Jorge Clarke De La Cerda

A Ciprian

Tudor
email: tudor@math.univ-lille1.fr

Wiener integrals with respect to the Hermite random field and applications to the wave equation

Keywords: 2000 AMS Classification Numbers: 60F05, 60H05, 60G18 Hermite process, Hermite sheet, Wiener integral, stochastic wave equation

The Hermite random field has been introduced as a limit of some weighted Hermite variations of the fractional Brownian sheet. In this work we define it as a multiple integral with respect to the standard Brownian sheet and introduce Wiener integrals with respect to it. As an application we study the wave equation driven by the Hermite sheet. We prove the existence of the solution and we study the regularity of its sample paths, the existence of the density and of its local times.

Introduction

The random fields or multiparameter stochastic processes have focused a significant amount of attention among scientists due to the wide range of applications that they have.

Particularly, self-similar random fields find some of their applications in various kind of phenomena, going from hydrology and surface modeling to network traffic analysis and mathematical finance, to name a few. From other side, this type of processes are also quite interesting when they appear as solutions to Stochastic Partial Differential Equations (SPDE's) in several dimensions, such as the wave or heat equations.

A class of processes that lies in the family described above are the Hermite random fields or Hermite sheets (from now on). Inside this class we can find the well-studied fractional Brownian sheet and the Rosenblatt processes, among others.

The Hermite processes of order q ≥ 1 are self-similar with stationary increments and live in the qth Wiener chaos, that is, they can be expressed as q times iterated integrals with respect to the Wiener process. The class of Hermite processes includes the fractional Brownian motion (fBm) which is the only Gaussian process in this family. Their practical aspects are striking: they provide a wide class of processes that allow to model long memory, self-similarity and Hölder-regularity, enabling a significant deviation from the fBm and other Gaussian processes. Since they are non-Gaussian and self-similar with stationary increments, the Hermite processes can also be an input in models where self-similarity is observed in empirical data which appears to be non-Gaussian.

The Hermite sheet of order q is only known in his representation as a non-central limit of a particularly normalized Hermite variations of the fractional Brownian sheet, see [START_REF] Réveillac | Hermite variations of the fractional Brownian sheet[END_REF] for the two-parameter case and [START_REF] Breton | On the rate of convergence in non-central asymptotics of the Hermite variations of fractional Brownian sheet[END_REF] for the general d-parametric case. In both cases the authors also prove self-similarity, stationary increments and Hölder continuity.

In the present work we deal directly with the multi-parametric case building the Hermite sheet as a natural extension of the expression for the Hermite process studied as a non-central limit in [START_REF] Dobrushin | Non-central limit theorems for non-linear functionals of Gaussian fields[END_REF] and [START_REF] Taqqu | Convergence of integrated processes of arbitrary Hermite rank[END_REF].

Fix d ∈ N\ {0}, define t = (t 1 , t 2 , . . . , t d ) ∈ R d and let H = (H 1 , H 2 , . . . , H d ) ∈ ( 1 2 , 1) d a Hurst multi-index

Z q H (t) = c(H, q) R d•q t 1 0 . . . t d 0   q j=1
(s 1 -y 1,j ) - 1 2 +

1-H 1 q + . . . (s d -y d,j )

-1 2 + 1-H d q +   ds d . . . ds 1 dW (y 1,1 , . . . , y d,1 ) . . . dW (y 1,q , . . . , y d,q ) = c(H, q) R d•q t 0 q j=1 (s -y j ) -1 2 + 1-H q + ds dW (y 1 ) . . . dW (y q ), (1)
the bold characters are for multidimensional quantities as indicated below in Section 2.

The above integrals are Wiener-Itô multiple integrals of order q with respect to the d-parametric standard Brownian sheet (W (y)) y∈R d (see [START_REF] Nualart | Malliavin Calculus and Related Topics[END_REF] for the definition) and c(H, q) is a positive normalization constant depending only on H and q. We designate the process Z q H (t) t∈R d as the Hermite sheet or Hermite random field.

From expression (1) it is possible to note that for d = 1 we recover the Hermite process which represents a family that has been recently studied in [START_REF] Chronopoulou | Self-similarity parameter estimation and reproduction property for non-Gaussian Hermite processes[END_REF], [START_REF] Maejima | Wiener Integrals with respect to the Hermite process and a Non-Central Limit Theorem[END_REF] and [START_REF] Pipiras | Regularization and integral representations of Hermite processes[END_REF]. As a particular case (q = 1) we recover the most known element of this family, the fractional Brownian motion, which has been largely studied due to its various applications. Recently, a rich theory of stochastic integration with respect to this process has been introduced and stochastic differential equations driven by the fractional Brownian motion have been considered for several purposes. The process obtained in (4) for d = 1, q = 2 is known as the Rosenblatt process, it was introduced by Rosenblatt in [START_REF] Rosenblatt | Independence and dependence[END_REF] and it has been called in this way by Taqqu in [START_REF] Taqqu | Weak convergence to the fractional Brownian motion and to the Rosenblatt process[END_REF]. Lately, this process has been increasingly studied by his different interesting aspects like wavelet type expansion or extremal properties, parameter estimations, discrete approximations and others potential applications (see [START_REF] Albin | A note on the Rosenblatt distributions[END_REF], [START_REF] Albin | On extremal theory for self similar processes[END_REF], [START_REF] Bardet | A wavelet analysis of the Rosenblatt process: chaos expansion and estimation of the self-similarity parameter[END_REF], [START_REF] Chronopoulou | Variations and Hurst index estimation for a Rosenblatt process using longer filters[END_REF], [START_REF] Tudor | Analysis of the Rosenblatt process[END_REF]).

As far as we know, the only well-known multiparameter process that can be obtained from ( 1) is the fractional Brownian sheet (d > 1 and q = 1). This process has been recently studied as a driving noise for stochastic differential equations and stochastic calculus with respect to it has been developed. We refer to [START_REF] Ayache | Drap brownien fractionnaire[END_REF], [START_REF] Kim | Stratonovich calculus with respect to fractional Brownian sheet[END_REF], [START_REF] Tudor | Itô Formula and Local Time for the Fractional Brownian Sheet[END_REF] for only a few works on various aspects of the fractional Brownian sheet.

In one hand the purpose of this article is to study the basic properties of the multiparameter Hermite process and then to introduce Wiener integrals with respect to the Hermite sheet in order to generalize and continue the line introduced in [START_REF] Maejima | Wiener Integrals with respect to the Hermite process and a Non-Central Limit Theorem[END_REF] putting a new brick in the construction of stochastic calculus driven by this class of processes in several dimensions. As in [START_REF] Breton | On the rate of convergence in non-central asymptotics of the Hermite variations of fractional Brownian sheet[END_REF] the covariance structure of the Hermite sheet is like the one of the fractional Brownian sheet, enabling the use of the same classes of deterministic integrands as in the fractional Brownian sheet profiting its well-known properties.

Also in the aim of this work lives the idea of making an approach to the study of stochastic partial differential equations in several dimensions driven by non-Gaussian noises, giving a specific expression for the driving noise allowing to use in a better way the properties of the equations by taking advantage of the results already existing in the literature. It is in this sense that, inspired by the works [START_REF] Balan | The stochastic wave equation with fractional noise: A random field approach[END_REF], [START_REF] De | Hitting times for the stochastic wave equation with fractional-colored noise[END_REF] or [START_REF] Dalang | Criteria for hitting probabilities with applications to systems of stochastic wave equations[END_REF] and exploiting these, we present a stochastic wave equation with respect to the Hermite sheet in spatial dimension d ≥ 1 and we study the existence, regularity, and other properties of the solution, including the existence of local times and of the joint density.

We organize our paper as follows. Section 2 presents the necessary notations and prove several properties of the Hermite sheet. In Section 3, we construct Wiener integrals with respect to this process. Section 4 is devoted to present the wave equation and discuss the existence and regularity of the solution and other properties.

Notation and the Hermite sheet

Throughout the work we use the notation introduced in [START_REF] Breton | On the rate of convergence in non-central asymptotics of the Hermite variations of fractional Brownian sheet[END_REF]. Fix d ∈ N\ {0} and consider multi-parametric processes indexed in R d . We shall use bold notation for multi-indexed quantities, i.e., a = (a 1 , a 2 , . . . , a d ), ab = (a

1 b 1 , a 2 b 2 , . . . , a d b d ), a/b = (a 1 /b 1 , a 2 /b 2 , . . . , a d /b d ), [a, b] = d i=1 [a i , b i ], (a, b) = d i=1 (a i , b i ), i∈[0,N] a i = N 1 i 1 =0 N 2 i 2 =0 . . . N d i d =0 a i 1 ,i 2 ,...,i d , a b = d i=1 a b i
i , and

a < b iff a 1 < b 1 , a 2 < b 2 , . . . ,
R H (s, t) = E[B H s B H t ] = d i=1 R H i (s i , t i ) = d i=1 s 2H i i + t 2H i i -|t i -s i | 2H i 2 . ( 2
)
The d-parametric standard Brownian sheet is the Gaussian process {W t : t = (t 1 , . . . , t d ) ∈ R d } equal to zero on the hyperplanes {t :

t i = 0}, 1 ≤ i ≤ d,
and its covariance function is given by

R(s, t) = E[W s W t ] = d i=1 R(s i , t i ) = d i=1 s i ∧ t i . (3)
Let q ∈ N, and the Hurst multi-index

H = (H 1 , H 2 , . . . , H d ) ∈ ( 1 2 , 1) d .
The Hermite sheet of order q is given by

Z q H (t) = c(H, q) R d•q t 1 0 . . . t d 0   q j=1 (s 1 -y 1,j ) -1 2 + 1-H 1 q + . . . (s d -y d,j ) -1 2 + 1-H d q +   ds d . . . ds 1 dW (y 1,1 , . . . , y d,1 ) . . . dW (y 1,q , . . . , y d,q ) = c(H, q) R d•q t 0 q j=1 (s -y j ) -1 2 + 1-H q + ds dW (y 1 ) . . . dW (y q ) (4)
where x + = max(x, 0). For a better understanding about multiple stochastic integrals we refer to [START_REF] Nualart | Malliavin Calculus and Related Topics[END_REF]. As pointed out before, when q = 1, ( 4) is the fractional Brownian sheet with Hurst multi-index

H = (H 1 , H 2 , . . . , H d ) ∈ ( 1 2 , 1) d .
For q ≥ 2 the process Z q H (t) is not Gaussian and for q = 2 we denominate it as the Rosenblatt sheet. Now let us calculate the covariance R q H (s, t) of the Hermite sheet. Using the isometry of multiple Wiener-Itô integrals and Fubini theorem one gets

