a complete probability space. This model can be seen as an extension of the Germ-Grain models [START_REF] Chiu | Stochastic Geometry and Its Applications[END_REF], and thus of the Boolean model, where the x i are the germs while the kernel plays the role of the grains, but with the difference that we are interested in the random fields formed by the sum of the grains and not just the random set formed by their unions. These fields were first introduced by Daley [START_REF] Daley | The definition of a multi-dimensional generalization of shot noise[END_REF], although they are based on the one-dimensional process [START_REF] Campbell | The study of discontinuous phenomena[END_REF][START_REF] Campbell | Discontinuities in light emission[END_REF][START_REF] Rice | Mathematical analysis of random noise[END_REF], and have been used in many fields [START_REF] Baccelli | Stochastic Geometry and Wireless Networks, Volume I-Theory, volume 3, No 3-4 of[END_REF][START_REF] Klüppelberg | Explosive Poisson shot noise processes with applications to risk reserves[END_REF][START_REF] Schmidt | Shot-noise processes in finance[END_REF][START_REF] Galerne | Random phase textures: Theory and synthesis[END_REF]. On one side, there is a strong relation between the shot noise framework and the Gaussian framework. The main result is due to Heinrich and Schmidt [START_REF] Heinrich | Normal Convergence of Multidimensional Shot Noise and Rates of This Convergence[END_REF], showing asymptotic normality in high intensity, in the sense of finite dimensional distribution, and even in the topology of uniform convergence on compact sets. Since then, this strong result has aroused a lot of interest in application [START_REF] Klüppelberg | Explosive Poisson shot noise processes with applications to risk reserves[END_REF][START_REF] Chan | Calculating the outage probability in a CDMA network with spatial Poisson traffic[END_REF][START_REF] Aljuaid | Investigating the Gaussian convergence of the distribution of the aggregate interference power in large wireless networks[END_REF] and in research [START_REF] Biermé | On the Perimeter of Excursion Sets of Shot Noise Random Fields[END_REF][START_REF] Lachièze-Rey | Asymptotics for the critical level and a strong invariance principle for high intensity shot noise fields[END_REF]. On the other side, one purpose of this paper is to show that the low intensity behaviour is really distinct from the Gaussian framework but through its visual definition, based on simple random tools, it allows a certain malleability in the calculations [START_REF] Biermé | Mean Geometry for 2D random fields: Level perimeter and level total curvature integrals[END_REF][START_REF] Lachièze-Rey | Two-Dimensional Kac-Rice Formula. Application to Shot Noise Processes Excursions[END_REF][START_REF] Lachièze-Rey | Normal convergence of nonlocalised geometric functionals and shot-noise excursions[END_REF][START_REF] Lerbet | Statistical inference on stationary shot noise random fields[END_REF] that is rarely found outside the Gaussian case.

All random field information is contained in the excursion sets, defined for any u ∈ R and open subset (the viewing window) T ⊂ R 2 by E X (u, T ) := {x ∈ T | X(x) ≥ u} Statistically, it is interesting to study geometric characteristics of these random sets, mainly their area, perimeter and Euler characteristic. It is very rare to know the law of these random functions of the level but it happens that we can calculate their mean values. As it is often the case, under certain assumptions of stationnarity and regularity, the Gaussian framework allows the explicit computation of these 3 geometric quantities [START_REF] Adler | Random Fields and Geometry[END_REF][START_REF] Azaïs | Level Sets and Extrema of Random Processes and Fields[END_REF]. With this exception there are only a few results on the geometry of excursion sets [START_REF] Adler | High Level Excursion Set Geometry for Non-Gaussian Infinitely Divisible Random Fields[END_REF][START_REF] Lachièze-Rey | Bicovariograms and Euler characteristic of random fields excursions[END_REF], and they usually require assumptions of regularity and bounded density existence.

Especially for shot noise random fields, by adopting a functional point of view, several results for the computation of mean geometry were obtained in [START_REF] Biermé | Mean Geometry for 2D random fields: Level perimeter and level total curvature integrals[END_REF]. More precisely, without bounded density assumption, they obtained results in two distinct frameworks : the one where the kernel is smooth and the one where the kernel is elementary, that may be unified in the framework of special bounded variation function [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF]. The quantity of interest for studying the regularity of excursion sets is the perimeter u → Per(E X (u, T )) view as a L 1 (R) function when X is a.s. a locally special bounded variation function. We emphasize that under stronger assumptions, Per(E X (u, T )) corresponds to the length of the level set {x ∈ T ; X(x) = u} [START_REF] Azaïs | Level Sets and Extrema of Random Processes and Fields[END_REF], also related to the Lipschitz-Killing curvature of order 1 of the excursion set [START_REF] Biermé | Lipschitz-Killing curvatures of excursion sets for two-dimensional random fields[END_REF]. Since we consider stationary random sets, by ergodicity, one should have Per(E X (u,T ))

L(T )

→ Per(E X (u)) a.s. as T tends to R 2 . Note that asymptotic normality have also been investigated in a Gaussian framework in [START_REF] Berzin | Estimation of local anisotropy based on level sets[END_REF] for a fixed level under regularity assumptions and for shot noise random fields in a weak framework in [START_REF] Lachièze-Rey | Normal convergence of nonlocalised geometric functionals and shot-noise excursions[END_REF]. Hence our function of interest is the mean density perimeter function u → Per(E X (u)) := E[Per(E X (u, T ))] L(T ) .

However, even for a simple smooth kernel as a Gaussian, the formulas of [START_REF] Biermé | Mean Geometry for 2D random fields: Level perimeter and level total curvature integrals[END_REF] are not fully explicit and in order to find the mean density perimeter, a Fourier inversion is needed. Through a kernel discretization approach, another aim of this paper will be to obtain formulas that are more easily usable in practice. Finally, we will use the area and perimeter of the excursion sets as a classifier to distinguish a Gaussian and shot noise field with the same first and second order statistics, especially for visually indistinguishable pattern. The paper is organized as follows. After reviewing the well-known formulas for smooth Gaussian and shot noise fields, we give in Section 2 the two distinct behaviors of the mean densities of the level perimeter integral for shot noise smooth random fields : the high regime for large λ values and the low regime for small λ values. Section 3 is devoted to the case where the kernel is a Gaussian function and its own properties, with in particular its scale invariances. We will study in Section 4 the discretization of this model and the limit of the mean perimeter when the discretization step tends to 0. We will make the link between the identified limit and the low regime. Under normalisation by its total variation, this will lead to more explicit formulas for perimeter than those already known. Finally, using these formulas, in Section 5 we will give a classification method to distinguish a shot noise field from a Gaussian field with the same moments of order 1 and 2.

Previous results

General formula.

For T an open bounded subset of R 2 and X a "nice" stationary random field on R 2 . By stationarity we easily get that the mean density area of excursion sets is given by the tail function of the common distribution. More precisely, for all u ∈ R,

Area(E X (u)) = E(L(E X (u, T )) L(T ) = P(X(0) ≥ u).
The computation of the perimeter is more involved and we use the weak functional approach developed in [START_REF] Biermé | On the Perimeter of Excursion Sets of Shot Noise Random Fields[END_REF]. Following the notation of [START_REF] Biermé | Mean Geometry for 2D random fields: Level perimeter and level total curvature integrals[END_REF] we define, when it exists, the level perimeter integral by the linear form

LP X (•, T ) : C b (R) -→ R (2) h -→ R h(u)Per(E X (u), T )du The knowledge of E[LP X (h, T )] for all h ∈ C b (R) allows us to get information of E[Per(E X (u), T )]
for almost every level u ∈ R. In particular computing (2) for ν ∈ R and h ν given by

(3) h ν (u) = e iνu , E[LP X (h ν , T )] corresponds to the Fourier transform of u → E[Per(E X (u), T )] at frequency ν.
Moreover, the total variation of X may be computed using h = 1 or ν = 0 and its expectation is given by

E[TV(X, T )] = E[LP X (1, T )]
. By stationarity, the quantities of interest are related to the mean densities

LP X (h) = E[LP X (h, T )] L(T ) = R h(u)
Per(E X (u)) du and TV(X) = LP X (1).

Note that we might expect that for nice ergodic random fields and nice growing sequences of T

LP X (h, T ) L(T ) -→ LP X (h) and Per(E X (u), T ) L(T ) -→ Per(E X (u)) a.s.
Assuming that X is a C 1 random field with X(0) and ∂ j X(0) having finite expectations for all j = 1, 2, by Theorem 2 of [START_REF] Biermé | Mean Geometry for 2D random fields: Level perimeter and level total curvature integrals[END_REF] for a.e. u ∈ R, the random variables Per(E X (u), T ) have finite expectation such that for all h bounded continuous function on R, one has

E(LP X (h, T )) = R h(u)E(Per(E X (u), T )) dt = E (h(X(0)) ∇X(0) ) L(T ).
In particular, LP X (h) = E (h(X(0)) ∇X(0) ) and TV(X) = E[ ∇X(0) ].

It follows that h → LP X (h) is a continuous linear form on C b (R) with subordinated norm given by TV(X). Moreover, when X is also isotropic, the computation may be simplified as

(4) LP X (h) = π 2 E (h(X(0))|∂ 1 X(0)|) and TV(X) = π 2 E[|∂ 1 X(0)|].
Following the case of shot noise fields developed in [START_REF] Biermé | On the Perimeter of Excursion Sets of Shot Noise Random Fields[END_REF] that are infinitely divisible random fields for which the use of characteristic functions is more tractable, we state the following result. Proposition 1. Let X be a stationary isotropic C 1 random field on R 2 , such that E(|X(0)|) < +∞ and E(|X 1 (0)| 1+ε ) < +∞ for some ε > 0, and let us denote for

(ν, v) ∈ R 2 , (5) ϕ(ν, v) = E[e iνX(0)+iv∂1X(0) ].
Then

LP X (h ν ) = - ∞ 0 1 v ∂ϕ ∂v (ν, v)dv, and
TV(X) = - ∞ 0 1 v ∂ϕ ∂v (0, v)dv, (6)
where h ν is given by (3) and the improper integrals are defined as lim

V →+∞ V 0 . Proof. Let ν ∈ R since X be a stationary isotropic C 2 random field on R 2 , by (4) we have LP X (h ν ) = π 2 E (h ν (X(0))|∂ 1 X(0)|) .
According to Proposition 2 of [START_REF] Biermé | Mean Geometry for 2D random fields: Level perimeter and level total curvature integrals[END_REF], since |h ν (X(0

))| ≤ 1 and E(|X 1 (0)| 1+ε ) < +∞ for some ε > 0 we get E (h ν (X(0))|∂ 1 X(0)|) = 2 π +∞ 0 1 v E (h ν (X(0))∂ 1 X(0) sin(v∂ 1 X(0))) dv. But ϕ(ν, v) = ϕ(ν, -v) so that ϕ(ν, v) = E (h ν (X(0)) cos(v∂ 1 X(0))) and therefore ∂ϕ ∂v (ν, v) = -E (h ν (X(0))∂ 1 X(0) sin(v∂ 1 X(0))) ,
from which we deduce the result.