R q H (s, t) = E[Z q H (s)Z q H (t)] = E    c(H, q) 2 R d•q s 0 q j=1 (u -y j ) -1 2 + 1-H q + du dW (y 1 ) . . . dW (y q ) • R d•q t 0 q j=1 (v -y j ) -1 2 + 1-H q + dv dW (y 1 ) . . . dW (y q )    = c(H, q) 2 R d•q    s 1 0 . . . s d 0 q j=1 d i=1 (u i -y i,j ) -1 2 + 1-H i q + du d . . . du 1 • t 1 0 . . . t d 0 q j=1 d i=1 (v i -y i,j ) -1 2 + 1-H i q + dv d . . . dv 1    dy 1,1 . . . dy d,1 . . . dy 1,q . . . dy d,q = c(H, q) 2 t 1 0 s 1 0 R q q j=1 (u 1 -y 1,j ) -1 2 + 1-H 1 q + (v 1 -y 1,j ) -1 2 + 1-H 1 q + dy 1,1 . . . dy 1,q du 1 dv 1 . . . t d 0 s d 0 R q q j=1 (u d -y d,j ) -1 2 + 1-H d q + (v d -y d,j ) -1 2 + 1-H d q + dy d,1 . . . dy d,q du d dv d but R q q j=1 (u -x j ) -1 2 + 1-H q + (v -x j ) -1 2 + 1-H q + dx 1 . . . dx q = R (u -x) -1 2 + 1-H q + (v -x) -1 2 + 1-H q + dx q , (5)
so

R q H (s, t) = c(H, q) 2 t 1 0 s 1 0 R (u 1 -y 1 ) -1 2 + 1-H 1 q + (v 1 -y 1 ) -1 2 + 1-H 1 q + dy 1 q du 1 dv 1 . . . t d 0 s d 0 R (u d -y d ) -1 2 + 1-H d q + (v d -y d ) -1 2 + 1-H d q + dy d q du d dv d .
Recalling that the Beta function β(p, q) = 1 0 z p-1 (1 -z) q-1 dz, p, q > 0, satisfies the following identity

R (u -y) a-1 + (v -y) a-1 + dy = β(a, 1 -2a)|u -v| 2a-1 (6) if a ∈ (0, 1 2 ), we see that, since H k ≥ 1 2 for every k = 1, .., d, R q H (s, t) = c(H, q) 2 t 1 0 s 1 0 β 1 2 - 1 -H 1 q , 2(1 -H 1 ) q q • |u 1 -v 1 | 2(H 1 -1) du 1 dv 1 . . . t d 0 s d 0 β 1 2 - 1 -H d q , 2(1 -H d ) q q • |u d -v d | 2(H d -1) du d dv d = c(H, q) 2 β 1 2 - 1 -H 1 q , 2(1 -H 1 ) q q 1 2H 1 (2H 1 -1) s 2H 1 1 + t 2H 1 1 -|t 1 -s 1 | 2H 1 . . . β 1 2 - 1 -H d q , 2(1 -H d ) q q 1 2H d (1 -2H d ) s 2H d d + t 2H d d -|t d -s d | 2H d .
So now we choose

c(H, q) 2 =   β 1 2 -1-H 1 q , 2(1-H 1 ) q q H 1 (2H 1 -1)   -1 . . .   β 1 2 -1-H d q , 2(1-H d ) q q H d (2H d -1)   -1 (7)
in this way we get

E Z q H (t) 2 = t 2H = t 2H 1 1 . . . t 2H d d , and finally R q H (s, t) = 1 2 s 2H 1 1 + t 2H 1 1 -|t 1 -s 1 | 2H 1 . . . s 2H d d + t 2H d d -|t d -s d | 2H d = d i=1 s 2H i i + t 2H i i -|t i -s i | 2H i 2 = d i=1 R H i (s i , t i ) = R H (s, t). ( 8
)
Remark 1 From the previous development we see that:

• The covariance structure is the same for all q ≥ 1, so it coincides with the covariance of the fractional Brownian sheet.

• In order to all the quantities are well defined, the condition H k ∈ ( 1 2 , 1), k = 1, . . . , d must be satisfied.

We will next prove the basic properties of the Hermite sheet: self-similarity, stationarity of the increments and Hölder continuity.

Let us first recall the concept of self-similarity for multiparameter stochastic processes.

Definition 1 A stochastic process (X t ) t∈T , where T ⊂ R d is called self-similar with self- similarity order α = (α 1 , . . . , α d ) > 0 if for any h = (h 1 , . . . , h d ) > 0 the stochastic process ( Xt ) t∈T given by Xt = h α X t h = h α 1 1 ...h α d d X t 1 h 1 ,..., t d h d
has the same finite dimensional distributions as the process X.

Proposition 1 The Hermite sheet is self-similar of order H = (H 1 , . . . , H d ).

Proof: The scaling property of the Wiener sheet implies that for every 0 < c = (c 1 , . . . , c d ) ∈ R d the processes (W (ct) t≥0 ) and √ cW (t) t≥0 have the same finite dimensional distributions. Therefore, if 1 = (1, . . . , 1) ∈ R d , using obvious changes of variables in the integrals with respect to ds and dW , Ẑq

H (t) = h H Z q t h = c(H, q)h H R d•q t h 0 q j=1 (s -y j ) -1 2 + 1-H q + ds dW (y 1 ) . . . dW (y q ) = c(H, q)h H-1 R d•q t 0 q j=1 ( s h -y j ) -1 2 + 1-H q + ds dW (y 1 ) . . . dW (y q ) = c(H, q)h H-1 R d•q t 0 q j=1 ( s h - y j h ) -1 2 + 1-H q + ds dW (h -1 y 1 ) . . . dW (h -1 y q ) = c(H, q)h H-1 h q 1 2 + 1-H q R d•q t 0 q j=1 (s -y j ) -1 2 + 1-H q + ds dW (h -1 y 1 ) . . . dW (h -1 y q ) = (d) c(H, q)h H-1 h q 1 2 + 1-H q h -q 2 R d•q t 0 q j=1 (s -y j ) -1 2 + 1-H q +
ds dW (y 1 ) . . . dW (y q ) = Z q H (t) where = (d) means equivalence of finite dimensional distributions.

Let us recall the notion of the increment of a d-parameter process X on a rectangle [s, t] ⊂ R d , s = (s 1 , . . . , s d ), t = (t 1 , . . . , t d ), with s ≤ t. This increment is denoted by ∆X [s,t] and it is given by ∆X

[s,t] = r∈{0,1} d (-1) d-d i=1 r i X s+r•(t-s) . (9)
When

d = 1 one obtains ∆X [s,t] = X t -X s while for d = 2 one gets ∆X [s,t] = X t 1 ,t 2 -X t 1 ,s 2 - X s 1 ,t 2 + X s 1 ,s 2 . Definition 2 A process (X t , t ∈ R d ) has stationary increments if for every h > 0, h ∈ R d the stochastic processes (∆X [0,t] , t ∈ R d ) and (∆X [h,h+t] , t ∈ R d ) have the same finite dimensional distributions.
Proposition 2 The Hermite sheet (Z q (t)) t≥0 has stationary increments.

Proof: Developing the increments of the process using the definition of the Hermite sheet and proceding as in the proof of Proposition 1 using the change of variables

s ′ = s -h, it is immediate to see that for every h > 0, h ∈ R d , ∆Z q [h,h+t] = (d) ∆Z q [0,t]
for every t.

Proposition 3 The trajectories of the Hermite sheet (Z q (t), t ≥ 0) are Hölder continuous of any order δ = (δ 1 , . . . , δ d ) ∈ [0, H) in the following sense: for every ω ∈ Ω, there exists a constant C ω > 0 such that for every s, t ∈ R d , s, t ≥ 0,

|∆Z q [s,t] | ≤ C ω |t 1 -s 1 | δ 1 • • • |t d -s d | δ d = C ω |t -s| δ .
Proof: Using the Cencov's criteria (see [START_REF] Cencov | Wiener random fields depending on several parameters[END_REF]) and the fact that the process Z q is almost surely equal to 0 when t i = 0, it suffices to check that

E ∆Z q [s,t] p ≤ C (|t 1 -s 1 | • • • |t d -s d |) 1+γ (10)
for some p ≥ 2 and γ > 0. From the self-similarity and the stationarity of the increments of the process Z q , we have for every p ≥ 2

E ∆Z q [s,t] p = E |Z 1 | p (|t 1 s 1 | • • • |t d -s d |) pH
and this obviously implies (10) by taking p arbitrary large.

Remark 2 In the one-parameter case, there exists several representations of the Hermite process (spectral domain representation, finite interval representation, positive half-axis representation, time domain representation, see [START_REF] Pipiras | Regularization and integral representations of Hermite processes[END_REF]). It has been shown in [START_REF] Pipiras | Regularization and integral representations of Hermite processes[END_REF] that all these representations of the (one-parameter) Hermite process have the same finite dimensional distributions. It would be interesting to generalize this study to the multi-parameter case.

Wiener integrals with respect to the Hermite sheet

Now we are well positioned to present Wiener integrals with respect to the d-parametric Hermite sheet. Let us consider a Hermite sheet Z q H (t) t∈R d . Denote E the family of elementary functions on R d of the form

f (u) = n l=1 a l 1 (t l ,t l+1 ] (u) (11) = n l=1 a l 1 (t 1,l ,t 1,l+1 ]×...×(t d,l ,t d,l+1 ] (u 1 , . . . , u d ), t l < t l+1 , a l ∈ R, l = 1, . . . , n.
For functions like f above we can naturally define its Wiener integral with respect to the Hermite sheet Z q H as

R d f (u)dZ q H (u) = n l=1 a l (∆Z q H ) [t l ,t l+1 ] (12)
where (∆Z q H ) [t l ,t l+1 ] (see ( 9)) stands for the generalized increments of Z q H on the rectangle

∆ t l := [t l , t l+1 ] = d i=1 [t i,l , t i,l+1 ] .
In the case d = 1, we simply have

(∆Z q H ) [t l ,t l+1 ] = Z q H (t 1,l+1 -t 1,l ) while for d = 2 (∆Z q H ) [t l ,t l+1 ] = Z q H (t 1,l+1 , t 2,l+1 ) -Z q H (t 1,l , t 2,l+1 ) -Z q H (t 1,l+1 , t 2,l ) + Z q H (t 1,l , t 2,l
). With the purpose of extending the definition [START_REF] Chronopoulou | Self-similarity parameter estimation and reproduction property for non-Gaussian Hermite processes[END_REF] to a larger family of integrands, we will point out some observations before. Let us consider the mapping J on the set of functions f : R d → R to the set of functions f : R d•q → R such that

J(f )(y 1 , . . . , y q ) = c(H, q) R d f (u) q j=1 (u -y j ) -1 2 + 1-H q + du (13) = c(H, q) R d f (u 1 , . . . , u d ) q j=1 d i=1 (u i -u i,j ) -1 2 + 1-H i q + du 1 . . . du d .
Using the mapping J we see that definition (4) can be re-expressed as follows

Z q H (t) = R d•q
J(1 [0,t 1 ]×...×[0,t d ] )(y 1 , . . . , y q )dW (y 1 ) . . . dW (y q ). (

) 14
Since J is clearly linear, definition [START_REF] Chronopoulou | Self-similarity parameter estimation and reproduction property for non-Gaussian Hermite processes[END_REF] can be tailored to

R d f (u)dZ q H (u) = n l=1 a l ∆Z q H [t l ,t l+1 ] = n l=1 a l   ξ∈{0,1} d (-1) d-d i=1 ξ i Z q H (t 1,l+ξ 1 , . . . , t d,l+ξ d )   = n l=1 a l ξ∈{0,1} d (-1) d-d i=1 ξ i R d•q J(1 [0,t 1,l+ξ 1 ]×...×[0,t d,l+ξ d ] )(y 1 , . . . , y q )dW (y 1 ) . . . dW (y q ) = n l=1 a l R d•q J(1 [t 1,l ,t 1,l+1 ]×...×[t d,l ,t d,l+1 ] )(y 1 , . . . , y q )dW (y 1 ) . . . dW (y q ) = R d•q
J(f )(y 1 , . . . , y q )dW (y 1 ) . . . dW (y q ). ( 15)

In this way we introduce the space

H = f : R d → R : R d•q (J(f )(y 1 , . . . , y q )) 2 dy 1 . . . dy q < ∞ ( 16
)
equipped with the norm

f 2 H = R d•q
(J(f )(y 1 , . . . , y q )) 2 dy 1 . . . dy q .

(17

)
Working the expression for the norm we see that

f 2 H = c(H, q) 2 R d•q      R d f (u) q j=1
(uy j )

-1 2 + 1-H q + du   •   R d f (v) q j=1 (v -y j ) -1 2 + 1-H q + dv      dy 1 . . . dy q .
Using ( 5), ( 6) and ( 7) we get that

f 2 H = c(H, q) 2 R d R d f (u 1 , . . . , u d )f (v 1 , . . . , v d )    d i=1 R q q j=1 (u i -y i,j ) -1 2 + 1-H i q + (v i -y i,j ) -1 2 + 1-H i q + dy i,1 . . . dy i,q    du 1 . . . du d dv 1 . . . dv d = c(H, q) 2 R d R d f (u 1 , . . . , u d )f (v 1 , . . . , v d ) • d i=1 R (u i -y) -1 2 + 1-H i q + (v i -y) -1 2 + 1-H i q + dy q du 1 . . . du d dv 1 . . . dv d = R d R d f (u 1 , . . . , u d )f (v 1 , . . . , v d ) d i=1 H i (2H i -1)|u -v| 2H i -2 du 1 . . . du d dv 1 . . . dv d = H(2H -1) R d R d f (u)f (v)|u -v| 2H-2 dudv, (18)
hence

H = f : R d → R : R d R d f (u)f (v)|u -v| 2H-2 dudv < +∞ ( 19
)
and

f 2 H = H(2H -1) R d R d f (u)f (v)|u -v| 2H-2 dudv.
The mapping

f → R d f (u)dZ q H (u) ( 20
)
provides an isometry from E to L 2 (Ω). Indeed, for f like in [START_REF] Chronopoulou | Variations and Hurst index estimation for a Rosenblatt process using longer filters[END_REF] it holds that

E R d f (u)dZ q H (u) 2 = n-1 k,l=0 a k a l E ∆Z q H [t k ,t k+1 ] • ∆Z q H [t l ,t l+1 ] (21) = n-1 k,l=0 a k a l ξ∈{0,1} d (-1) d-d i=1 ξ i ρ∈{0,1} d (-1) d-d j=1 ρ j E Z q H (t k+ξ )Z q H (t l+ρ ) = n-1 k,l=0 a k a l ξ∈{0,1} d (-1) d-d i=1 ξ i ρ∈{0,1} d (-1) d-d j=1 ρ j R H (t k+ξ , t l+ρ ) = n-1 k,l=0 a k a l H 1 (2H 1 -1) . . . H d (2H d -1) t 1,k+1 t 1,k . . . t d,k+1 t d,k • t 1,l+1 t 1,l . . . t d,l+1 t d,l |u 1 -v 1 | 2H 1 -2 . . . |u d -v d | 2H d -2 du 1 . . . du d dv 1 . . . dv d = n-1 k,l=0 a k a l < 1 [t 1,k ,t 1,k+1 ]ו••×[t d,k ,t d,k+1 ] , 1 [t 1,l ,t 1,l+1 ]ו••×[t d,l ,t d,l+1 ] > H = < f, f > H , (22)
where we have made a slight abuse of notation, t k+ξ = (t 1,k+ξ 1 , . . . , t d,k+ξ d ).

On the other hand, from what shown in [START_REF] Pipiras | Integration questions related to the fractional Brownian motion[END_REF] section 4, it follows that the set of elementary functions E is dense in H. As a consequence the mapping (13) can be extended to an isometry from H to L 2 (Ω) and relation [START_REF] Dobrushin | Non-central limit theorems for non-linear functionals of Gaussian fields[END_REF] still holds.

Remark 3

The elements of H may be not functions but distributions; it is therefore more practical to work with subspaces of H that are sets of functions. Such a subspace is

|H| = f : R d → R R d R d |f (u)||f (v)||u -v| 2H-2 dvdu < ∞ .
Then |H| is a strict subspace of H and we actually have the inclusions

L 2 (R d ) ∩ L 1 (R d ) ⊂ L 1 H (R d ) ⊂ |H| ⊂ H, (23)
where

L p denotes L p 1 ⊗ . . . ⊗ L p d .
The space |H| is not complete with respect to the norm • H but it is a Banach space with respect to the norm

f 2 |H| = R d R d |f (u)||f (v)||u -v| 2H-2 dvdu
Remark 4 Expression (15) presents a useful interpretation for the Wiener integrals with respect to the Hermite sheet; as elements in the q-th Wiener chaos generated by the d-parametric standard Brownian field.

Application: The Hermite stochastic wave equation

In this section we presents the linear stochastic wave equation as an example of equations driven by a Hermite sheet. We show the existence of the solution and study some properties of it thanks to the definition of the Wiener integrals with respect to the Hermite sheet.

Consider the linear stochastic wave equation driven by an infinite-dimensional Hermite sheet Z q H with Hurst multi-index H ∈ (1/2, 1) (d+1) . That is

∂ 2 u ∂t 2 (t, x) = ∆u(t, x) + Żq H (t, x), t > 0, x ∈ R d (24)
u(0, x) = 0, x ∈ R d ∂u ∂t (0, x) = 0, x ∈ R d .
Here ∆ is the Laplacian on R d and Z q H = {Z q H (t, x); t ≥ 0, x ∈ R d } is the (d + 1)-parametric Hermite sheet whose covariance is given by

E Z q H (s, x)Z q H (t, y) = R H (t, s)R H 0 (x, y) if H = (H, H 1 , . . . , H d )
and we denoted by H 0 = (H 1 , . . . , H d ) and Żq stands for the formal derivative of Z (q) . Equivalently we can write

E Żq H (s, x) Żq H (t, y) = H(2H -1)|t -s| 2H-2 d i=1 (H i (2H i -1) • |x i -y i | 2H i -2 ). (25)
Let G 1 be the fundamental solution of ∂ 2 u ∂t 2 -∆u = 0. It is known that G 1 (t, •) is a distribution in S ′ (R d ) with rapid decrease, and

FG 1 (t, •)(ξ) = sin(t|ξ|) |ξ| , (26)
for any ξ ∈ R d , t > 0 and d ≥ 1, where F denotes the Fourier transform (see e.g. [START_REF] Treves | Basic Linear Partial Differential Equations[END_REF]). In particular,

G 1 (t, x) = 1 2 1 {|x|<t} , if d = 1 G 1 (t, x) = 1 2π 1 t 2 -|x| 2 1 {|x|<t} , if d = 2 G 1 (t, x) = c d 1 t σ t , if d = 3,
where σ t denotes the surface measure on the 3-dimensional sphere of radius t.

The mild solution of ( 24) is a square-integrable process u = {u(t, x); t ≥ 0, x ∈ R d } defined by:

u(t, x) = t 0 R d G 1 (t -s, x -y)Z q H (ds, dy). (27)
The above integral is a Wiener integral with respect to the Hermite sheet, as introduced in Section 2.

Existence and regularity of the solution

By definition, u(t, x) exists if and only if the stochastic integral above is well-defined, i.e.

g t,x := G 1 (t -•, x -•) ∈ H. In this case, E|u(t, x)| 2 = g t,x 2
H . We state the result on the existence and the regularity of the solution of [START_REF] Taqqu | Weak convergence to the fractional Brownian motion and to the Rosenblatt process[END_REF].

Proposition 4 Let Z q H (t, x) be the (d + 1)-parametric Hermite sheet of order q. Denote by

β = d - d i=1 (2H i -1). ( 28
)
Then, the stochastic wave equation ( 24) admits an unique mild solution (u(t, x)) t≥0,x∈R d if and only if

β < 2H + 1. ( 29
)
If in addition we have that

β ∈ (2H -1, d ∧ 2H + 1) (30)
then, for fixed 0 < t 0 < T , the following statements are true:

a.-Let x ∈ R d fixed.
Then there exist positive constants c 1 , c 2 such that for every s, t ∈ [t 0 , T ]

c 1 |t -s| 2H+1-β ≤ E |u(t, x) -u(s, x)| 2 ≤ c 2 |t -s| 2H+1-β .
Also for every fixed x ∈ R d the application Fix M > 0 and assume [START_REF] Xiao | Sample path properties of anisotropic Gaussian random fields: A minicourse on stochastic partial differential equations[END_REF]. For every t, s ∈ [t 0 , T ] and x, y ∈ [-M, M ] d there exist positive constants C 1 , C 2 such that

t → u(t, x)
C 1 ∆ ((t, x); (s, y)) ≤ E |u(t, x) -u(s, y)| 2 ≤ C 2 ∆ ((t, x); (s, y)) . (32)
Proof: By the isometry of the Wiener integral with respect to the Hermite sheet, the L 2 norm will be

Eu(t, x) 2 = α H t 0 du t 0 dv|u -v| 2H-2 R d R d dydzG 1 (t -u, x -y)G 1 (t -v, x -z) × d i=1 (H i (2H i -1))|y i -z i | 2H i -2 = α H t 0 du t 0 dv|u -v| 2H-2 R d sin(u|ξ|) sin(v|ξ|) |ξ| 2 µ(dξ)
where

µ(dξ) = c H d i=1 |ξ i | -(2H i -1) (33)
with ξ = (ξ 1 , . . . , ξ d ), α H = H(2H -1) and c H = H(2H -1). This is, u(t, x) has the same L 2 norm as in the case q = 1, i.e. when the noise of the equation is a fractional Brownian sheet. It therefore follows from [START_REF] Balan | The stochastic wave equation with fractional noise: A random field approach[END_REF], Theorem 3.1 that the above integral is finite if and only if

R d 1 1 + |ξ| 2 H+ 1 2 µ(dξ) < ∞
with µ given by (33). The above condition is equivalent to β < 2H + 1, see Example 3.4 in [START_REF] Balan | The stochastic wave equation with fractional noise: A random field approach[END_REF].