2.2. Isotropic smooth Gaussian random fields. When the random field X is an isotropic smooth (at least C 1 ) Gaussian random field, we write ϕ X , respectively ϕ ∂1X , the characteristic function of X(0), respectively of ∂ 1 X(0). Since X(0) and ∂ 1 X(0) are non correlated by stationarity and thus independent by Gaussian distribution, the characteristic function in [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF]

is ϕ(ν, v) = ϕ X (ν)ϕ ∂1X (u). Hence - ∞ 0 1 v ∂ϕ ∂v (ν, v)dv = ϕ X (ν) - ∞ 0 1 v ∂ϕ ∂1X (v) ∂v dv , with ϕ ∂1X (v) = exp(-Var[∂ 1 X(0)]v 2 /2). It follows that for all ν ∈ R, (7) LP X (h ν ) = π 2 Var[∂ 1 X(0)]ϕ X (ν).
Corollary 1. Let X be a stationary isotropic C 1 Gaussian random field then for almost all u ∈ R

Per(E X (u)) = π 2 Var[∂ 1 X(0)] 1 2πVar[X(0)] exp - (u -E(X(0))) 2 2Var[X(0)] , and
TV(X) = π 2 Var[∂ 1 X(0)].
Note that under the stronger assumption that X is a.s. C 2 the previous stated equality will hold for any level u ∈ R. We refer to Theorem 6.8 of [START_REF] Azaïs | Level Sets and Extrema of Random Processes and Fields[END_REF] for instance.

2.3.

Isotropic smooth shot noise fields. Considering now a shot noise field X given by (1), with a radial kernel g ∈ C 2 (R 2 ), we can also go ahead in computations. We recall the notation g m = mg and introduce a random variable M of distribution F . For p > 0, we denote [START_REF] Berzin | Estimation of local anisotropy based on level sets[END_REF] µ p (g M ) :=

R 2 R |g m (x)| p F (dm)dx = E(|M | p ) R 2 |g(x)| p dx.
For g ∈ C 2 (R 2 ) and i = 1, 2 we note the partial derivative ∂ i g := ∂g ∂xi and for j = (j 1 , j 2 ) ∈ Z 2 , |j| ≤ 2, we note also D j g :=

∂ |j| g ∂ j 1 x1∂ j 2 x2 such that D (1,0) g = ∂ 1 g.
Actually, under the assumption that µ 1 (D j g M ) < +∞ for all |j| ≤ 2, the shot noise field X is a.s a stationary isotropic C 1 random field and the characteristic function of (X(0),

∂ 1 X(0)) is explicitly given for (ν, v) ∈ R 2 by (9) ϕ(ν, v) = exp λ R 2 R [e iνmg(x)+ivm∂1g(x) -1]F (dm)dx . Hence we get ∂ϕ ∂v (ν, v) = -S 0 (ν, v)ϕ(ν, v), with S 0 (ν, v) = -iλ R 2 R m∂ 1 g(x)e iνmg(x)+ivm∂1g(x) F (dm)dx = λ R 2 R m∂ 1 g(x)e iνmg(x) sin (vm∂ 1 g(x)) F (dm)dx,
and, assuming moreover that µ 2 (D j g M ) < +∞ for all |j| ≤ 2, , by Proposition 1 with ε = 1 (see also Theorem 3 of [START_REF] Biermé | Mean Geometry for 2D random fields: Level perimeter and level total curvature integrals[END_REF]), we obtain

∀ν ∈ R, LP X (h ν ) = ∞ 0 1 v ϕ(ν, v)S 0 (ν, v)dv. ( 10
)
Lemma 1. Under the assumption that g is a C 1 radial kernel and that µ 1 (∂ 1 g M ) < +∞ and

µ 2 (∂ 1 g M ) < +∞, we get ∞ 0 1 v S 0 (ν, v)dv = λ π 2 R 2 R |m∂ 1 g(x)|e iνmg(x) F (dm)dx.
Proof. We follow the proof of Proposition 2 in [START_REF] Biermé | Mean Geometry for 2D random fields: Level perimeter and level total curvature integrals[END_REF]. Let V > 0 and recall that for y ∈ R we have

∞ 0 sin(vy) v dv := lim V →+∞ V 0 sin(vy) v dv = π 2 sgn(y),
and

C := sup V >0 V 0 sin(v)
v dv < +∞. Hence we obtain that for a.e x ∈ R 2 and F a.e. m ∈ R,

m∂ 1 g(x)e iνmg(x) V 0 sin (vm∂ 1 g(x)) v dv -→ V →+∞ π 2 |m∂ 1 g(x)|e iνmg(x) .
But for all V > 0

m∂ 1 g(x)e iνmg(x) V 0 sin (vm∂ 1 g(x)) v dv ≤ C |m∂ 1 g(x)| , with R 2 R |m∂ 1 g(x)| F (dm)dx < +∞.
By Lebesgue's theorem we can conclude that lim

V →+∞ R 2 R m∂ 1 g(x)e iνmg(x) V 0 sin (vm∂ 1 g(x)) v dvF (dm)dx = π 2 R 2 R |m∂ 1 g(x)|e iνmg(x) F (dm)dx.
Now it remains to justify that for all V > 0,

V 0 1 v S 0 (ν, v)dv = R 2 R m∂ 1 g(x)e iνmg(x) V 0 sin (vm∂ 1 g(x)) v dvF (dm)dx.
But it simply follows from Fubini's theorem once remarked that m∂ 1 g(x)e iνmg(x) sin (vm∂

1 g(x)) v ≤ |m∂ 1 g(x)| 2 .
2.4. Asymptotics for high and small λ. In this section we investigate asymptotics of the level perimeter function of shot noise random fields with respect to λ. In the sequel we denote X λ the shot noise field X given by (1) for an homogeneous Poisson point process of intensity λ > 0.

We assume that g is a C 2 radial kernel such that µ 1 (D j g M ) < +∞ and µ 2 (D j g M ) < +∞ for all |j| ≤ 2. Since µ 2 (g M ) < +∞, we let B λ be a stationary Gaussian field with the same second order statistics than X λ meaning that for all y, y ∈ R 2 [START_REF] Biermé | Mean Geometry for 2D random fields: Level perimeter and level total curvature integrals[END_REF] using Campbell's Theorem [START_REF] Kingman | Completely Random Measures[END_REF] or [START_REF] Chiu | Stochastic Geometry and Its Applications[END_REF].

E(B λ (y)) = E(X λ (y)) = λ R 2 R g m (-x)dxF (dm) = λE(M ) R 2 g(x)dx, and
Cov(B λ (y), B λ (y )) = Cov(X λ (y), X λ (y )) = λ R 2 R g m (y -x)g m (y -x)F (dm)dx = λE(M 2 ) R 2 g(x)g(x -(y -y ))dx,
Under our assumptions, up to choose a modification, we can assume that both B λ and X λ are a.s. C 1 . It follows that ∂ 1 B λ and ∂ 1 X λ are centered, stationary and

Cov(∂ 1 B λ (y), ∂ 1 B λ (y )) = Cov(∂ 1 X λ (y), ∂ 1 X λ (y )) = R 2 R ∂ 1 g m (y -x)∂ 1 g m (y -x)F (dm)dx = λE(M 2 ) R 2 ∂ 1 g(x)∂ 1 g(x -(y -y ))dx, Moreover, writing B(y) = B λ (y)-E(B λ (y)) √ λ
, and Z λ (y) = X λ (y)-E(X λ (y))

√ λ

, as λ → +∞, we obtain as in [START_REF] Heinrich | Normal Convergence of Multidimensional Shot Noise and Rates of This Convergence[END_REF],

(Z λ (y), ∂ 1 Z λ (y)) d -→ λ→+∞ (B(y), ∂ 1 B(y)) .
Note that this asymptotic normality for high intensity can be strenghen with an invariance principle and even a coupling between Z λ and B for a strong invariance principle as in [START_REF] Lachièze-Rey | Asymptotics for the critical level and a strong invariance principle for high intensity shot noise fields[END_REF]. Hence it is not surprising to observe a Gaussian behavior for geometry of excursion sets in high intensity. This is the object of our first result, which proof is postponed to Appendix 6.1.

Theorem 1 (High regime). Let us assume that g is a C 2 radial kernel and that µ p (D j g M ) < +∞ for all |j| ≤ 2 and 1 ≤ p ≤ 4. Then, there exists

C + 0 , C 1 + and C 2 + positive constants such that for |ν| ≤ C 0 + LP X λ (h ν ) -LP B λ (h ν ) = LP X λ (h ν ) - √ λϕ B λ (ν) π 2 µ 2 (∂ 1 g M ) ≤ C + 1 + (1 -δ 0 (ν))C + 2
, where

ϕ B λ (ν) = exp iνλE(M ) R 2 g(x)dx - λµ 2 (g M )ν 2 2 ,
δ 0 (ν) = 1 if ν = 0 and δ 0 (ν) = 0 otherwise and

C + 0 ≤ 1 6 min µ 4 (g M ) µ 2 (g M ) 1/2 , 5µ 2 (∂ 1 g M ) µ 4 (g M ) 1/2 µ 4 (∂ 1 g M ) 1/2 C + 1 ≤ 10 µ 4 (∂ 1 g M ) µ 2 (∂ 1 g M ) 1/2 C + 2 ≤ √ π 3µ 4 (g M ) 1/2 µ 2 (∂ 1 g M ) 1/2 µ 2 (g M ) + µ 4 (∂ 1 g M ) µ 2 (∂ 1 g M ) 1/2
.

Observe the interest of the result since ∀ν ∈ R the two quantities LP X λ (h ν ) and LP B λ (h ν ) tend to infinity when λ → ∞. Note also that, using

LP Z λ (h ν ) = e -iνE(X λ (x))/ √ λ √ λ LP X λ (h ν/ √ λ ) and LP B (h ν ) = e -iνE(X λ (x))/ √ λ √ λ LP B λ (h ν/ √ λ ),
our result extends Theorem 4 of [START_REF] Biermé | On the Perimeter of Excursion Sets of Shot Noise Random Fields[END_REF] for the normalized shot noise field : for any ν ∈ R,

LP Z λ (h ν ) -→ λ→+∞ LP B (h ν ).
This implies the following weak convergence for u ∈ R,

Per(E X λ (E(X λ (x)) + u √ λ)) → λ→+∞ Per(E B (u)),
as well as (taking ν = 0)

TV(X λ ) √ λ → λ→+∞ TV(B) = π 2 µ 2 (∂ 1 g M ).
But Theorem 1 actually gives a rate for convergence, following ideas in [START_REF] Biermé | Crossings of smooth shot noise processes[END_REF], since

TV(X λ ) √ λ - π 2 µ 2 (∂ 1 g M ) = TV(X λ ) √ λ - TV(B λ ) √ λ ≤ C + 1 √ λ .
More surprisingly we observe an other behavior for small values of λ. This is the object of the following theorem which proof is also postponed to Appendix Section 6.2.

Theorem 2 (Low regime). Let us assume that g is a C 2 radial kernel and that µ p (D j g M ) < +∞ for all |j| ≤ 2 and 1 ≤ p ≤ 2. Assume moreover that there exists β ∈ (0, 1)

such that µ β (∂ 1 g M ) < +∞, then, for C - 0 > 0 and λ ≤ C - 0 , there exists C - 1 > 0 such that LP X λ (h ν ) -λϕ X λ (ν) R LP gm (h ν , R 2 )F (dm) ≤ C - 1 λ 2 1+β , with ϕ X λ (ν) = exp λ R 2 R e iνgm(x) -1 F (dm)dx , LP gm (h ν , R 2 ) := R h ν (u)Per(E gm (u), R 2 )du = π 2 R 2 |∂ 1 g m (x)|e iνgm(x) dx, and
C - 1 ≤ β -1 C - β µ 1 (∂ 1 g M ) exp(C - β C - 0 1-β 1+β ) + 2 + 2C - β C - 0 1-β 1+β , for C - β = 2 1-β µ β (∂ 1 g M ). A particular consequence of this result is that, in view of Lemma 1, taking ν = 0 we get TV(X λ ) λ - π 2 µ 1 (∂ 1 g M ) ≤ C - 1 λ 1-β 1+β ,
and therefore

TV(X λ ) λ → λ→0 π 2 µ 1 (∂ 1 g M ).
The next section is devoted to a special kernel g for which we will be able to identify this limit.