The proof of the other items is strongly held in the covariance structure of the Hermite sheet, which is the same as for the fractional Brownian sheet. By a careful revision of the proofs of Theorem 3.1 in [START_REF] Balan | The stochastic wave equation with fractional noise: A random field approach[END_REF], Propositions 1, 2, 3 and Corollary 1 in [START_REF] De | Hitting times for the stochastic wave equation with fractional-colored noise[END_REF], it is possible to appreciate that the computations are also valid for any process with a covariance structure like the one presented in these articles, in particular in our case.

• The bounds for the increments in time are consequence of Proposition 1 in [START_REF] De | Hitting times for the stochastic wave equation with fractional-colored noise[END_REF], and the Hölder regularity comes from Corollary 1 in [START_REF] De | Hitting times for the stochastic wave equation with fractional-colored noise[END_REF], this proves a.

• The bounds for the increments in the space variable are deduced from Proposition 2 in [START_REF] De | Hitting times for the stochastic wave equation with fractional-colored noise[END_REF], and the space Hölder regularity is direct from Proposition 3 in [START_REF] De | Hitting times for the stochastic wave equation with fractional-colored noise[END_REF], this proves b.

• Point c follows from a and b by following the lines of the proof of Theorem 2 in [START_REF] De | Hitting times for the stochastic wave equation with fractional-colored noise[END_REF].

Existence of local times

We will show that the solution of ( 24), viewed as a process in (t, x), admits a square integrable local time.

Let us define the local time of a stochastic process (X t ) t∈J . Here J denotes a subset of R d . For any Borel set I ⊂ J the occupation measure of X on I is defined as

µ I (A) = λ (t ∈ I, X t ∈ A) , A ∈ B(R)
where λ denotes the Lebesgue measure. If µ I is absolutely continuous with respect to the Lebesgue measure, we say that X has local time on I. The local time is defined as the Radon-Nikodym derivative of µ I L(I, x) = dµ I dλ (x), x ∈ R.

We will use the notation

L(t, x) := L([0, t], x), t ∈ R d + , x ∈ R.
The local time satisfies the occupation time formula

I f (X t )dt = R f (y)L(I, y)dy (34)
for any Borel set I in T and for any measurable function f : R → R.

Proposition 5 Let u(t, x), t ≥ 0, x ∈ R d be the solution to [START_REF] Taqqu | Weak convergence to the fractional Brownian motion and to the Rosenblatt process[END_REF] and assume (30) where β is given by ( 28). Then on each set Proof: It is well known from [START_REF] Berman | Local times and sample function properties of stationary Gaussian processes[END_REF] (see also Lemma 8.1 in [START_REF] Xiao | Sample path properties of anisotropic Gaussian random fields: A minicourse on stochastic partial differential equations[END_REF]) that, for a jointly measurable zero-mean stochastic process X = (X(t), t ∈ [0, T]) (T belongs to R d ) with bounded variance, the condition for every x ∈ R.

[a, b] × [A, B] ⊂ [0, ∞) × R d the process u(t, x), t ≥ 0, x ∈ R d

Existence of the joint density for the solution in the Rosenblatt case

It is possible to obtain the existence of the joint density of the random vector (u(t, x), u(s, y)) with s = t or x = y in the case when the wave equation ( 24) is driven by a Hermite sheet of order q = 2 (the Rosenblatt sheet). The result is based on a criterium for the existence of densities for vectors of multiple integrals which has recently been proven in [START_REF] Nourdin | Absolute continuity and convergence of densities for random vectors on Wiener chaos[END_REF].

Let us state our result.

Proposition 6 Let u(t, x), t ≥ 0, x ∈ R d be the mild solution to [START_REF] Taqqu | Weak convergence to the fractional Brownian motion and to the Rosenblatt process[END_REF]. Then for every (t, x) = (s, y), (t, x), (s, y) ∈ (0, ∞) × R d , the random vector (u(t, x), u(s, y)) admits a density.

Proof: Note that for every t ≥ 0 and x ∈ R d , the random variable u(t, x) is a multiple integral of order 2 with respect to the d-parametric Brownian sheet. A result present in [START_REF] Nourdin | Absolute continuity and convergence of densities for random vectors on Wiener chaos[END_REF] states that a two-dimensional vector of multiple integrals of order 2 admits a density if and only if the determinant of the covariance matrix is strictly positive. Denote by C(t, s, x, y) the covariance matrix of (u(t, x), u(s, y)). The determinant of this matrix is the same for every q ≥ 1, from the covariance structure of the Hermite sheet. It is clear that for q = 1 obviously det C(t, s, x, y) is strictly positive, since the vector (u(t, x), u(s, y)) is a Gaussian vector and hence admits a density when (t, x) = (s, y). This implies that det C(t, s, x, y) is also strictly positive for q = 2 and so the vector (u(t, x), u(s, y)) admits a density also for q = 2.

2 . 2 ∧ 1 .

 221 is almost surely Hölder continuous of any order δ ∈ 0, 2H+1-β b.-Fix M > 0 and t ∈ [t 0 , T ]. Then there exist positive constants c 3 , c 4 such that for anyx, y ∈ [-M, M ] d c 3 |x -y| 2H+1-β ≤ E |u(t, x) -u(t, y)| 2 ≤ c 4 |x -y| 2H+1-β .Also, for any t ∈ [t 0 , T ] the applicationx → u(t, x)is almost surely Hölder continuous of any order δ ∈ 0,2H+1-β

c. -

 - Denote by ∆ the following metric on[0, T ] × R d ∆ ((t, x); (s, y)) = |t -s| 2H+1-β + |x -y| 2H+1-β .(31)

  admits a local time (L([a, b] × [A, B], y), y ∈ R) which is square integrable with respect to y E R L([a, b] × [A, B], y) 2 dy < ∞ a.s.

Remark 5

 5 X(t) -X(s)] 2 ) -1/2 dsdt < ∞ is sufficient for the local time of X to exist on [0, T] almost surely and to be square integrable as a function of y. According to the inequality (32), for allI = [a, b] × [A, B] included in [0, ∞) × R d we have, I I (E [u(t, x) -u(s, y)] 2 ) -1/2 dtdxdsdy < C I I |t -s| 2H+1-β + |x -y| 2H+1-β -1 2 dtdxdsdy and this is finite for β > 2H -1. Thus almost surely the local time of u exists and is square integrable. It follows as a consequence of Lemma 8.1 in [29] that the local time of the solution u admits the following L 2 representation L([a, b] × [A, B], x) = 1 2π R dze -izx [a,b]×[A,B] dsdye iu(s,y)z

  a d < b d (analogously for the other inequalities). Before introducing the Hermite sheet we briefly recall the fractional Brownian sheet and the standard Brownian sheet.The d-parametric anisotropic fractional Brownian sheet is the centered Gaussian process {B H t : t = (t 1 , . . . , t d ) ∈ R d } with Hurst multi-index H = (H 1 , . . . , H d ) ∈ (0, 1) d . It is equal to zero on the hyperplanes {t : t i = 0}, 1 ≤ i ≤ d, and its covariance function is given by

* Partially supported by BECAS CHILE 2011 and the CONICYT-ECOS program C10E03 † Supported by the CNCS grant PN-II-ID-PCCE-2011-2-0015. Associate member of the team Samm, Université de Panthéon-Sorbonne Paris 1. Supported by the ANR grant "Masterie" BLAN 012103 is also acknowledged.

1


Aurela Shehu
email: aurela.shehu@inria.fr

Jinlong Yang
email: jinlong.yang@inria.fr

Jean-Sébastien Franco
email: jean-sebastien.franco@inria.fr

Franck Hétroy-Wheeler

Stefanie Wuhrer
email: stefanie.wuhrer@inria.fr

Computing temporal alignments of human motion sequences in wide clothing using geodesic patches

In this paper, we address the problem of temporal alignment of surfaces for subjects dressed in wide clothing, as acquired by calibrated multi-camera systems. Most existing methods solve the alignment by fitting a single surface template to each instant's 3D observations, relying on a dense point-to-point correspondence scheme, e.g. by matching individual surface points based on local geometric features or proximity. The wide clothing situation yields more geometric and topological difficulties in observed sequences, such as apparent merging of surface components, misreconstructions, and partial surface observation, resulting in overly sparse, erroneous point-to-point correspondences, and thus alignment failures. To resolve these issues, we propose an alignment framework where point-to-point correspondences are obtained by growing isometric patches from a set of reliably obtained body landmarks. This correspondence decreases the reliance on local geometric features subject to instability, instead emphasizing the surface neighborhood coherence of matches, while improving density given sufficient landmark coverage. We validate and verify the resulting improved alignment performance in our experiments.

Introduction

Capturing dynamic scenes, such as human body motion and performance, is a task of interest with a broad range of applications for entertainment, design, medical, and cultural heritage purposes. Typically, performance is observed with a set of calibrated cameras, yielding a set of individual raw 3D reconstructions with no temporal coherence. An important and challenging task in this context is to perform temporal alignment of the sequence, i.e. expressing the en- * Joint first authors tire 3D sequence as the deformation of a single deformed template mesh. This representation has large benefits as it allows for efficient storage, transmission, reverse scene analysis and semantic characterization of the scene as one moving body.

How to obtain this alignment has been widely studied in the context of humans with tight clothing, where only the motion of humans needs to be characterized. This yields a large family of template-based surface tracking methods, which follow a similar resolution canvas. First a surface template is chosen and obtained, that can be e.g. a generic human model, a particular reconstruction among those observed, or a pre-obtained scan of the human subject. Second, a point-to-point correspondence scheme is devised, using point proximity in Euclidean space, in a geometric or appearance feature space, or on a learned manifold. Third, because the correspondences so obtained are often insufficiently dense, non-uniformly distributed over the body, and erroneous, a deformation model with a reduced control parameter set is used to constrain the estimation of the full body alignment and reduce the search space of deformation, typically based on human kinematics, piecewise rigid or affine.

Wide clothing acquisitions generally fail tight clothing alignment method assumptions. In particular the correspondence becomes much more challenging since such sequences exhibit much stronger geometric and topological noise, more occlusions, and non-rigid behavior, due to the inherent geometric variability of clothing. The sensitivity of correspondences to the representativeness of template topology and geometry is also drastically increased.

We address this problem with a full alignment solution, centered on a correspondence model suitable for clothes. We ground this work in two key assumptions. First, we assume that the topology of the human in clothing is fixed over time. This assumption holds for the human body itself as well as many clothing styles such as t-shirts, trousers and skirts. Second, we assume that the geometry of the model deforms in a near-isometric way. This is true when considering locomotions of the human and clothing that are not very elastic. In practice, the acquired object violates both assumptions due to aforementioned acquisition noise. That is, the acquired topology may change due to the merging of close-by body parts, which in turn completely changes the intrinsic geometry of the model. To design a method that is robust to this type of acquisition noise, we use a deformation model based on partial near-isometric patches. These patches are grown from a set of preselected landmarks that can be robustly found in each frame, and ensure consistent densification of correspondences over the surface. To ease correspondence over all frames, we also provide an automatic template selection method among raw 3D models of the input sequence, maximizing topological adequacy.

Our experiments demonstrate the success of the method for temporal alignment of a variety of clothing sequences. In particular we favorably compare our method against two state of the art temporal alignment methods [START_REF] Allain | On mean pose and variability of 3d deformable models[END_REF][START_REF] Allain | An efficient volumetric framework for shape tracking[END_REF] based on more restrictive locally rigid deformation assumptions, on both pretracked and real datasets. Results show that our method outperforms previous work for human characters with wide or layered clothing.

Related work

Most methods to temporally align sequences of 3D data require a prior on the shape to track. More general techniques without prior knowledge on the geometry also exist, but they assume a strong temporal coherence. These methods use nearest neighbor techniques as a prior to aid the alignment. They usually suffer from drift and cannot follow fast motion [START_REF] Budd | Global nonrigid alignment of surface sequences[END_REF]. Dou et al. [START_REF] Dou | 3d scanning deformable objects with a single rgbd sensor[END_REF] reduce drifts errors by detecting loop closures and distributing the alignment error over the loop. Newcombe et al. [START_REF] Newcombe | Dynamicfusion: Reconstruction and tracking of non-rigid scenes in real-time[END_REF] reconstruct a non-rigidly deforming surface while estimating a per frame volumetric warp field that transforms this surface into the live frame. Tevs et al. [START_REF] Tevs | Animation cartography: Intrinsic reconstruction of shape and motion[END_REF] solve the problem of following fast motion by first computing a few landmark correspondences and then propagating to a dense matching, assuming isometric deformation. Landmarks are computed as geometric feature points. Contrary to [START_REF] Tevs | Animation cartography: Intrinsic reconstruction of shape and motion[END_REF], our landmarks are not computed as geometric features on the cloth surface. Instead, we estimate the body shape under the cloth and then automatically transfer the anatomical landmarks from the body shape to the cloth. Many methods for human body shape estimation under clothing have been proposed [START_REF] Jain | MovieReshape: tracking and reshaping of humans in videos[END_REF][START_REF] Wuhrer | Estimation of human body shape and posture under clothing[END_REF][START_REF] Neophytou | A layered model of human body and garment deformation[END_REF][START_REF] Pons-Moll | DYNA: a model of dynamic human shape in motion[END_REF]. We use the recent method of Yang et al. [START_REF] Yang | Estimation of human body shape in motion with wide clothing[END_REF] since it works for moving shapes and is fully automatic.

The majority of works for temporal alignment of human motion sequences make use of a template showing a specific subject in a specific type of clothing. The main ad-vantage of such methods is that they are particularly robust to severe acquisition artifacts. However, a template is required for tracking. For example, the method of Bradley et al. [START_REF] Bradley | Markerless garment capture[END_REF], which is designed for the temporal alignment of a moving cloth, constructs the template from a photograph of the garment. Aguiar et al. [START_REF] De Aguiar | Performance capture from sparse multiview video[END_REF][START_REF] Stoll | Video-based reconstruction of animatable human characters[END_REF] create the template by a full-body laser scan of the subject in its current clothes before performing capture. A physically-based cloth model is then used for temporal alignment. Budd et al. [START_REF] Budd | Global nonrigid alignment of surface sequences[END_REF] introduce a way to align frames that are not necessarily temporally adjacent, but most similar according to a volume-based shape similarity measure, which makes the template-based alignment more robust to large motions between adjacent frames. Allain et al. [START_REF] Allain | On mean pose and variability of 3d deformable models[END_REF] perform shape tracking assuming a locally rigid deformation model, in order to compute both the mean pose and correspondences over time. The template is manually selected as one of the meshes of the input sequence. While this method is surface-based, the follow-up method [START_REF] Allain | An efficient volumetric framework for shape tracking[END_REF] proposes a volumetric parametrization of the tracking which shows better results than surfacebased methods. The shape is represented by a centroidal Voronoi tessellation, which enables volume conservation. This method also assumes a template is provided. Our method is template-based for robustness purposes. However, no additional information nor user interaction is necessary to build the template since it is automatically selected among the meshes of the sequence, using a method similar in spirit to the one of Letouzey and Boyer [START_REF] Letouzey | Progressive shape models[END_REF].

Most temporal alignment methods are well adapted to human characters in tight clothes. Methods handling loose clothing usually assume near-isometric deformations [START_REF] Bradley | Markerless garment capture[END_REF][START_REF] Tevs | Animation cartography: Intrinsic reconstruction of shape and motion[END_REF]. This is because most clothes do not usually stretch during motion [START_REF] Goldenthal | Efficient simulation of inextensible cloth[END_REF]. However, computing approximate isometries can be computationally expensive [START_REF] Brunton | A low-dimensional representation for robust partial isometric correspondences computation[END_REF][START_REF] Rodolà | Dense non-rigid shape correspondence using random forests[END_REF]. In our method, we reduce the computation cost by selecting anatomically-motivated landmarks to discover approximate isometries.

Method overview

Given an unstructured temporal sequence showing a clothed human performing a motion, our method produces a consistently deforming model. To be robust to acquisition and topological noise our method has three steps as shown in Figure 1: a human body model is used to find good initial starting points for the partial near-isometries; point trajectories are computed using a partial near-isometric deformation model; and the point trajectories are used as assignments in a template-based deformation to find a coherent model that deforms over time. We now provide more detail for each step.

First, we estimate the undressed human body shape in motion for the given sequence. This step reduces significantly the search space of the problem by providing a good First, a statistical model of undressed body shape is fitted to the frame (c). Pre-marked anatomical points on the fitted human body model are mapped to the input frame (d). Second, these anatomical markers are used to guide a partial near-isometric correspondence computation between an automatically selected template (green model in (a), and shown in top row) and the frame (e). Third, the resulting correspondences are used as assignments to deform the template to the input frame (f).

initialization for the computation of near-isometric partial matches. This step takes advantage of recent robust methods that use statistical human body models to estimate the naked human body shape under clothing [START_REF] Jain | MovieReshape: tracking and reshaping of humans in videos[END_REF][START_REF] Wuhrer | Estimation of human body shape and posture under clothing[END_REF][START_REF] Neophytou | A layered model of human body and garment deformation[END_REF][START_REF] Pons-Moll | DYNA: a model of dynamic human shape in motion[END_REF][START_REF] Yang | Estimation of human body shape in motion with wide clothing[END_REF]. In our implementation, we use an automatic method that estimates the naked body shape in motion under clothing based on a 3D input sequence [START_REF] Yang | Estimation of human body shape in motion with wide clothing[END_REF]. For the specific frame that is shown in blue in Figure 1(a) and enlarged in (b), the estimated human body shape is shown in Figure 1(c). The input frame is shown in grey and the estimated human body under clothing in brown. Next, we transition from the naked human body to surface of the clothing. We once manually select a small set of anatomical points on the statistical model and automatically transfer these points on the clothed sequence as shown in Figure 1(d).

Second, we use a partial near-isometric deformation model that is robust with respect to topological acquisition noise to compute point trajectories on the input sequence. Following Letouzey and Boyer [START_REF] Letouzey | Progressive shape models[END_REF], we proceed by first taking advantage of the assumption that the real topology stays fixed over time to automatically select a template frame from the sequence based on topological and geometric criteria. The automatically selected template is shown in green in Figure 1(a) and enlarged in the first row. Second, for each frame of the sequence, we independently compute correspondence information between the template and the frame. This computation finds partial near-isometric matches between two frames [START_REF] Brunton | A low-dimensional representation for robust partial isometric correspondences computation[END_REF]. To increase robustness and make the algorithm efficient, we use the previously computed anatomical points on each frame for initialization. The partial matches are then merged into a global correspondence. The computed correspondence between template and a frame is shown in Figure 1(e). This correspondence information between the template and every frame of the sequence results in point trajectories over time.

Third, we find a consistent topology that deforms over time by extending a standard template fitting technique [START_REF] Allen | The space of human body shapes: Reconstruction and parameterization from range scans[END_REF] to take advantage of the previously computed point trajectories and operate on motion sequences. Since the previous step computes the correspondence information between the template and every other frame independently, the resulting point trajectories may be interrupted and the correspondence of a particular point on the template may not be known on every frame. To remedy this we add this final step that deforms the template to the entire motion sequence using previously computed point trajectories as assignments. The output of this step is shown in Figure 1(f).

Clothed human alignment

This section provides details of the three steps of our proposed method.

Human body estimation and mapping anatomical points to clothed human

We start by estimating the human body shape for the given input sequence. The result of this step allows to map Human body estimation Given a human motion sequence in wide clothing, we start by estimating the undressed human body shape for the entire sequence using a recent automatic method [START_REF] Yang | Estimation of human body shape in motion with wide clothing[END_REF]. This method takes advantage of a learned statistical human body model [START_REF] Pishchulin | Building statistical shape spaces for 3d human modeling[END_REF]. In particular, the statistical body model represents the variation in identity across different people in a 100-dimensional shape space learned using principal component analysis, and the variations in posture for a fixed identity using a linear blend skinning that is found using a rigging tool [START_REF] Baran | Automatic rigging and animation of 3D characters[END_REF]. This model allows to generate a human model M(β, Θ), where β denotes the parameters controlling identity and Θ for posture.

The method proceeds by finding a single β along with Θ i for each input frame by minimizing the energy

E body = w l E l + w n E n + w w E w (1)
consisting of a weighted sum of three terms, where E l is an energy that aligns a small set of automatically detected landmarks, E n is an energy that pulls each vertex on M(β, Θ i ) to its nearest neighbor on the i-th frame, and E w is designed to cope with wide clothing by encouraging M(β, Θ i ) to be located inside all observed frames. The weights w l , w n , and w w allow to trade off the influence of the different energy terms, and we follow the weight schedule proposed by Yang et al. [START_REF] Yang | Estimation of human body shape in motion with wide clothing[END_REF].

Figure 2 shows the estimated human body shape and posture for two frames of a sequence. Note that all meshes M(β, Θ) generated this way share the same vertex ordering and mesh topology.