Gaussian shot noise random field and previous results

From now on, we consider for λ, µ, σ ∈ (0, +∞), the Gaussian shot noise random field with exponential marks defined by

(12) X λ,µ,σ (x) = i g mi,σ (x -x i ),
with (x i ) i an homogeneous Poisson point process of intensity λ in R 2 , the (m i ) i are independent marks of exponential law of parameter µ and g m,σ = mg σ with the Gaussian density g σ (13)

g σ (x) = 1 2πσ 2 exp - x 2 2σ 2 .
Let us emphasize that, g σ is of course a radial C ∞ kernel and moreover, if M denotes a random variable with exponential law of parameter µ whose distribution is denoted by F , then for any p > 0

µ p (g M,σ ) = E(M p ) 1 p(2πσ 2 ) p-1 = Γ(p + 1) µ p 1 p(2πσ 2 ) p-1 = Γ(p + 1) p(2π) p-1 σ 2-2p µ p , while µ p (∂ 1 g M,σ ) = E(M p ) 1 (2πσ 4 ) p R |x 1 | p e -px 2 1 2σ 2 dx 1 R e -px 2 2 2σ 2 dx 2 = E(M p ) 1 (2πσ 4 ) p × Γ p + 1 2 2σ 2 p p+1 2 × 2π σ 2 p = 2 1-p/2 Γ(p + 1)Γ p+1 2 p 1+p/2 π p-1/2 σ 2-3p µ p .
Hence, the model satisfies all the assumptions of the previous section. Moreover, in [START_REF] Biermé | Mean Geometry for 2D random fields: Level perimeter and level total curvature integrals[END_REF], the characteristic function of (X λ,µ,σ (x),

∂ 1 X λ,µ,σ (x)) is explicitly given for (ν, v) ∈ R 2 by ϕ(ν, v) = E[e iνX λ,µ,σ (x)+iv∂1X λ,µ,σ (x) ] = exp   λ ∞ 0 2π 0 ir(νσ 2 -vr cos(θ))e -r 2 /2σ 2 2πσ 4 µ -i(νσ 2 -vr cos(θ))e -r 2 /2σ 2 dθdr   (14)
and in particular, taking v = 0, this shows that X λ,µ,σ (x) follows a Gamma law of parameters 2πµσ 2 and 2πλσ 2 . As a consequence we simply have

E(X λ,µ,σ (x)) = λ µ and Var(X λ,µ,σ (x)) = λµ 2 (g M,σ ) with µ 2 (g M,σ ) = 1 2πσ 2 µ 2 .
We can also simply compute the covariance function in [START_REF] Biermé | Mean Geometry for 2D random fields: Level perimeter and level total curvature integrals[END_REF], that remains a Gaussian function, namely

Cov(X λ,µ,σ (x), X λ,µ,σ (y)) = λ 2πσ 2 µ 2 exp -||x -y|| 2 4σ 2 .
Finally, in view of Equation ( 10), we have also

S 0 (ν, v) = iλ ∞ 0 2π 0 2πσ 4 µr 2 cos(θ)e -r 2 /2σ 2 2πσ 4 µ -i(νσ 2 -vr cos(θ))e -r 2 /2σ 2 2 dθdr ( 15
)
So we can numerically approach ν → LP X λ,µ,σ (h ν ) and a Fourier inversion allows us to obtain the desired function u → Per(E X (u)). We refer to Appendix Section 6.3 for algorithm and illustration.

The model satisfies several invariances as stated in the next proposition.

Proposition 2. Let κ > 0 and k > 0. Then we have

(X λ,µ,σ (x), ∂ 1 X λ,µ,σ (x)) | x ∈ R 2 f dd = X λk 2 ,µk 2 , σ k x k , 1 k ∂ 1 X λk 2 ,µk 2 , σ k x k | x ∈ R 2 ,
and

{(X λ,µ,σ (x), ∂ 1 X λ,µ,σ (x)) | x ∈ R 2 } f dd = 1 κ X λ, µ κ ,σ (x), ∂ 1 X λ, µ κ ,σ (x) | x ∈ R 2 .
Proof. Let n ∈ N, x 1 , . . . , x n ∈ R 2 and u 1,1 , u 1,2 , . . . , u n,1 , u n,2 ∈ R. For x ∈ R 2 we note x (1) and x (2) its spatial coordinates. By Campbell's Theorem [START_REF] Kingman | Completely Random Measures[END_REF] we have that

-log E exp i n l=1 u l,1 X λk 2 ,µk 2 , σ k ( x l k ) + u l,2 k ∂ 1 X λk 2 ,µk 2 , σ k ( x l k ) = R 2 R+ 1 -exp i n l=1 u l,1 mk 2 2πσ 2 e -k 2 || x l k -c|| 2 2σ 2 + u l,2 mk 4 x (1) l 2πk 2 σ 4 e -k 2 || x l k -c|| 2 2σ 2 λk 2 µk 2 e -µk 2 m dmdc = R 2 R+ 1 -exp i n l=1 u l,1 m 2πσ 2 e -||x l -c || 2 2σ 2 + u l,2 m x (1) l 2πσ 2 e -||x l -c || 2 2σ 2 λµe -µm dm dc = -log E exp i n l=1 u l,1 X λ,µ,σ (x l ) + u l,2 ∂ 1 X λ,µ,σ (x l )
using the change of variables c = kc et m = k 2 m, which proves the first equality. The same arguments allow us to check the second equality, observing that

R 2 R+ 1 -exp i n l=1 u l,1 m 2πκσ 2 e -||x l -c|| 2 2σ 2 + u l,2 mx (1) l 2πκσ 4 e -||x l -c|| 2 2σ 2 λ µ κ e -µm κ dmdc = R 2 R+ 1 -exp i n l=1 u l,1 m 2πσ 2 e -||x l -c|| 2 2σ 2 + u l,2 m x (1) l 2πσ 4 e -||x l -c|| 2 2σ 2
λµe -µm dm dc using m = m κ .

Note that introducing the Gaussian random field B λ,µ,σ sharing the same second order statistics than X λ,µ,σ , it will also verify the previous invariances. Moreover, since µ

1 (g M,σ ) = 1 µ , µ 2 (g M,σ ) = 1 2πσ 2 µ 2 and µ 2 (∂ 1 g M,σ ) = 1 4πσ 4 µ 2 ,
we can compute by ( 7)

LP B λ,µ,σ (h ν ) = 1 2σ 2 µ λ 2 e i λ µ ν exp - ν 2 2 λ 2πσ 2 µ 2 .
and we can restate Theorem 1 in terms of resolution σ. Proposition 3. There exist c 0 (µ), c 1 (µ) > 0 such that for |ν| ≤ σc 0 (µ), one has

LP X λ,µ,σ (h ν ) -LP B λ,µ,σ (h ν ) ≤ c 1 (µ) σ 3 . Proof. Using Proposition 2 and (4), it follows that LP X λ,µ,σ (h ν ) = 1 σ LP X λσ 2 ,µσ 2 ,1 (h ν ) = 1 σ 3 LP X λσ 2 ,µ,1 (h ν/σ ),
and similarly for the Gaussian field. We can take c 0 (µ

) = C + 0 and c 1 (µ) = C + 1 + C + 2 for M ∼ E(µ) and σ = 1, according to Theorem 1, to get for |ν/σ| ≤ c 0 (µ) LP X λσ 2 ,µ,1 (h ν/σ ) -LP B λσ 2 ,µ,1 (h ν/σ ) ≤ c 1 (µ),
that gives the result.

Theorem 1 tells us that for µ > 0 and σ > 0 fixed, we have that LP X λ,µ,σ (h ν ) is close to LP B λ,µ,σ (h ν ) and grows in high intensity. The previous proposition allows us to interpret this result in high resolution. For λ > 0 and µ > 0 fixed, we have that LP X λ,µ,σ (h ν ) is close to LP B λ,µ,σ (h ν ), decreases in high resolution (σ → ∞) and the difference between it tends to 0 faster (in 1/σ 3 ) than their decreases (in 1/σ 2 ).

Elementary approximation

4.1. The elementary Gaussian function. The idea of this paper is to discretize the Gaussian kernel to use the other framework of elementary functions in the sense of [START_REF] Biermé | Mean Geometry for 2D random fields: Level perimeter and level total curvature integrals[END_REF]. To take into account that the range of g σ depends of σ, we write k σ (x) = e -x 2 /2σ 2 . The range of k σ is now (0, 1], independently from parameters.

For n ≥ 1 we denote the elementary Gaussian function by

k (n-1) σ (x) = 1 n n-1 i=1 1 I B(0,r (n) i ) (x), for r (n) i = σ -2 log(i/n) in such a way that (r (n) i ) 1≤i≤n-1 is strictly decreasing with r (n) n
= 0 and that when x ∈ B(0, r

(n) i ) B(0, r (n) i+1 ) one has i n ≤ k σ (x) < i+1 n and k (n) σ (x) = i/n such that k σ -k (n) σ ∞ = 1/n. The function k (n)
σ is an elementary function with discontinuity points given by its regular points

R k (n) σ = n-1 ∪ i=1 ∂B(0, r (n) i ),
meaning that it is piecewize constant with a regular discontinuity set (see Definition 5 in [START_REF] Biermé | Mean Geometry for 2D random fields: Level perimeter and level total curvature integrals[END_REF]). It also implies that k

(n)
σ is a special bounded variation function [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF]. An illustration is given in Figure 1.

As a discretization of the Gaussian kernel we have the following asymptotics in the Lebesgue L p (R 2 ) spaces for > 0.

Lemma 2. Let σ > 0. Then for all p > 0, we have we have

||k σ -k (n) σ || p -→ n→∞ 0, where || • || p is the classical norm (p > 1) or quasi-norm (0 < p < 1) in L p (R 2 ).
||k σ -k (n) σ || p p = R 2 k σ (x) -k (n) σ (x) p dx = n-1 i=1 B(0,r (n) i ) B(0,r (n) i+1 ) k σ (x) -k (n) σ (x) p dx + B(0,r (n) 1 ) c k σ (x) p dx ≤ n-1 i=1 1 n p L B(0, r (n) i ) B(0, r (n) i+1 ) + B(0,r (n) 1 ) c k σ (x) p dx. But for all 1 ≤ i ≤ n -1 we have L B(0, r (n) i ) B(0, r (n) i+1 ) = π(r (n) i ) 2 -π(r (n) i+1 ) 2 = πσ 2 -2 ln i n + 2 ln i + 1 n = 2πσ 2 (ln(i + 1) -ln(i)) .
Therefore

||k σ -k (n) σ || p p ≤ 2πσ 2 n p ln(n) + B(0,r (n) 1 ) c k σ (x) p dx.
Since p > 0 and r

(n) 1 -→ n→∞ +∞ we get ||k σ -k (n) σ || p p -→ n→∞ 0.
Now let us remark that both k σ and k

(n)
σ are functions of SBV(R 2 ), the space of special functions of bounded variation as defined in [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF] with total variation given by

TV(k σ , R 2 ) = R 2 ∇k σ (x) dx and TV(k (n) σ , R 2 ) = n-1 i=1 1 n H 1 (B(0, r (n) i )) = 2πσ n n-1 i=1 -2 ln i n .
Then we can introduce the level perimeter integrals LP kσ (•, R 2 ) and LP

k (n) σ (•, R 2 ) that define continuous linear form on (C b (R), • ∞ ). Proposition 4. Let σ > 0. Then TV(k σ , R 2 ) = π √ 2πσ
and for all n ∈ N,

TV(k (n) σ , R 2 ) ≤ TV(k σ , R 2 ). Moreover, for any h ∈ C b (R), one has LP k (n) σ (h, R 2 ) -→ n→∞ LP kσ (h, R 2 ),
where

LP kσ (h, R 2 ) = 1 0 h(s)2πσ -2 log(s)ds.
Proof. We use the general co-area formula obtained in Theorem 1 of [START_REF] Biermé | On the Perimeter of Excursion Sets of Shot Noise Random Fields[END_REF]. Let h ∈ C b (R). It follows that on the one hand,

LP kσ (h, R 2 ) := R h(u)Per(E kσ (u), R 2 )du = R 2 h(k σ (x)) ∇k σ (x) dx,
from which we can compute by a change of variables in polar coordinates that [START_REF] Cao | The geometry of correlation fields with an application to functional connectivity of the brain[END_REF] LP kσ (h, R 2 ) = 2π

+∞ 0 h e -r 2 2σ 2
r 2 σ 2 e -r 2 2σ 2 dr.