Mapping anatomical points from human body to surface of clothing We manually select a small set of oriented anatomical points on the statistical model M(β, Θ), where an oriented point is a 3D location on the surface along with a direction in its tangent plane. An oriented point is in practice selected by choosing two points: a starting point and a close-by neighbor that defines the direction in the tangent plane. Note that these points only need to be chosen once for any β and Θ, as all body models share the same mesh structure.

To automatically map an oriented anatomical point t on M(β, Θ i ) to the clothing surface of the i-th frame S i , we intersect the line through t along the normal direction of t with S i . If there's any intersection outside of M(β, Θ i ), and the distance of the one closest to t is within a threshold τ o , this intersection point is considered a valid mapping. Otherwise we look for the intersection that lies inside M(β, Θ i ), and chose the closest one to be a valid mapping if the distance to t is smaller than τ i . In case no valid mapping is found, t is removed from consideration for S i .

An example of all valid oriented anatomical points on two frames of a sequence is shown in Figure 2, where points are shown in blue and orientations in black.

Computation of point trajectories

Here we describe how to compute point trajectories given the input sequence and oriented anatomical points. First, we automatically select a template from all frames of the sequence. Second, we establish a dense correspondence between the template and each frame of the sequence with the help of a partial near-isometric deformation model.

Automatic template selection

Often in an acquired sequence the topology changes over time due to acquisition noise. In particular, tiny holes frequently appear and large contacts appear when limbs are close-by, see Figure 3 for an example. To remedy this problem during alignment, we automatically detect in the sequence a template T and register T to every other frame S of the sequence.

The template selection is based on topological and geometric criteria. Since we assume that the scene has a fixed topology and that observed topology changes come from acquisition artifacts, it follows that observed topological properties of the shape can only grow [START_REF] Letouzey | Progressive shape models[END_REF]. That is, observed splits are accepted as changes of the real shape while observed merges are ignored from consideration as the real shape cannot merge. To account for acquisition artifacts, splits are only considered if they are persistent over time and small components are filtered. As observed splits are accepted, we first select as candidate templates all frames that have a maximum number of components. As observed splits and merges are directly linked to the genus number, we further select from the candidate templates the frames with minimum genus of the largest component.

The aforementioned topological criteria might provide several candidate templates. The final selection is based on a geometric quality criterion. From the candidate templates, we select the one with minimum area ratio (maximum point area/minimum point area, where a point area is the area of the Voronoi region around a vertex) of the largest component, as the rest of our pipeline benefits from a template whose vertices are as uniform in area as possible. Figure 3 shows the selected template T for one of our test sequences.

Dense correspondence computation

To align T to a frame S , we use a partial near-isometric deformation model that is robust to topological acquisition noise. Nearisometric models have previously been used successfully for cloth modeling [START_REF] Popa | Wrinkling captured garments using spacetime data-driven deformation[END_REF]. Ideally, T and S should be mapped by a global near-isometric mapping. In practice, due to acquisition noise, such a global mapping may not exist.

To remedy this, we use a partial near-isometric model to account for topological acquisition noise as in Brunton et al. [START_REF] Brunton | A low-dimensional representation for robust partial isometric correspondences computation[END_REF]. To this end, we consider every frame S to be a set of smooth, orientable 2-manifolds embedded in threedimensional space. In practice, S is discretized by a set of points that are connected by a neighborhood graph. We denote the geodesic distance between two points s i , s j ∈ S by d S (s i , s j ).

Consider a mapping f : U → S , where U is a subset of T . The mapping function f is a near-isometry if:

|d U (t i , t j ) -d S (f (t i ), f (t j ))| ≤ (2)
where t i , t j are vertices in U and refers to the allowed stretching threshold. Since U ⊂ T refers to a shape part, we seek parts of T that can be mapped to parts of S without much stretching. Brunton et al. [START_REF] Brunton | A low-dimensional representation for robust partial isometric correspondences computation[END_REF] use the partial near-isometry model for pairwise frame alignment. They show that a correspondence between an oriented point on T and an oriented point on S is sufficient to recover an isometric mapping. In the following, we use t and s to denote both points and oriented points on T and S , respectively.

Due to the acquisition noise in the acquired data, it is necessary to define several corresponding oriented point pairs to recover a full alignment between T and S . It is known that the automatic computation of corresponding oriented point pairs is a computational bottleneck for the original method that renders the processing of motion sequences impractical.

We overcome this problem by using the automatically computed oriented anatomical points on each of the frames. For these oriented points, the correspondence information is known across frames. We use them as starting points for the partial near-isometric correspondence computation between T and S . To speed up the computation time, we minimize point correspondence information coming from many overlapping partial near-isometries that is discovered from nearby starting points. For this, we stop when the discovered near-isometric part has an intrinsic radius bigger than a distance threshold τ d . To increase robustness, we ignore from consideration near-isometric parts that have an intrinsic radius smaller than a threshold τ s .

After finding multiple near-isometric parts between T and S , we merge the parts into a global alignment by assigning to each t in T that is mapped to at least one point on S the geodesic average of all computed assignments on S . Figure 4 shows a color-coded alignment computed between T and S for one of the test sequences.

After this step, we have computed pairwise correspondence information between T and every frame S of the sequence. This allows us to follow the trajectory of a particular point on T over time.

Consistent topology alignment

In the previous step, we computed frame correspondences between T and every other frame S . To find a consistent topology that deforms over time we extend a standard template fitting technique [START_REF] Allen | The space of human body shapes: Reconstruction and parameterization from range scans[END_REF] to take advantage of the previously computed point trajectories. That is, the previously computed frame correspondence serves as assignment information to guide the template deformation to allow for fast motion and reduce drift.

In particular, let T contain N T vertices, and let the motion sequence contain N f frames S 1 , S 2 , . . . , S N f . Without loss of generality, let T be frame S j . We fit the frames of the sequence starting at T by processing adjacent frames in (backward and forward temporal) order as S j-1 , . . . , S 1 and S j+1 , . . . , S N f . After S i has been fitted, we update T to the fitting result of S i . This initializes the location and shape of the template to be close to the next frame to be fitted, thereby allowing for a simple template fitting scheme.

The template fitting follows a standard technique [START_REF] Allen | The space of human body shapes: Reconstruction and parameterization from range scans[END_REF], and fits T to S by optimizing the following energy

E template = w c E c + w d E d + w s E s + w t E t , (3)
which consists of the weighted linear combination of four simple energy terms: correspondence energy E c , nearest neighbor energy E d , deformation smoothness energy E s , and temporal smoothness energy E t , which are weighted by w c , w d , w s , and w t , respectively. To model the deformation, each vertex t i of T is expressed in homogeneous coordinates and transformed using a 4 × 4 matrix A i that represents an affine transformation in R 3 . The first energy term modifies the marker energy proposed by Allen et al. [START_REF] Allen | The space of human body shapes: Reconstruction and parameterization from range scans[END_REF] to use the previously computed correspondences from T to S as assignments. The energy is expressed as

E c = 1 N T i=1 w corr,i N T i=1 w corr,i A i t i -s c (t i ) 2 2 , (4)
where w corr,i is a weight equal to 1 if t i has a correspondence on S and equal to 0 otherwise, s c (t i ) is the point of S that corresponds to t i and • 2 is the Euclidean distance. Using this energy guides the deformation in case of large motion between adjacent frames and reduces drift. Note that unlike the marker term, our energy does not rely on any manually provided information on S , but takes advantage of the correspondences found using the partial near-isometric deformation model. The second energy term is a standard data term that pulls each t i to its nearest neighbor on S as

E d = 1 N T i=1 w N N,i N T i=1 w N N,i A i t i -s n (t i ) 2 2 , (5)
where w N N,i is a weight equal to 1 if the nearest neighbor is valid and 0 otherwise, and s n (t i ) is the point on S that is the nearest neighbor of t i . We consider the nearest neighbor valid if the surface normals at each point are less than 90 • apart and the distance between two points is at most 0.2m. When used without the correspondence energy (Equation ( 4)), this nearest neighbor energy is known to suffer from drift [START_REF] Budd | Global nonrigid alignment of surface sequences[END_REF]. To avoid this problem, we only activate this energy once the deformed template is close to S , thereby reducing drift. The third energy term is the standard deformation smoothness energy

E s = 1 N T i=1 |N (i)| j∈N (i) A i -A j 2 F , (6)
where N (i) is the 1-ring neighborhood of t i and • F is the Frobenius norm. This term encourages a smooth deformation field across T .

As the fourth term, we add a simple temporal smoothness energy to prevent very large deformations between adjacent frames as

E t = 1 N T N T i=1 A i -I 2 2 , ( 7
)
where I is the identity matrix. This energy discourages large displacements of individual vertices between adjacent frames and helps to prevent jittering in case of slightly inaccurate correspondences between T and S . Figure 5 shows an example of a template-fitted frame for one of our sequences, and Figure 6 shows the template fitting with and without E c , for an example where the target frame is only four frames away from T .

Implementation Details

For the human body estimation under clothing, we use the code of Yang et al. [START_REF] Yang | Estimation of human body shape in motion with wide clothing[END_REF]. We manually select 40 oriented anatomical points on the statistical model and map these points to the clothed sequence. To map points from human body to clothed human we set τ i = 0.2m and τ o = 0.05m.

To compute T over a given sequence we do not count as components shells with an area smaller than 0.1m 2 . To compute dense correspondence we use the code of Brunton et al. [START_REF] Brunton | A low-dimensional representation for robust partial isometric correspondences computation[END_REF] with = 0.25m. To increase robustness and computational efficiency, we set τ s = 0.1m and τ d = 0.25m.

The weights (w c , w d , w s , w t ) in the template fitting step are fixed by solving the optimization problem in several stages with different energy weights at each stage as proposed by Allen et al. [START_REF] Allen | The space of human body shapes: Reconstruction and parameterization from range scans[END_REF]. So that when the template is far away from the target frame, correspondence and smoothness terms can lead the deformation and we set the weights to (1, 400, 0, 0.1). Now the template is already closely enough to the target, we can use nearest neighbor information. We decrease the weight of spatial smoothness and increase the data term and set the weights to (1, 200, 10, 0.1). Next, we turn off the weights of the dense correspondence term and optimize the energy based on the other three weights. This is to address inaccurate dense correspondence information. We set the weights to (0, 100, 10, 0.1) and finally to (0, 50, 10, 0.1). The nearest neighbors are recomputed for each vertex every 20 iterations of optimization.

The energy E template optimized using a quasi-Newton method [START_REF] Liu | On the limited memory BFGS method for large scale optimization[END_REF].

Experiments

For better visualization, please refer to the supplemental video * .