Therefore, with the change of variables s = e -r 2 2σ 2 we get

LP kσ (h, R 2 ) = 1 0 h(s)2πσ -2 log(s)ds.
Note that we could also just remark that E kσ (u) = B(0, σ -2 ln(u)) and Per(E kσ (u), R 2 ) = 2πσ -2 ln(u), for any u ∈ (0, 1]. From ( 16) we obtain for h = 1 that TV(k σ , R 2 ) = π √ 2πσ. On the other hand for any n ≥ 1,

LP k (n) σ (h, R 2 ) := R h(u)Per(E k (n) σ (u), R 2 )du = n-1 i=1 ∂B(0,r (n) i ) i/n (i-1)/n h(s)dsH 1 (dx). Therefore, LP k (n) σ (h, R 2 ) = n-1 i=1 2πr (n) i i/n (i-1)/n h(s)ds.
Let us note f (s) = -2 log(s). This function is well defined on ]0, 1], non-negative and strictly decreasing. Moreover we have for all n ∈ N *

1/n 0 f (s)ds = 2 log(n) n + +∞ √ 2 log(n) e -u 2 /2 du ≤ 2 log(n) n + 1 n 2 log(n) ,
using the inequalities on the tail distribution of the Gaussian (see p175 of [START_REF] Feller | An Introduction to Probability Theory and Its Applications[END_REF]). So we have

LP kσ (h, R 2 ) -LP k (n) σ (h, R 2 ) = 1 0 h(s)2πσ -2 log(s)ds - n-1 i=1 2πσ -2 log(i/n) i/n (i-1)/n h(s)ds ≤ 2πσ||h|| ∞ (A n + B n + C n ) with A n = 1/n 0 -2 log(s) --2 log(1/n) ds ≤ 1/n 0 -2 log(s)ds ≤ 2 2 log(n) n , B n = n-1 i=2 i/n i-1 n -2 log(s) --2 log(i/n) ds ≤ n-1 i=2 i/n i-1 n -2 log i -1 n --2 log(i/n) ds = 1 n n-1 i=2 -2 log i -1 n --2 log(i/n) = 1 n -2 log(1/n) --2 log n -1 n ,
and

C n = 1 n-1 n -2 log(s) ds ≤ 1 n -2 log n -1 n .
Hence,

LP kσ (h, R 2 ) -LP k (n) σ (h, R 2 ) ≤ 2πσ||h|| ∞ 3 2 log(n) n + -→ n→∞ 0
Finally, the uniform upper bound for TV(k

(n)
σ , R 2 ) can be deduced by taking h = 1 or by remarking that, for all u ∈ R,

Per(E k (n) σ (u), R 2 ) ≤ Per(E kσ (u), R 2 ).
Let us conclude this part by remarking that for k m,σ = mk σ we have for any level u ∈ R,

Per(E km,σ (u), R 2 ) = Per E kσ u m , R 2 .
Defining similarly for n ∈ N and m > 0, ( 17)

k (n) m,σ = mk (n)
σ , we can state the following corollary.

Corollary 2. Let σ > 0 and m > 0. Then TV(k m,σ , R 2 ) = mTV(k σ , R 2 ) = mπ √ 2πσ and for all n ≥ 1, TV(k (n) m,σ , R 2 ) ≤ TV(k m,σ , R 2 ). Moreover, for any h ∈ C b (R), one has LP k (n) m,σ (h, R 2 ) -→ n→∞ LP km,σ (h, R 2 ),
where

LP km,σ (h, R 2 ) = 1 0 mh(ms)2πσ -2 log(s)ds.

Proof. It simply follows from the fact that for

h ∈ C b (R), LP km,σ (h, R 2 ) = R h(u)Per(E km,σ (u), R 2 )du = R h(u)Per(E kσ (u/m), R 2 )du = R mh(mv)Per(E kσ (v), R 2 )dv, by a change of variable. Hence, denoting [h] m := mh(m•) we get LP km,σ (h, R 2 ) = LP kσ ([h] m , R 2 ) and similarly LP (n) km,σ (h, R 2 ) = LP (n) kσ ([h] m , R 2 ).
4.2. The elementary Gaussian shot noise random field. Let λ, µ, σ ∈ (0, +∞) and X λ,µ,σ be a Gaussian shot noise random field defined by [START_REF] Biermé | The effect of discretization on the mean geometry of a 2D random field[END_REF] and note that

X λ,µ,σ (x) = i m i 2πσ 2 k σ (x -x i ),
with marks { mi 2πσ 2 } with distribution E(2πσ 2 µ) denoted as F σ . Proposition 5. Let n ≥ 1 and σ > 0. For λ, µ > 0 and {x i } i∈I an homogeneous Poisson point process of intensity λ on R 2 independently marked with {m i } i∈I of exponential distribution F σ of parameter 2πσ 2 µ, the random field

(18) X (n) λ,µ,σ (x) := i m i k (n) σ (x -x i ),
is an elementary field on any open bounded T ⊂ R 2 , in the sense of [START_REF] Biermé | Mean Geometry for 2D random fields: Level perimeter and level total curvature integrals[END_REF]. Moreover for all h ∈ C b (R) one has

LP X (n) λ,µ,σ (h) = λ R LP k (n) m,σ (h X (n) λ,µ,σ (0) , R 2 )F σ (dm), where hX (n) λ,µ,σ (0) (s) = E[h(X (n) λ,µ,σ (0) + s)] for s ∈ R.
Proof. Let us check assumptions of Theorem 5 of [START_REF] Biermé | Mean Geometry for 2D random fields: Level perimeter and level total curvature integrals[END_REF]. Note that k

(n) m,σ = mk (n) σ
is an elementary function with compact support in B(0, r

(n) 1 ) satisfying R R 2 |k (n) m,σ (x)|dxF σ (dm) < +∞, since R |m|F σ (dm) < +∞. Moreover, by Corollary 2 R TV(k (n) m,σ )F σ (dm) ≤ R TV(k m,σ )F σ (dm) = R 2 ∇k σ (x) dx × R |m|F σ (dm) < +∞,
and, using [START_REF] Cao | The geometry of correlation fields with an application to functional connectivity of the brain[END_REF] in Proposition 4,

R LTaC(k (n) m,σ )F σ (dm) ≤ 2π × R |m|F σ (dm) < +∞,
that finishes to prove [START_REF] Bernardino | Statistics for Gaussian random fields with unknown location and scale using Lipschitz-Killing curvatures[END_REF]. Equation ( 22) simply follows from the fact that the supports do not depend on m and again that R |m|F σ (dm). Now, let us note that all discontinuity points are regular points, ensuring (24) while (25) comes from the fact that

(19) H 0 R k (n) m ,σ ∩ τ x R k (n) m,σ ≤ 2(n -1) 2 1 I x ≤2r (n) 1 .
The last point ( 26) is obtained remarking that for almost all x ∈ R 2 the regular sets can not intersect in a tangency position. This allows to obtain the fact that X (n) λ,µ,σ is an elementary field in T according to Theorem 4 of [START_REF] Biermé | Mean Geometry for 2D random fields: Level perimeter and level total curvature integrals[END_REF]. To conclude it only remains to check (31) of Theorem 5, that follows from [START_REF] Cotsakis | Statistical properties of a perimeter estimator for spatial excursions observed over regular grids[END_REF].

We couple this elementary field with our smooth shot noise field by setting [START_REF] Daley | The definition of a multi-dimensional generalization of shot noise[END_REF] X λ,µ,σ (x) :

= i m i k σ (x -x i ),
with the same marked Poisson point process with marks of distribution E(2πσ 2 µ).

Proposition 6. Let λ, µ, σ ∈ (0, +∞), n ∈ N, X

λ,µ,σ given by (18) and X λ,µ,σ defined by [START_REF] Daley | The definition of a multi-dimensional generalization of shot noise[END_REF]. Then, X (n) λ,µ,σ (x) converges towards X λ,µ,σ (x) in L 2 (Ω) (and therefore also in L 1 (Ω)), for all x ∈ R 2 .

Proof. By stationarity we can consider x = 0. It follows that

E[|X λ,µ,σ (0) -X (n) λ,µ,σ (0)| 2 ] = Var X λ,µ,σ (0) -X (n) λ,µ,σ (0) + E X λ,µ,σ (0) -X (n) λ,µ,σ (0) 2 = λE[M 2 σ ]||k σ -k (n) σ || 2 2 + λ 2 E[M σ ] 2 ||k σ -k (n) σ || 2 1
, which tends to 0, according to Lemma 2. Here we introduce M σ a random variable of distribution E(2πσ 2 µ).

Theorem 3. Let λ, µ, σ ∈ (0, +∞), n ∈ N, X (n)
λ,µ,σ given by (18) and X λ,µ,σ defined by [START_REF] Daley | The definition of a multi-dimensional generalization of shot noise[END_REF]. For h ∈ C b (R) and Lipschitz on R, we have

(21) LP X (n) λ,µ,σ (h) -→ n→∞ LP X λ,µ,σ (h) := λ R LP km,σ ( hX λ,µ,σ (0) , R 2 )F σ (dm)
where hX λ,µ,σ (0) (s) = E[h(X λ,µ,σ (0) + s)] for s ∈ R and F σ is the exponential distribution of parameter 2πσ 2 µ.

Proof. According to Proposition 5, we know that

(22) LP X (n) λ,µ,σ (h) = λ R LP k (n) m,σ ( hX (n) λ,µ,σ (0) , R 2 )F σ (dm)
where hX (n) λ,µ,σ (0

) (s) = E[h(X (n)
λ,µ,σ (0) + s)] for s ∈ R. Proof. Let n ≥ 1. It is clear that for h ∈ C b (R), the function hX (n) λ,µ,σ (0) remains bounded and moreover, for all s ∈ R,

| hX (n) λ,µ,σ (0) (s)| ≤ E h(X (n) λ,µ,σ (0) + s) ≤ h ∞ .
Moreover, for s, t ∈ R we have

hX (n) λ,µ,σ (0) (s) -hX (n) λ,µ,σ (0) (t) ≤ E h(X (n) λ,µ,σ (0) + s) -h(X (n) λ,µ,σ (0) + t) ≤ E [Lip(h)|s -t|] = Lip(h)|s -t|,
that concludes the proof.