Pretracked dataset We have tested our method against the methods by Allain et al. [START_REF] Allain | On mean pose and variability of 3d deformable models[END_REF][START_REF] Allain | An efficient volumetric framework for shape tracking[END_REF] on four sequences (crane, march2, samba, squat1) from Vlasic et al. [START_REF] Vlasic | Articulated mesh animation from multi-view silhouettes[END_REF]. Since this dataset is temporally registered we can use it as ground truth for our evaluation. Our method takes as input the original pretracked dataset and we compute as error the Euclidean distance between our estimated vertex position and the ground truth. Both methods by Allain et al. require preprocessing of the data, which involves manual selection of the template and downsampling of the input mesh. We first map the processed template to the pretracked template by a nearest neighbor approach and then proceed with the evaluation as described above. The preprocessing is done in favor of the reference algorithms.

All three methods share similar average quantitative results. The average vertex error over all sequences is 3.6cm, 3.9cm and 3.9cm for [START_REF] Allain | On mean pose and variability of 3d deformable models[END_REF], [START_REF] Allain | An efficient volumetric framework for shape tracking[END_REF] and our method respectively. The percentage of the vertices within 10cm error is 96.22, 94.68, and 96.83 respectively. Figure 7 shows the per-frame error on each sequence and for each method. Our method is * https://hal.inria.fr/hal-01367791 Figure 8. Results on sequences march2 (top row, frame #55) and samba (bottom row, frame #90) from [START_REF] Vlasic | Articulated mesh animation from multi-view silhouettes[END_REF]. From left to right: S , T , result with [START_REF] Allain | On mean pose and variability of 3d deformable models[END_REF], result with [START_REF] Allain | An efficient volumetric framework for shape tracking[END_REF], and our result. See also the supplementary video.

Figure 9. Quantitative results on sparse markers (dataset of [START_REF] Yang | Estimation of human body shape in motion with wide clothing[END_REF]). Top: cumulative error curves for [START_REF] Allain | On mean pose and variability of 3d deformable models[END_REF], [START_REF] Allain | An efficient volumetric framework for shape tracking[END_REF] and our method. Bottom: average marker error per frame of our method for all eight sequences. generally more robust in the sense the maximum error for a sequence is lower. As shown in Figure 8, our method also better follows the deformation of wide clothes and makes good use of the prior knowledge of the underlying human whereas the two other methods loose tracking of the knee.

Dataset with sparse markers

We have also tested our method on eight sequences from the dataset of Yang et al. [START_REF] Yang | Estimation of human body shape in motion with wide clothing[END_REF]. These sequences include two subjects, one male Figure 7. Per-frame result on 4 sequences from [START_REF] Vlasic | Articulated mesh animation from multi-view silhouettes[END_REF]. From left to right: our method, Allain et al. [START_REF] Allain | On mean pose and variability of 3d deformable models[END_REF] and Allain et al. [START_REF] Allain | An efficient volumetric framework for shape tracking[END_REF]. and one female, wearing either layered or wide clothes, either performing a walk or rotating the body. The mesh sequences were obtained using visual hull reconstructions from a 68-camera-system. Reconstruction artifacts like holes are present. Along with the mesh sequence, marker trajectories are also provided. Each subject has 14 markers placed on anatomically significant locations: forehead, shoulders, elbows, wrists, belly, knees, feet and heels. If these locations are covered by cloth while the subject is at resting pose then markers are placed on the clothes. We map each marker position to the nearest template vertex and use this mapping and correspondence information to compute the distance error of the alignment. We compute as error the Euclidean distance between our estimated location and the ground truth location of the marker.

Figure 9 shows a quantitative evaluation of our method on these sparse markers. The cumulative error curve, from which the template frame is excluded, shows that our method performs as well as the methods by Allain et al.

The bottom curves show that our method does not drift in time, because our dense correspondence is calculated from the template to each frame independently and is based on a partial near-isometric deformation model.

Figure 10 shows alignment results on representative sequences of the dataset of [START_REF] Yang | Estimation of human body shape in motion with wide clothing[END_REF]. As for the pretracked dataset, these results also demonstrate that our method is more robust than [START_REF] Allain | On mean pose and variability of 3d deformable models[END_REF] and [START_REF] Allain | An efficient volumetric framework for shape tracking[END_REF], where limbs may be switched or the tracking of clothing may get lost. The color-coding shows that our method does not suffer from significant drift. Artifacts produced in the visual hull reconstruction even have a chance to be repaired such as the tunnel appearing on the left leg for the last example.

Conclusions

In this paper we have presented a method for the temporal alignment of motion sequences of humans in wide clothing. In order to be robust to geometric and topological artifacts, we use reliable body landmarks which are mapped to the clothed surface and serve as starting points to grow partial near-isometric patches. These patches allow to compute pairwise correspondence information between Figure 10. Alignment result on three representative sequences of [START_REF] Yang | Estimation of human body shape in motion with wide clothing[END_REF]. From left to right: S , T , result with [START_REF] Allain | On mean pose and variability of 3d deformable models[END_REF], result with [START_REF] Allain | An efficient volumetric framework for shape tracking[END_REF] and our result.

an automatically selected template frame, based on topological and geometric criteria, and every other frame of the sequence. Correspondences are then used to guide the deformation from the template to the other frames.

Given a statistical human model with a sparse set of annotated anatomical points, our method is fully automatic. It relies on a few assumptions, which are valid in most practical cases. In particular, cloth is supposed to be inelastic and not to slip much along the body. The topology of the scene is supposed to stay constant, and at least one frame in the sequence needs to capture the correct topology.

Figure 1 .

 1 Figure 1. Method overview. Given an input sequence of 3D meshes shown in (a), each frame is processed in three steps, which are shown for the frame indicated in blue and shown in (b).First, a statistical model of undressed body shape is fitted to the frame (c). Pre-marked anatomical points on the fitted human body model are mapped to the input frame (d). Second, these anatomical markers are used to guide a partial near-isometric correspondence computation between an automatically selected template (green model in (a), and shown in top row) and the frame (e). Third, the resulting correspondences are used as assignments to deform the template to the input frame (f).

Figure 2 .

 2 Figure 2. Left: estimated human body M(β, Θ i ) shown in brown overlaid with two input frames. Right: oriented anatomical points on two meshes of a sequence. The points are shown in blue, and their orientations in black.

Figure 3 .

 3 Figure 3. Some frames of a sequence and the computed template T highlighted in green.

Figure 4 .

 4 Figure 4. Color-coded correspondence information computed between T (right) and S (left) using a partial near-isometric deformation model.

Figure 5 .

 5 Figure 5. Template T (left) and template-fitted frame S (right). Correspondence is color-coded.

Figure 6 .

 6 Figure 6. Our E c energy term prevents loss of tracking. From left to right: S , T , template deformation without E c , and with E c .

† http://kinovis.inrialpes.fr

Acknowledgments

This work has been supported by the ANR through the ACHMOV project (ANR-14-CE24-0030). We would like to thank Benjamin Allain for providing comparison code, Daniel Vlasic for providing the pre-tracked dataset and the Kinovis platform † for the sparse marker dataset.


Hakim Arezki

Mohamed Boutchich
email: mohamed.boutchich@upmc.fr

David Alamarguy

Ali Madouri

José Alvarez

Pere Roca I Cabarrocas

Jean-Paul Kleider

Fei Yao

Young Hee Lee

Electronic properties of embedded graphene: doped amorphous silicon/CVD graphene heterostructures

Large-area graphene film is of great interest for a wide spectrum of electronic applications, such as field effect devices, displays, and solar cells, among many others. Here, we fabricated heterostructures composed of graphene (Gr) grown by chemical vapor deposition (CVD) on copper substrate and transferred to SiO2/Si substrates, capped by n or p-type doped amorphous silicon (a-Si:H) deposited by plasma-enhanced chemical vapor deposition.

Using Raman scattering we show that despite the mechanical strain induced by the a-Si:H deposition, the structural integrity of the graphene is preserved. Moreover, Hall effect measurements directly on the embedded graphene show that the electronic properties of CVD graphene can be modulated according to the doping type of the a-Si:H as well as its phase i.e. amorphous or nanocrystalline. The sheet resistance varies from 360 Ω/sq to 1260 Ω/sq for the (p)-a-Si:H/Gr (n)-a-Si:H/Gr, respectively. We observed a temperature independent hole mobility of up to 1400 cm²/Vs indicating that charge impurity is the principal mechanism limiting the transport in this heterostructure. We have demonstrated that embedding CVD graphene under a-Si:H is a viable route for large scale graphene based solar cells or displays applications.