Now we write for

h ∈ C b (R) and Lipschitz, LP k (n) m,σ ( hX (n) λ,µ,σ (0) , R 2 ) -LP km,σ ( hX λ,µ,σ (0) , R 2 ) ≤ LP k (n) m,σ ( hX (n) λ,µ,σ (0) -hX λ,µ,σ (0) , R 2 ) + LP k (n) m,σ ( hX λ,µ,σ (0) , R 2 ) -LP km,σ ( hX λ,µ,σ (0) , R 2 ) (23)
Let > 0. By Proposition 6, there exists N ∈ N such that ∀n ≥ N we have

E X λ,µ,σ (0) -X (n) λ,µ,σ (0) ≤ 1 + Lip(h) .
Let n ≥ N , then for all u ∈ R,

| hX λ,µ,σ (0) (u) -hX (n) λ,µ,σ (0) (u)| ≤ E h(X λ,µ,σ (0) + u) -h(X (n) λ,µ,σ (0) + u) ≤ Lip(h)E X λ,µ,σ (0) -X (n) λ,µ,σ (0) ≤ . It follows that || hX λ,µ,σ (0) -hX (n) λ,µ,σ (0) || ∞ -→ n→∞ 0.
Thus we have

LP k (n) m,σ ( hX (n) λ,µ,σ (0) -hX λ,µ,σ (0) , R 2 ) ≤ TV(k (n) m,σ , R 2 )|| hX λ,µ,σ (0) -hX (n) λ,µ,σ (0) || ∞ ≤ TV(k m,σ , R 2 )|| hX λ,µ,σ (0) -hX (n) λ,µ,σ (0) || ∞ by Corollary 2. It follows that LP k (n) m,σ ( hX (n) λ,µ,σ (0) -hX λ,µ,σ (0) , R 2 ) -→ n→+∞ 0.
For the second term, we simply use again Corollary 2 with h := hX λ,µ,σ (0) ∈ C b (R) and combining both we can conclude that

LP k (n) m,σ ( hX (n) λ,µ,σ (0) , R 2 ) -→ n→∞ LP km,σ ( hX λ,µ,σ (0) , R 2 ).
Since we also have At that point it is natural to ask if LP X λ,µ,σ (h) = LP X λ,µ,σ (h). Let us look at this with this next corollary.

|LP k (n) m,σ ( hX (n) λ,µ,σ (0) , R 2 )| ≤ h ∞ TV(k (n) m,σ , R 2 ) ≤ |m| h ∞ TV(k σ , R 2 ),
Corollary 3. Let λ, µ, σ ∈ (0, +∞), n ∈ N, X (n) λ,µ,σ given by (18) and X λ,µ,σ defined by [START_REF] Daley | The definition of a multi-dimensional generalization of shot noise[END_REF]. Then we have [START_REF] Ghojogh | Linear and quadratic discriminant analysis: Tutorial[END_REF] TV(X

(n) λ,µ,σ ) -→ n→∞ TV(X λ,µ,σ ) := LP X λ,µ,σ (1) = λ σµ π 2 .
Proof. By taking h constant equal to 1 we also have hX (n) λ,µ,σ (0) constant equal to 1. Then by Theorem 3 we have TV(X

(n) λ,µ,σ ) -→ n→∞ λ R LP km,σ (1, R 2 )F σ (dm) = λ R 1 0 m2πσ -2 log(s)dsF σ (dm) = 2πσλ 2πσ 2 µ 1 0 -2 ln(s)ds = λ σµ π 2 .
The simplicity of this result has the defect of representing only the low "regime" with small λ of Theorem 2 since we can now identify the limit. Proposition 7. Let λ, µ, σ ∈ (0, +∞) and ν ∈ R. Then

LP X λ,µ,σ (h ν ) = λϕ X λ,µ,σ (ν) R LP gm,σ (h ν , R 2 )F (dm).
Proof. Actually, by definition of LP X λ,µ,σ in [START_REF] Bernardino | Statistics for Gaussian random fields with unknown location and scale using Lipschitz-Killing curvatures[END_REF] we have that

LP X λ,µ,σ (h ν ) = λϕ X λ,µ,σ (ν) R LP km,σ (h ν , R 2 )F σ (dm).
Now, by the general co-area formula obtained in Theoreme 1 of [START_REF] Biermé | On the Perimeter of Excursion Sets of Shot Noise Random Fields[END_REF] we have

LP X λ,µ,σ (h ν ) = λϕ X λ,µ,σ (ν) R R 2 h ν (k m,σ (x)) ∇k m,σ (x) dxF σ (dm),
and a change of variable in m allows us to get

LP X λ,µ,σ (h ν ) = λϕ X λ,µ,σ (ν) R R 2 h ν (g m,σ (x)) ∇g m,σ (x) dxF (dm),
where we recognize

LP gm,σ (h ν , R 2 ) = R 2 h ν (g m,σ (x)) ∇g m,σ (x) dx.
Then similarly to what we have done for high regime, we can restate Theorem 2 in terms of resolution σ. Proposition 8. Assuming λσ 2 ≤ c 0 , for all β ∈ (0, 1), there exists c β (µ) > 0 such that for all

ν ∈ R LP X λ,µ,σ (h ν ) -LP X λ,µ,σ (h ν ) ≤ c β (µ)λ 2 1+β σ 1-3β 1+β .
Proof. Using change of variables we can also check that it follows that

LP X λ,µ,σ (h ν ) = 1 σ LP X λσ 2 ,µσ 2 ,1 (h ν ) = 1 σ 3 LP X λσ 2 ,µ,1 (h ν/σ ).
For β ∈ (0, 1), we can take c β (µ) = C -1 for M ∼ E(µ) and σ = 1, according to Theorem 2, to get for all ν ∈ R,

LP X λσ 2 ,µ,1 (h ν/σ ) -LP B λσ 2 ,µ,1 (h ν/σ ) ≤ c β (µ) λσ 2 2 1+β ,
that gives the result.

Hence we introduce the following quantities that will be used for numerical computations.

Proposition 9. Let λ, µ, σ ∈ (0, +∞), and let f λ,µ,σ be the density of a gamma law of parameters 2πµσ 2 and 2πλσ 2 , then we define the mean elementary perimeter function as

(25) Per(E X λ,µ,σ (u)) := λ R 1 0 f λ,µ,σ (u -ms)m2πσ -2 ln(s)dsF σ (dm),
where F σ denotes the exponential distribution of parameter 2πσ 2 µ. Then, for all h ∈ C b (R), one has

LP X λ,µ,σ (h) = R h(u)Per(E X λ,µ,σ (u))du.
Moreover, its Fourier transform is given for all ν ∈ R by

(26) LP X λ,µ,σ (h ν ) = λ 2πµσ 2 2πµσ 2 -iν 2πλσ 2 2πσ 1 0 2πσ 2 µ (iνs -2πσ 2 µ) 2 -2 ln(s)ds Proof. Let h ∈ C b (R). By Theorem 3 and Corollary 2 it follows that LP X λ,µ,σ (h) = λ R LP km,σ ( hX λ,µ,σ (0), R 2 )F σ (dm) = λ R 1 0 E[h(X λ,µ,σ (0) + ms)]m2πσ -2 ln(s)dsF σ (dm) = λ R 1 0 R h(t + ms)f λ,µ,σ (t)m2πσ -2ln(s)dtdsF σ (dm) = λ R 1 0 R h(u)f λ,µ,σ (u -ms)m2πσ -2ln(s)dudsF σ (dm) = R h(u) λ R 1 0 f λ,µ,σ (u -ms)m2πσ -2ln(s)dsF σ (dm) du,
where we recognize Per(E X λ,µ,σ (u)) under integral. Let h ν (s) = e iνs for s, ν ∈ R. By Corollary 2 we have that

LP X λ,µ,σ (h ν ) = λϕ X λ,µ,σ (ν) R 1 0 e iνms m2πσ -2 ln(s)dsF (dm) = λ 2πµσ 2 2πµσ 2 -iν 2πλσ 2 2πσ 1 0 -2 ln(s) R me iνms F (dm)ds = λ 2πµσ 2 2πµσ 2 -iν 2πλσ 2 2πσ 1 0 2πσ 2 µ (iνs -2πσ 2 µ) 2 -2 ln(s)ds
Let us remark that the integral giving u → Per(E X λ,µ,σ (u)) is not easy to compute but can be efficiently numerically approximated as well as its Fourier transform (see Appendix). We strongly believe that the difference between the theoretical perimeter and the elementary perimeter comes from the total variation. That is why, for X a random field and u ∈ R, let us define the normalized perimeter function by

Per * (E X λ,µ,σ (u)) := Per(E X λ,µ,σ (u))
TV(E X λ,µ,σ (u)) ,

and make the following conjecture, motivated by Figure 2.

Conjecture : for almost all u ∈ R we have Per * (E X λ,µ,σ (u)) = Per(E X λ,µ,σ (u))

TV(E X λ,µ,σ (u))

.

Classification between Gaussian and shot noise fields

We have already seen that there is a closed correspondence between Gaussian and shot noise random fields. In practice, this can be used in insurance risk management [START_REF] Klüppelberg | Explosive Poisson shot noise processes with applications to risk reserves[END_REF] or in wireless networks [START_REF] Chan | Calculating the outage probability in a CDMA network with spatial Poisson traffic[END_REF][START_REF] Aljuaid | Investigating the Gaussian convergence of the distribution of the aggregate interference power in large wireless networks[END_REF]. This section has a specific aim : the classification between these two random fields. For (λ, µ, σ) ∈]0, +∞[ 3 , considering a shot noise field X λ,µ,σ (i.e. a shot noise with a Gaussian kernel and exponential mark), by Campbell's Theorem [START_REF] Kingman | Poisson processes, volume 3 of Oxford Studies in Probability[END_REF], the mean is E[X λ,µ,σ (0)] = λ µ and the isotropic covariance function is itself Gaussian, since we have for all r > 0

C X λ,µ,σ (r) = λ 2πσ 2 µ 2 exp(- r 2 4σ 2 ),
where C X λ,µ,σ (r) = Cov(X λ,µ,σ (x), X λ,µ,σ (y)) for all x, y ∈ R 2 with ||x -y|| = r. That is why, the Gaussian random field of interest B λ,µ,σ has the same mean M B λ,µ,σ = λ µ and the same covariance function C B λ,µ,σ (r) = C X λ,µ,σ (r). When there is no ambiguity, we write X = X λ,µ,σ and B = B λ,µ,σ . Conversely, let B a Gaussian random field with Gaussian covariance function, then there exist (λ, µ, σ) ∈]0, +∞[ 3 and a shot noise random field X λ,µ,σ with the same mean, the same variance and the same covariance function than B. Indeed, if we note C B (r) = Var[B(0)]e -r 2 4σ 2 the covariance function and M B = E[B(0)] the mean of the Gaussian random field, the parameters of the corresponding shot noise X λ,µ,σ are

           λ = M 2 B 2πσ 2 Var[B(0)] µ = M B 2πσ 2 Var[B(0)] σ = σ.
Some simulations are presented in Figure 3, 4, 5 to compare a Gaussian field with a shot noise field. Each random field is simulated on a window T = [0, 1000] 2 with the same mean, same variance and same correlation. We have chosen, arbitrarily, the same intensity of the Poisson process λ = 0.001 and the same mark µ = 0.01 for all simulations. Although the intensity parameter λ may seem small, it represents on average 1000 points on T . Also, let us not forget that the parameter of an exponential law is the inverse of its mean, hence a mean of 100 for the mark. The difference between Figures 3,4, 5 is in the σ parameter and note that, similarly to the high intensity convergence [START_REF] Heinrich | Normal Convergence of Multidimensional Shot Noise and Rates of This Convergence[END_REF], we can less and less distinguish the fields with the increase of σ.

We will start by introducing the classification based on the geometries of Gaussian fields. By Corollary 1, since the second spectral moment of B is

λ 2 = Var[∂B(0)] = ∂ 2 1 Cov B (0) = λ 4πµ 2 σ 4 , for u ∈ R we have that (27) Per(E B (u)) = 1 √ 8σ exp -πµ 2 σ 2 (u -λ µ ) 2 λ
and we have also [START_REF] Klüppelberg | Explosive Poisson shot noise processes with applications to risk reserves[END_REF] Area To compute ( 27) and ( 28), it is necessary to estimate the parameters. To do this, we use the method of moments, similar to what we did for the shot noise random fields in [START_REF] Lerbet | Statistical inference on stationary shot noise random fields[END_REF]. More precisely, suppose that we observe our field on a rectangle T ⊂ Z 2 and let us introduce the following moment estimators :

(E B (u)) = ∞ u 1 2πVar[B(0)] exp - (x -λ/µ) 2 2Var[B(0)] dx
M 1 = 1 |T | x∈T X(x) (29)
M 2 = 1 |T | x∈T X(x) 2 (30) h(w) = 1 |T w| x∈T w X(x)X(x + w), (31)
where w ∈ Z 2 and T w = {x ∈ T ; x + w ∈ T }. Since the field is assumed to be isotropic, the estimators of mixed moment of second order h(w) can be grouped according to the norm of w which allows to improve the estimator. That is why we consider, for r ∈ Z + ,

h(r) = 1 2 ( h(re 1 ) + h(re 2 )
where (e 1 , e 2 ) denotes the canonical basis of R 2 .

From these estimators, let us introduce the estimators of variance

V = M 2 -M 1 2
and of the isotropic correlation function ρ(r) = ( h(r)-M1 2 )

V

. Then, to estimate (λ, µ, σ), we can use for any r ∈ Z 2 the estimators :

(32)                      λ r = -2 M 1 2 log( ρ(r)) πr 2 ( M 2 -M 1 2 ) µ r = -2 M 1 log( ρ(r)) πr 2 ( M 2 -M 1 2 ) σ r = -r 2 4 log( ρ(r))
By property of association of a Gaussian field with Gaussian covariance function, they are consistent estimators and asymptotically normal, similar to what we did for the shot noise fields [START_REF] Lerbet | Statistical inference on stationary shot noise random fields[END_REF].

Following this work, we can optimize the choice of r by taking r * = argmin

r∈N * | ρ(r) -exp(- 1
2 )|, the distance where the covariance function decreases fastest. These good estimates of the parameters allow us to compute the theoretical formulas [START_REF] Klüppelberg | Explosive Poisson shot noise processes with applications to risk reserves[END_REF][START_REF] Kingman | Poisson processes, volume 3 of Oxford Studies in Probability[END_REF] with ( λ r * , µ r * , σ r * ) instead of (λ, µ, σ). To stay in the spirit of the article, we are interested in the normalized perimeter function

Per * (E F (u)) = Per(E F (u))
TV(E F (u)) .

We can also estimate these geometries. For f a random field data observed on a rectangle T ⊂ Z 2 , let us introduce u min the minimum of f and u max its maximum. Let val a regular discretization of [u min , u max ] of size lval ≥ 2 and note sval = umax-umin lval-1 the distance between two successive points of val. For all u ∈ val, the area function estimator is

Area(E f (u)) = 1 |T | x∈T 1 f (x)≥u .
To estimate the perimeter function we simply use the estimator Per(E f (u), T ) of [START_REF] Abaach | Testing marginal symmetry of digital noise images through the perimeter of excursion sets[END_REF], also defined as P

f in [START_REF] Cotsakis | Statistical properties of a perimeter estimator for spatial excursions observed over regular grids[END_REF], multiplied by π/4 to correct the bias according to Proposition 4.4 in [START_REF] Biermé | The effect of discretization on the mean geometry of a 2D random field[END_REF] in view of isotropy of fields under study. Then we normalized according to the level to set Per * (E f (u)).

The geometric statistic that will be used as classifier is

d B (f ) = (d B area (f ), d B per (f )) ∈ R 2 with (33) d B area (f ) = sval u∈val (Area(E B (u)) -Area(E f (u))) 2 1/2
the discretized L 2 distance between the emperical area and the theoretical Gaussian area, and

(34) d B per (f ) = sval u∈val (Per * (E B (u)) -Per * (E f (u))) 2 1/2
the discretized L 2 distance between the empirical perimeter and the theoretical normalized Gaussian perimeter.

Our classification approach is that of the supervised learning. To do this, we will use the quadratic discriminant analysis [START_REF] Ghojogh | Linear and quadratic discriminant analysis: Tutorial[END_REF]. To use the classical notation of classification theory, let us note C 1 the class of Gaussian random field with Gaussian covariance and C 2 the class of shot noise random field with Gaussian covariance. We use several simulations of Gaussian fields and shot noise fields to understand the behavior of the two distances ( 33)-( 34) under the two different classes. More precisely, we simulate N ∈ N * independent Gaussian random fields B 1 , . . . , B N in C 1 of same distribution than B to compute

d B (B 1 ), . . . , d B (B N ) ,
the distances under the Gaussian assumption. In the same way, we simulation N independent shot noise random fields X 1 , . . . , X N in C 2 with same second order statistics than B to compute the distances under the shot noise assumption d B (X 1 ), . . . , d B (X N ) . To do this, we will use the quadratic discriminant analysis [START_REF] Ghojogh | Linear and quadratic discriminant analysis: Tutorial[END_REF]. For f a random field data and k = 1, 2, we assume that, conditional on f ∈ C k , d B (f ) corresponds to the realisation of a two-dimensional random variable with Gaussian distribution N (µ k , Σ k ). Since it is accepted that Σ 1 = Σ 2 , the boundary of the classification decision rule is quadratic [START_REF] Ghojogh | Linear and quadratic discriminant analysis: Tutorial[END_REF]. Noting p B k the density of a Gaussian variable N (µ k , Σ k ), the classification rule is given by

f ∈ C 1 if p B 1 (d B (f )) > p B 2 (d B (f )) f ∈ C 2 if p B 1 (d B (f )) < p B 2 (d B (f ))
When making a classification decision, there is always the possibility of making a mistake. For d ∈ R 2 and k = 1, 2, we introduce the score function d → S B k (d) that represents the probability of choosing class k given that f ∈ C k . In the quadratic discriminant analysis, conditionally to the fact that f ∈ C k , the score function

S B k (d B (f )) = p B k (d B (f )) p B 1 (d B (f ))+p B 2 (d B (f ))
. Finally, the error of the classification if f ∈ C k is defined by 1 minus the score S B k (d B (f )).

For k = 1, 2, to compute the density p B k , it only remains to estimate the mean by ( 35)

µ k = 1 N N i=1 d B area (f i ), d B per (f i ) ,
and the covariance matrix by ( 36)

Σ k = 1 N N i=1 (d B area (f i ) -µ k (1)) 2 (d B area (B i ) -µ k (1))(d B per (f i ) -µ k (2)) (d B area (f i ) -µ k (1))(d B per (f i ) -µ k (2)) (d B per (f i ) -µ k (2)) 2
, where f 1 , . . . , f N are iid samples in C k . )) with 1 ≤ j ≤ M and k = 1, 2, where (d B (f 1,j )) j are the distances obtained for M = 50 Gaussian random fields data (f 1,j ) j and (d B (f 2,j )) j are the distances obtained for M = 50 shot noise random fields (f 2,j ) j . We would like to draw your attention in order not to confuse the role of M and N . For each of the M random fields, we simulate N Gaussian fields and N shot noise fields, which constitute the training set, using the parameters (λ, µ, σ) estimated from the initial field. We observe that the classification of Gaussian fields is very successful with 50/50 good results and a very high score with a average of about 0.95 and a standard deviation of about 0.07. The classifications of shot noise fields is not as successfull since only 25/50 gave the results and with a bad score with a average of about 0.49 and a standard deviation of about 0.24. Now, thanks to the study of the perimeter of Gaussian shot noise random fields, we can make exactly the same classification method, but this time based on shot noise geometries. Once the model parameters (λ, µ, σ) have been estimated by the method of moments presented in [START_REF] Lerbet | Statistical inference on stationary shot noise random fields[END_REF], we can compute the theoretical volume easily by

Area(E X (u)) = ∞ u x 2πλσ 2 -1 e -x2πµσ 2 (2πµσ 2 ) 2πλσ 2 Γ(2πλσ 2 ) dx,
given that X(0) is gamma distributed.

For the theoretical perimeter function, we approximate it by Algorithm 2 or 3. Figure 8 is the theoretical perimeter of a Gaussian and shot noise field, by increasing the values of σ and with the same values of λ and µ. Let us remember that, in view of Proposition 3, that it is consistent to observe a convergence of the curves with the increase of σ.

So, for f a random field data, the geometric statistic that will be used as classifier is d X (f ) = (D X area (f ), D X per (f )) defined by [START_REF] Lerbet | Statistical inference on stationary shot noise random fields[END_REF][START_REF] Rice | Mathematical analysis of random noise[END_REF] replacing B by X. With the same N Gaussian and N shot As before, we suppose that for f a random field data and k = 1, 2, conditional on the event f ∈ C k , d X (f ) correponds to the realisation of a two-dimensional random variable with Gaussian distribution and we note p X k its density. That is why the classification rule is

f ∈ C 1 if p X 1 (d X (f )) > p X 2 (d X (f )) f ∈ C 2 if p X 1 (d X (f )) < p X 2 (d X (f ))
where for i = 1, 2 the density p X i is compute with the analogue of ( 35) and ( 36) replacing B by X.

This time, we refer to this classification as the shot noise classification, because it is based on the geometrical formulas of the shot noise fields, and performed it on the same sample of M = 50 Gaussian fields (the class C 1 ) and M = 50 shot noise fields (the class C 2 ) than in the Gaussian classification. Figure 9 represents a histogram of the classifications score S X k (d X (f k,j )) with 1 ≤ j ≤ M and k = 1, 2, where (d X (f 1,j )) j are the distances obtained by the shot noise classification for the same M = 50 Gaussian random fields (f 1,j ) j than in the Gaussien classification, and (d X (f 2,j )) j are the distances obtained for the same M = 50 shot noise random fields (f 2,j ) j than in the Gaussien classification. We observe that the classification of Gaussian fields is bad with 26/50 good results and a bad score with a average of about 0.55 and a standard deviation of about 0.24. The classifications of shot noise fields is, this time, very successfull with 50/50 good results and with a high score with a average of about 0.95 and a standard deviation of about 0.06. This highlights the usefulness of the shot noise classification. These two classifications should be used together and when they do not have the same conclusion, the classification with the largest difference between the probability densities is preferred reducing the error as much as possible. In Figure 10 we compare the classification score for the sample of Gaussian fields. For better visibility, we have sorted the scores, so that they are ascending for the shot noise classification. A score below 1/2 means that this is not the choice retained by the classification procedure. We do the same for the 50 shot noise fields in Figure 11, where we have this time sorted the scores to be increasing for the Gaussian classification.

6. Appendix 6.1. Proof of Theorem 1. Recall that ϕ denotes the characteristic function of (X λ (0), ∂ 1 X λ (0)). According to [START_REF] Biermé | On the Perimeter of Excursion Sets of Shot Noise Random Fields[END_REF] we have for ν ∈ R, where the improper integral ∞ 0 is obtained as lim V →+∞ V 0 . Actually, for V > V > 0, integrating by parts,

LP X λ (h ν ) = ∞ 0 1 v ϕ(ν, v)S 0 (ν, v)dv,
V V 1 v ∂ϕ ∂v (ν, v)dv = ϕ(ν, V ) V - ϕ(ν, V ) V + V V ϕ(ν, v) v 2 dv.
Since ϕ is a characteristic function it is bounded by 1 and we can let V tends to ∞ and obtain (37 for l = 1 : n 2 do 3:

) +∞ V 1 v ∂ϕ ∂v dv ≤ 2 V .
Approximate ϕ(ν(k), v(l)) by numerically integration of formula [START_REF] Campbell | Discontinuities in light emission[END_REF] 4:

Approximate S 0 (ν(k), v(l)) by numerically integration of formula [START_REF] Campbell | The study of discontinuous phenomena[END_REF] 5:

end for 6: end for 7: Approximate ν → E[LP X (h ν , T )] from the formula [START_REF] Biermé | On the Perimeter of Excursion Sets of Shot Noise Random Fields[END_REF] 8: Approximate u → E[Per(E X (u), T )] by inverse fourier transform Beyond the choice of precisions n 1 , n 2 and n 3 ; it is the choice of r max , v max andν max that is not easy since they depend on the parameters of the field. The choices of u min and u max can be chosen with the minimum and maximum field. Let σ = 20, µ = 1 100 and λ = 9.5e -4 which corresponds to an average of 1000 points over an observation window [0, 1024] 2 . We choose a good precision n 1 = n 2 = n 3 = 2 10 , and we recommend to choose r max = 200 and v max = 6000. Using the tic toc command from Matlab, the algorithm lasts 1000 seconds or almost 3 hours. It is annoying for applications such as parameter applications or classification. The green curve in Figure 2 comes from this algorithm.

Recall the formula (25) obtained in Proposition 9 :

Per(E X λ,µ,σ (u)) = λ2πσE f λ,µ,σ (u -M σ V )M σ -2ln(V ) , with V , a random variable uniform on (0, 1] and independent from M σ with distribution F (dm), that is exponential distribution of parameter 2πσ 2 µ. Using Monte-Carlo approximation we can approximate the function u → λ2πσE f λ,µ,σ (u -M σ V )M σ -2ln(V ) by λ2πσ N N j f λ,µ,σ (u -M σ,j V j ) -2 ln(V j )M σ,j , choosing (V j , M σ,j ) 1≤j≤N iid of same law than (V, M σ ) and N sufficiently large.

Algorithm 2: Calculate the mean perimeter function from elementary framework with Monte Carlo

Choose n 3 the number of item of the vector u ; Choose u min and u max the minimum and maximum of the vector u ; u = u min + (0 : n 3 -1) umax-umin n3 ; Choose N the number of repeats in the Monte Carlo method ;

1: Draw (V j ) 1≤j≤N iid with uniform law on [0, 1] 2: Draw (M σ,j ) 1≤j≤N iid with exponential law of parameter 2πσ 2 µ 3: Approximate u → E[Per(E X λ,µ,σ (u), T )] by λ2πσ N N j=1 f λ,µ,σ (u -M σ,j V j ) -2 ln(V j )M σ,j

Figure 1 .

 1 Figure 1. Elementary approximation of a Gaussian function

Lemma 3 .

 3 Let h ∈ C b (R) and Lipschitz with Lipschitz constant Lip(h). Then, for all n ≥ 1, we also have hX (n) λ,µ,σ (0) ∈ C b (R) and Lipschitz with hX (n) λ,µ,σ (0) ∞ ≤ h ∞ and Lip hX (n) λ,µ,σ (0) ≤ Lip(h).

  and R |m|F σ (dm) = E(|M σ |) < +∞, we can use Lebesgue's theorem to obtain the result.

Figure 2 .

 2 Figure 2. Perimeter and normalized perimeter of a Gaussian shot noise random field with exponential marks. The paramaters are in the title. In red, the empirical perimeter from a simulation on F = [0, 1000] 2 and in green its theoretical formula u → Per(E X (u)) approximated by Algorithm 1. In blue and yellow, the elementary asymptotic perimeter u → Per(E X (u)) approximated by Algorithm 3 and 2.

Figure 3 .

 3 Figure 3. Gaussian and shot noise random fields simulated on F = [0, 1000] 2 with λ = 0.001, µ = 1 100 and σ = 10.

Figure 4 .

 4 Figure 4. Gaussian and shot noise random fields simulated on F = [0, 1000] 2 with λ = 0.001, µ = 1 100 and σ = 30.

Figure 5 .

 5 Figure 5. Gaussian and shot noise random fields simulated on F = [0, 1000] 2 with λ = 0.001, µ = 1 100 and σ = 50.

  In the classification lexicon, these two samples are called training set. It remains to determine to which sample the observed distance pairs d B (f ) = (d area B (f ), d per B (f )) best belong.

Figure 6 .

 6 Figure 6. Example of a classification, with the two learning samples of size N = 50 and the classification boundary. The initial Gaussien field B was simulated using λ = 0.001, µ = 0.01 and σ = 50.

Figure 6

 6 Figure 6 is an example of a classification where the initial field is Gaussian, and it can be seen that it is well classified in the class of Gaussian fields. We refer to this classification as the Gaussian classification, because it is based on the geometrical formulas of the Gaussian fields, and performed it on a sample of N = 50 Gaussian fields (the class C 1 ) and N = 50 shot noise fields (the class C 2 ).Figure 7 represents a histogram of the scores of the classification S B k (d B (f k,j)) with 1 ≤ j ≤ M and k = 1, 2, where (d B (f 1,j )) j are the distances obtained for M = 50 Gaussian random fields data (f 1,j ) j and (d B (f 2,j )) j are the distances obtained for M = 50 shot noise random fields (f 2,j ) j . We would like to draw your attention in order not to confuse the role of M and N . For each of the M random fields, we simulate N Gaussian fields and N shot noise fields, which constitute the training set, using the parameters (λ, µ, σ) estimated from the initial field. We observe that the classification of Gaussian fields is very successful with 50/50 good results and a very high score with a average of about 0.95 and a standard deviation of about 0.07. The classifications of shot noise fields is not as successfull since only 25/50 gave the results and with a bad score with a average of about 0.49 and a standard deviation of about 0.24.

  Figure 6 is an example of a classification where the initial field is Gaussian, and it can be seen that it is well classified in the class of Gaussian fields. We refer to this classification as the Gaussian classification, because it is based on the geometrical formulas of the Gaussian fields, and performed it on a sample of N = 50 Gaussian fields (the class C 1 ) and N = 50 shot noise fields (the class C 2 ).Figure 7 represents a histogram of the scores of the classification S B k (d B (f k,j)) with 1 ≤ j ≤ M and k = 1, 2, where (d B (f 1,j )) j are the distances obtained for M = 50 Gaussian random fields data (f 1,j ) j and (d B (f 2,j )) j are the distances obtained for M = 50 shot noise random fields (f 2,j ) j . We would like to draw your attention in order not to confuse the role of M and N . For each of the M random fields, we simulate N Gaussian fields and N shot noise fields, which constitute the training set, using the parameters (λ, µ, σ) estimated from the initial field. We observe that the classification of Gaussian fields is very successful with 50/50 good results and a very high score with a average of about 0.95 and a standard deviation of about 0.07. The classifications of shot noise fields is not as successfull since only 25/50 gave the results and with a bad score with a average of about 0.49 and a standard deviation of about 0.24.

Figure 7 .

 7 Figure 7. Results for the Gaussian classification. In red, a histogram of the classifications score S B 1 (d B (f 1,j )) j obtained from a sample of M = 50 Gaussian fields. In blue, the score S B 2 (d B (f 2,j )) j for M = 50 shot noise fields.

Figure 8 .

 8 Figure 8. Gaussian and shot noise perimeter computed with (27) and Algorithm 2. The parameters are λ = 0.001, µ = 1 100 and from top left to bottom right σ = 10, 30, 50, 70.

Figure 9 .

 9 Figure 9. Results for the shot noise classification. In red, a histogram of the classifications score S X 1 (d X (f 1,j )) j obtained from a sample of M = 50 Gaussian fields. In blue, the score S X 2 (d X (f 2,j )) j for a sample of M = 50 shot noise fields. The samples are the same as in Figure 7.

Figure 10 .

 10 Figure 10. The two scores S B 1 (d B (f 1,j )) j and S X 1 (d X (f 1,j )) j for the Gaussian sample.

Figure 11 .

 11 Figure 11. The two scores S B 2 (d B (f 2,j )) j and S X 2 (d X (f 2,j )) j for the Gaussian sample.

2 , 1 v 2 F 2 F 4 m ≤ 6 g 2 m ; • ν 2 g 2 m ∂ 1 g 2 m ≤ 5 6 ( 4 m + 6v 2 ν 2 g 2 m ∂ 1 g 2 m + v 4 ∂ 1 g 4 m≤ 6ν 2 g 2 m + 5v 2 ∂ 1 g 2 m + v 2 ∂ 1 g 2 m≤ 6 [ 1 :-νmax 2 +; 1 :

 212242642244222226121 We introduceψ(ν, v) := exp (iνE(X λ (0))) exp -λµ 2 (g M ) 2 ν 2 exp -λµ 2 (∂ 1 g M ) 2 vthe characteristic function of the Gaussian vector (B λ (0), ∂ 1 B λ (0)) and note that we haveLP B λ (h ν ) = -Moreover introducing S 1 (v) = λµ 2 (∂ 1 g M )v we can write ∂ψ ∂v (ν, v) = -S 1 (v)ψ(ν, v). It follows that LP X λ (h ν ) -LP B λ (h ν ) = |S 0 (ν, v)ϕ(ν, v) -S 1 (ν, v)ψ(ν, v)| + 4 V .Then we write|S 0 (ν, v)ϕ(ν, v) -S 1 (v)ψ(ν, v)| ≤ |S 0 (ν, v)||ϕ(ν, v) -ψ(ν, v)| + |S 0 (ν, v) -S 1 (v)||ψ(ν, v)|.For the first term, we can writeϕ(ν, v) -ψ(ν, v) = ψ(ν, v) [exp (z(ν, v)) -1] ,wherez(ν, v) := λ e i[νgm+v∂1gm] -1 -i[νg m + v∂ 1 g m ] -[νg m + v∂ 1 g m ] 2 (dm)dx,using the fact that, since ∂ 1 g m is an odd function ∂ 1 g m = 0 and g m ∂ 1 g m = 0. Hence, using finite increments inequality,|ϕ(ν, v) -ψ(ν, v)| ≤ |ψ(ν, v)||z(ν, v)| exp( (z(ν, v))),where(z(ν, v)) := λ cos[νg m + v∂ 1 g m ] -1 -[νg m + v∂ 1 g m ] 2 (dm)dx. Since 0 ≤ cos(s)-1+ s 2 2 ≤ 1 4! s 4 for all s ∈ R, we have 0 ≤ (z(ν, v)) ≤ λ 4! [νg m +v∂ 1 g m ] 4 F (dm)dx. Now let us choose V = µ2(∂1gm) µ4(∂1gm) 1/2 and ν with |ν| ≤ C + 0 . Then it follows that • ν 2 g ∂ 1 g m ) 2 ; • V 2 (∂ 1 g m ) 4 ≤ (∂ 1 g m )2 , where we used Cauchy-Schwarz inequality for the second inequality. It follows that for v ≤ V we get[νg m + v∂ 1 g m ] 4 = ν 4 g νg m + v∂ 1 g m ] 2 ,and therefore0 ≤ (z(ν, v)) ≤ λ 4 [νg m + v∂ 1 g m ] 2 F (dm)dx.6.3. Algorithms. Algorithm Calculate the mean perimeter function from smooth frameworkChoose n 1 the number of item of the vector ν ; Choose ν max maximum of the vector ν ; ν = (0 : n 1 -1) νmax n1-1 ; Choose n 2 the number of item of the vector v ; Choose v max the maximum of the vector v ; v = (1 : n 2 ) vmax n2 ; Choose r max the maximum in the integral of ϕ(ν, v) and S 0 (ν, v) ; Choose n 3 the number of item of the vector u ; Choose u min and u max the minimum and maximum of the vector u ; u = u min + (0 : n 3 -1) umax-umin n3 for k = 1 : n 1 do 2:

This work has been supported by the grant ANR-19-CE40-0005.

Hence we have

Moreover we can simply bound

and, since (38) S 0 (ν, v) = λ e iugm ∂ 1 g m sin(v∂ 1 g m )F (dm)dx, we simply bound

It follows that

where c 3 = 2 3 × 2 3 × sup s∈R |s| 3 e -s 2 = 4 √ 6e -3/2 if ν = 0 and c 3 = 0 else. Now let us focus on the second term and recall that S 0 (ν, v) = λ ∂ 1 g m e iνgm sin(v∂ 1 g m )F (dm)dx and S 1 (v) = λv (∂ 1 g m ) 2 F (dm)dx so that we write

It follows that

We get the result of upper bounds for C + 1 and C + 2 , combining previous inequalities with Cauchy-Schwarz inequalities. 6.2. Proof of Theorem 2. Let ν ∈ R and write again

Let V > 0 and remark that after using Fubini's theorem, since µ 2 (∂ 1 g m ) < +∞, we can also integrate by parts to obtain for

It follows choosing β ∈ (0, 1) we can use that 1

letting V tends to infinity we obtain

Now we write

where z(ν, v) := λ e iνgm e iv∂1gm -1 F (dm)dx = -λ e iνgm 2 sin 2 (v∂ 1 g m /2)F (dm)dx.

Hence for β ∈ (0, 2]

We choose

Here there is no choice except for the precision n 3 and the sample size N of the Monte Carlo method, since u min and u max can be chosen experimentally. Let the same example with σ = 20, µ = 1 100 and λ = 9.5e -4 . We choose a good precision n 3 = 2 10 and N = 2000. Using the tic toc command from Matlab, the algorithm lasts 12 seconds. It is an excellent performance for applications.

It is also possible to use the Fourier framework with the equation [START_REF] Kingman | Completely Random Measures[END_REF]. Here there is no choice except for the precision n 1 and n 3 , since ν max , u min and u max can be chosen experimentally. Let the same example with σ = 20, µ = 1 100 and λ = 9.5e -4 . We choose a good precision n 1 = n 3 = 2 10 . Using the tic toc command from Matlab, the algorithm lasts 1 seconds. It is a perfect performance for applications.


Soraya Benguerfi
email: soraya.benguerfi@gmail.com

F Reizine

F Eugène

P Tattevin

A Maamar

Remarkable resolution of COVID-19-associated cerebral vasculitis with methylprednisolone

The present research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Keywords. COVID-19, vasculitis, central nervous system

Although coronavirus disease 2019 (COVID-19) primarily manifests as a respiratory tract disease, reports of neurological complications are increasing [START_REF] Meppiel | Neurologic manifestations associated with COVID-19: a multicentre registry[END_REF][START_REF] Frontera | A Prospective Study of Neurologic Disorders in Hospitalized Patients With COVID-19 in New York City[END_REF]. The pathophysiology of central nervous system (CNS) injuries remains unclear. Data available suggest that endothelial lesions, coagulation disorders, and excessive inflammation are the primary mechanisms responsible for CNS manifestations. SARS-CoV-2 has limited potential for neuroinvasion, and immune complex-mediated mechanisms do not play a major role in the disease pathophysiology [START_REF] Ye | The pathogenesis and treatment of the `Cytokine Storm' in COVID-19[END_REF][START_REF] Solomon | Neurological infection with SARS-CoV-2 -the story so far[END_REF][START_REF] Thakur | COVID-19 neuropathology at Columbia University Irving Medical Center/New York Presbyterian Hospital[END_REF]. As a consequence, the management of COVID-19-related CNS manifestations is poorly defined. We report a case of CNS vasculitis in a critically ill patient with COVID-19, who dramatically improved with high-dose corticosteroids.

A 74-year-old man with no medical history presented with a 6-day history of fever, cough, and confusion. Breath sounds were diminished in the left lung with bilateral basilar crackles.

Neurological examination was normal.

A nasopharyngeal swab for SARS-CoV-2 testing was positive by PCR. Chest computed tomography (CT) showed bilateral diffused ground-glass opacities. Brain CT was normal. respiratory distress syndrome (ARDS). He required invasive mechanical ventilation, sedation with midazolam and morphine for 7 days, neuromuscular blockade for 3 days, preventive anticoagulation with low molecular weight heparin, and antimicrobial treatment with cefotaxime and voriconazole. ARDS resolved, but the patient remained comatose for more than 48 hours following discontinuation of all sedative agents. Contrast-enhanced brain magnetic resonance imaging (MRI) revealed several acute ischemic strokes affecting different vascular territories, right sulcal acute subarachnoid hemorrhage, and multiple bilateral microbleeds (Figure 1), highly suggestive of CNS vasculitis. Cerebrospinal fluid analysis showed white blood cell count of 9/mm 3 and red blood cell count of 905/mm 3 ; glucose level was 4.6 mmol/L (plasma glucose level, 6.2 mmol/L), and protein level was 0.24 g/L. No organism was observed on Gram stain and the culture was negative. PCR tests for SARS-CoV-2, herpes simplex virus, and varicella-zoster virus were negative. There were no clinical arguments for systemic vasculitis. Renal and hepatic functions were preserved. Antinuclear antibodies, antineutrophil cytoplasmic antibodies, rheumatoid factor, angiotensin-converting enzyme, and blood cultures were negative. A slight alteration of lupus anticoagulant was detected but without anti-beta-2-glycoprotein or anti-cardiolipin, suggesting a false positivity not uncommon in critical care patients.

Intravenous methylprednisolone was started on day 14 (1 g/day for 3 days, and then 1 mg/kg/day). Twelve hours after methylprednisolone was initiated, the patient woke up and could be extubated. He was discharged from the ICU on day 24 on oral systemic corticosteroids. Neurological examination returned to normal on day 28 and was still unremarkable after 6 months. recovery of neurological functions following initiation of high-dose corticosteroids. Of note, the patient was admitted before the benefit of dexamethasone was documented by the RECOVERY trial [START_REF] Recovery Collaborative Group | Dexamethasone in Hospitalized Patients with Covid-19[END_REF]. He therefore did not receive any corticosteroids before developing CNS vasculitis. Interestingly, Timmons et al. reported the only case of biopsy-confirmed CNS vasculitis that shortly followed a possible COVID-19 infection [START_REF] Timmons | CNS Lymphocytic Vasculitis in a Young Woman With COVID-19 Infection[END_REF]. However, the authors were unable to confirm the infection because SARS-CoV-2 PCR testing was unavailable during the initial episode. The patient was treated with high-dose intravenous corticosteroids and improved. Several viruses may induce vascular inflammation or occlusion by immune complex-mediated mechanisms [START_REF] Moore | Immune mechanisms in the primary and secondary vasculitides[END_REF]. SARS-CoV-2-mediated disease is largely driven by immunological and inflammation processes, the primary drivers of morbidity and mortality. [START_REF] Ye | The pathogenesis and treatment of the `Cytokine Storm' in COVID-19[END_REF]. In children, SARS-CoV-2 has been associated with Kawasaki-like disease, a multisystem inflammatory syndrome, mostly related to extensive vasculitis of medium-caliber vessels [START_REF] Verdoni | An outbreak of severe Kawasaki-like disease at the Italian epicentre of the SARS-CoV-2 epidemic: an observational cohort study[END_REF].

Although less documented, similar mechanisms may explain CNS vasculitis in adult patients [START_REF] Younger | Central Nervous System Vasculitis due to Infection[END_REF].

In the observation reported herein, introduction of high-dose corticosteroids was followed by remarkable recovery of the neurological function, which strengthens the hypothesis of an immune-mediated mechanism. Furthermore, the temporal sequence in our report and the absence of arguments for systemic vasculitis suggest that COVID-19 was the causal factor of CNS vasculitis. The non-specific neurological manifestations, especially in the ICU setting, highlight the importance of brain imaging to diagnose complications and to select an appropriate treatment. Corticosteroids might be the treatment of choice in patients with COVID-19-associated CNS vasculitis. Physicians should be aware of this severe complication associated with COVID-19 and of the potential benefit of high-dose corticosteroids.

Figure 1 .

 1 Figure 1. Man, 74 years old. Figure 1A, axial susceptibility-weighted imaging: multiple bilateral microbleeds (arrows). Figure 1B, axial susceptibility-weighted imaging: right sulcal acute subarachnoid hemorrhage (arrow).

Funding

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-forprofit sectors.

Availability of data and materials

Not applicable.

Availability of data and materials

Not applicable.

Ethics approval and consent to participate

The patient gave informed consent.

Consent to publish

The patient gave written informed consent for publication of case details and images.

Competing interests

The authors declare that they have no competing interest.

Authors' contributions

Soraya Benguerfi, Florian Reizine, Pierre Tattevin, and Adel Maamar conceived the study.

François Eugène analyzed the MRI results and selected Figure 1. All authors discussed the results and contributed to the final manuscript.

Ethical statement

Ethics approval and consent to participate

The patient gave his informed consent.

Consent to publish

The patient gave his informed consent.

Competing interests

The authors declare that they have no competing interests.


Jean-Pierre Gangneux

Jeff Morcet

Bruno Laviolle

Eric Dannaoui

Jean-François Timsit

Stéphane Ruckly

Jean-Ralph Zahar

Marie-Elisabeth Bougnoux

Time-dependent bias when analysing COVID-19-associated pulmonary aspergillosis -Authors' reply

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

ACCEPTED MANUSCRIPT -CLEAN COPY 1 Time-dependent bias when analysing COVID-19-associated pulmonary aspergillosis

Authors' reply

We thank Justin de Brabander and colleagues for their interest in our findings on COVID-19-associated pulmonary aspergillosis (CAPA) reported from the MYCOVID cohort. They raised a very interesting question on the factor time and competing risks in studying the relation between CAPA and mortality as exemplified by their own cohort.

Initially, we observed a statistically significant association between CAPA occurrence and mortality using a Cox model. Then, we decided to refine the Cox model analysis for mortality estimation with a landmark analysis, considering the mean day of proven or probable CAPA development (11 days in our cohort). 1 But, as suggested by de Brabander and colleagues, we performed a complementary analysis, considering the time-dependent bias due to CAPA occurrence on the estimation of mortality. 2 Austin and colleagues previously defined a competing risk as an event whose occurrence precludes the occurrence of the primary event of interest. 3 Here, the competing states were "time to death" or "discharge", both during intensive care unit (ICU) stay. No follow-up data were collected after ICU discharge as mentioned in the manuscript. 2 Using this model, the cause-specific hazard ratio (CS-HR) at discharge due to CAPA was 0•57 (95% CI 0•38-0•84) and the CS-HR of death due to CAPA was 2•08 (1•47-2•93), compared to 1•45 (1•03-2•03) found with the initial log-rank analysis. 1 Overall, we provide evidence that the magnitude effect of CAPA on mortality when considering the immortal bias and a cause-specific model increased the strength of the association between development of CAPA and mortality in the MYCOVID cohort.