Introduction

The recent advances in large-scale synthesis of graphene and alternative 2D materials [START_REF] Maitra | Strategies for the Synthesis of Graphene, Graphene Nanoribbons[END_REF][START_REF] Kumar | Synthesis and Biomedical Applications of Graphene[END_REF][START_REF] Boutchich | Atmospheric pressure route to epitaxial nitrogen-doped trilayer graphene on 4H-SiC[END_REF] offer new opportunities to investigate various heterostructures composing the building blocks of new devices and architectures [START_REF] Ho | A Self-Aligned High-Mobility Graphene Transistor: Decoupling the Channel with Fluorographene to Reduce Scattering[END_REF][START_REF] Brus | Stability of graphene-silicon heterostructure solar cells[END_REF]. However, the electronic properties of graphene differ according to the growth technology utilized [START_REF] Petridis | Solution processed reduced graphene oxide electrodes for organic photovoltaics Nanoscale Horiz[END_REF], the grain size, the interaction with the underlying substrate as well as the non-intended doping that systematically occurs during fabrication [START_REF] Criado | The Covalent Functionalization of Graphene on Substrates[END_REF]. In addition, graphene and 2D materials are extremely sensitive to environment, implying uncontrollable variations such as the threshold voltage of 2D based field effect devices [START_REF] Bartolomeo | Charge transfer and partial pinning at the contacts as the origin of a double dip in the transfer characteristics of graphene-based field-effect transistors[END_REF][START_REF] Bong | Facile graphene n-doping by wet chemical treatment for electronic applications[END_REF]. On the other hand, the modulation of the electronic properties of graphene is a key prerequisite to envision its integration as an active material but also as an electrode [START_REF] Loh | Graphene and Graphene-like Molecules: Prospects in Solar Cells[END_REF]. In that regard, a lot of work has been devoted to the study of more or less stable organic or inorganic materials deposited onto graphene [START_REF] Aktürk | Bismuth doping of graphene[END_REF][START_REF] Boutchich | Electronic Properties of Large Area Nitrogen Doped Trilayer Graphene on 4H-SiC[END_REF][START_REF] Dong | Doping Single-Layer Graphene with Aromatic[END_REF], [START_REF] Stylianakis | Efficient ternary organic photovoltaics incorporating a graphene-based porphyrin molecule as a universal electron cascade material[END_REF], [START_REF] Stylianakis | Photochemical Synthesis of Solution-Processable Graphene Derivatives with Tunable Bandgaps for Organic Solar Cells[END_REF], [START_REF] Georgakilas | Noncovalent Functionalization of Graphene and Graphene Oxide for Energy Materials[END_REF]. Field effect doping and dopant incorporation at these interfaces allow the modulation of the work function, the carrier density and sheet resistance as well as the mobility [START_REF] Kwon | Increased Work Function in Few-Layer Graphene Sheets via Metal Chloride Doping[END_REF][START_REF] Das | Effect of HNO3 functionalization on large scale graphene for enhanced tri-iodide reduction in dye-sensitized solar cells[END_REF][START_REF] Gunes | Tuning the work function of monolayer graphene on 4H-SiC (0001) with nitric acid[END_REF][20][START_REF] Kim | Buried graphene electrodes on GaN-based ultra-violet light-emitting diodes[END_REF], [START_REF] Yu | The enhanced efficiency of graphene-silicon solar cells by electric field doping[END_REF], [START_REF] Petrone | Chemical Vapor Deposition-Derived Graphene with Electrical Performance of Exfoliated Graphene[END_REF]. Kim et al. [21] demonstrated GaN-based light-emitting diodes where the conductivity of graphene was substantially improved when capped with silicon nitride (SiNx) but did not discuss the electrostatic impact of the fixed charges trapped in the SiNx. Gluba et al. [START_REF] Gluba | Embedded graphene for large-area silicon-based devices[END_REF] studied CVD graphene capped with a-Si:H and polycrystalline silicon. Coupling Raman spectroscopy and Hall effect measurements, they observed that the structural features of graphene where preserved and the mobility is limited at room temperature due to phonon scattering [START_REF] Chauhan | Inelastic Phonon Scattering in Graphene FETs[END_REF][START_REF] Chen | Charged-impurity scattering in graphene[END_REF][START_REF] Chen | Intrinsic and extrinsic performance limits of graphene devices on SiO2[END_REF], [START_REF] Yasaei | Bimodal Phonon Scattering in Graphene Grain Boundaries[END_REF], [START_REF] Alexeev | Two-phonon scattering in graphene in the quantum Hall regime[END_REF] . They concluded that large-scale CVD graphene applications might be hindered due this latter effect. Recently, Ho et al. [START_REF] Ho | Sunlight-activated graphene-heterostructure transparent cathodes: enabling high-performance n-graphene/p-Si Schottky junction photovoltaics[END_REF] demonstrated a graphene/TiOx photoactive heterostructure cathode to fabricate an (n) graphene/(p) Si Schottky junction with power conversion efficiency > 10 %. These recent advances [START_REF] Kiraly | Graphene-Silicon Heterostructures at the Two-Dimensional Limit[END_REF] shine light on the integration of graphene in solar cell applications that are already competitive with respect to organic based demonstrators but also on field effect devices where graphene can be implemented as a channel or a floating gate [START_REF] Boutchich | Atmospheric pressure route to epitaxial nitrogen-doped trilayer graphene on 4H-SiC[END_REF][START_REF] Hong | Graphene Flash Memory[END_REF].

Here, we investigate graphene/doped amorphous silicon heterostructures. The chemical elements present on the graphene were determined by X-ray spectroscopy (XPS) and the work function was measured by ultraviolet photoemission spectroscopy (UPS). The homogeneity of the transferred graphene layers was confirmed by Raman spectroscopy. The latter also showed that the mechanical strain hinders the field effect doping resulting from the fixed charges in the capping a-Si:H. Hall effect measurements as a function of temperature show that the doping nature of the a-Si:H influences the electronic properties such as the sheet resistance that drops down to 430 Ω/sq for the (p) a-Si:H/Gr for a hole mobility of 1400 cm²/Vs. Moreover, our measurements do not present any temperature dependence, that is in a sharp contrast with the work of Gluba et al. [START_REF] Gluba | Embedded graphene for large-area silicon-based devices[END_REF] on undoped a-Si:H. At last we have shown that the recrystallization of the doped a-Si:H film relaxes the mechanical strain [START_REF] Lee | Measurement of the Elastic Properties and Intrinsic Strength of Monolayer Graphene[END_REF], [START_REF] Lee | High-Strength Chemical-Vapor-Deposited Graphene and Grain Boundaries[END_REF] and only the field effect doping remains. We have demonstrated that embedding CVD graphene under a-Si:H is a route for large scale graphene based solar cells or display applications.

Heterostructures and experiments

Graphene layers were grown on commercially available 100 µm thick copper foil (from Nilaco, 99.96%) annealed at 1075 °C with 1000 sccm Ar and 500 sccm H2 for 2h by CVD at atmospheric pressure. The annealed copper foil was polished using the chemical mechanical polishing method (6 ml copper etchant was diluted in 100ml water). The polished Cu film was placed in a 2 inch quartz tube chamber and heated up to 1075 °C with 1000 sccm Ar and 500 sccm H2 for annealing for 2h to remove residuals on the copper surface [START_REF] Güneş | Large-area graphene-based flexible transparent conducting films[END_REF][START_REF] Wu | Controllable poly-crystalline bilayered and multilayered graphene film growth by reciprocal chemical vapor deposition[END_REF]. During the atmospheric CVD growth, the H2 gas was reduced to 170 sccm and 200 sccm of CH4 (0.1% diluted in Ar was injected) for 25 min.

After growth, the CH4 was turned off and the chamber was cooled down to room temperature. The copper foil was subsequently etched in FeCl3 etchant and the graphene was transferred onto SiO2/Si substrate using PMMA resist.

To fabricate the heterostructure, p-type and n-type a-Si:H films were deposited by standard plasma-enhanced chemical vapor deposition at 175 °C using a multiplasma-monochamber reactor [START_REF] Roca I Cabarrocas | A fully automated hot-wall multiplasma-monochamber reactor for thin film deposition[END_REF], from the dissociation of silane mixed with trimethyl-boron or phosphine. The oxygen content of the films, as deduced from secondary ion mass spectrometry measurements is ~ 2x10 19 cm -3 resulting from the residual impurities in the reactor [START_REF] Vignoli | Over-coordination and order in hydrogenated nanostructured silicon thin films: their influence on strain and electronic properties[END_REF].

We have studied both configurations : i) the graphene on top of a-Si:H and graphene coated by the a-Si:H thin films which have the advantage of providing a encapsulating layer making the measurements less sensitive to the atmosphere conditions.

We have investigated the electronic properties of graphene based heterostructures: (i) graphene on top of doped a-Si:H on silicon oxide and (ii) graphene on SiO2 capped with 10 nm doped a-Si:H, Figure 1. The Raman measurements were performed with a frequency-doubled NdYag laser operating at 532 nm as the excitation source with up to 20 mW continuous wave output power. High laser power can induce crystallization of the a-Si:H, so the laser power was judiciously tuned to investigate the Raman signature before and after crystallization of the a-Si:H.

Results and discussion

Let us first address the case where the graphene films are deposited on top of the a-Si:H layers. Figure 2 presents the Raman spectra of CVD graphene transferred onto 10 nm n-and p-type a-Si:H. All samples show a peak at 480 cm -1 , characteristic of a-Si:H, as well as the G and 2D peaks of graphene. However, the G peak shift is 7 cm -1 and the 2D peak shift is 3 cm -1 on the p-type a-Si:H with respect to the n-type a-Si:H. This indicates that the graphene on p-type a-Si:H/Gr/SiO 2 Gr/SiO 2 a-Si:H has been slightly n-type doped. This doping may result from contamination subsequent to the transfer process or possibly from a charge transfer at the heterostructure interface, Figure 2. Table 1: Transport properties for graphene transferred onto amorphous silicon/silicon oxide stacks. These data represent average between 1.3 K and 300 K.

Table 1 shows the temperature dependence of the Hall-effect mobility, charge carrier density, and sheet resistance of transferred CVD graphene (Gr) on top of SiO2, as well as on (p)-a-Si:H/ SiO2 and (n)-a-Si:H/SiO2 stacks. We do not observe any dominant scattering mechanism and the as-transferred layers exhibit hole conduction with electrical properties that are nearly temperature independent. Note that the sheet resistance of graphene on both types of a-Si:H films presents values that become interesting for transparent electrode application without any additional doping process. UPS measurements confirm that the heterostructures are electronically different. WF is equal to 4.35 eV for the Gr/(p)-a-Si:H heterostructure and 4.15 eV for the Gr/(n)-a-Si:H one. However, it is questionable to assign this 0.2 eV difference to a charge transfer at the interface of the heterostructure. The carrier density is doubled between Gr/(p)-a-Si:H and Gr/(n)-a-Si:H while the mobility decreases from 2100 to 1800 cm²/Vs. It is likely that this change is a consequence of the unintended doping during the transfer and fabrication processes.

In order to elucidate whether there could be charge transfer between a-Si:H and graphene, we have studied capped graphene i.e. (n)-a-Si:H/Gr and (p)-a-Si:H/Gr. Here, a 10 nm thick doped a-Si:H layer has been deposited onto Gr and Hall bars have been patterned. The graphene has been contacted first and the a-Si:H film has been etched to reach the contact pads. Figure 3 shows the Raman spectra of the different heterostructures benchmarked to the graphene onto SiO2. The measurements are carried out directly on different locations of the Hall bar device to take into account the homogeneity of the sample.

Figure 3: Raman spectra of the graphene on SiO2 substrate capped with doped a-Si:H. For clarity, the spectra are shifted vertically.

Interestingly, the D, G and 2D Raman modes of the graphene are preserved after deposition of a-Si:H indicating that the capping process is nondestructive although the low intensity of the D mode is slightly higher for the (n)-a-Si:

H/Gr compared to the (p)-a-Si:H/Gr heterostructure. However compared to the Gr/SiO2 sample, the G and 2D modes of both a-Si:H based heterostructures exhibit a shift to lower frequencies, Figure 3. Moreover, for the (p)-a-Si:H/Gr heterostructure, the 2D/G ratio decreases. This trend could result from mechanical strain induced by the boron or phosphorus doped a-Si:H layers [START_REF] Mohiuddin | Uniaxial strain in graphene by Raman spectroscopy: G peak splitting, Grüneisen parameters, and sample orientation[END_REF][START_REF] Huang | Probing Strain-Induced Electronic Structure Change in Graphene by Raman Spectroscopy[END_REF], or from field effect doping from the charges at the interface.

Although field effect doping cannot be excluded, it is clearly hindered by the mechanical strain that is the dominant 1400 16001800 2100 2650 2700 2750

ω G =1594cm -1 ω G = 1592cm -1 ω G = 1600cm -1 Intensity (a.u) (n)-a-Si:H / Gr/SiO 2 (p)-a-Si:H / Gr/SiO 2 Gr / SiO 2 ω a-Si:H = 2000 cm -1
Raman Shift (cm -1 )

ω 2D = 2690cm -1 ω 2D = 2677cm -1 ω 2D = 2691cm -1
mechanism as can be seen by the downshift observed on both the G and 2D modes for the two types of heterostructures [START_REF] Casiraghi | Probing disorder and charged impurities in graphene by Raman spectroscopy[END_REF][START_REF] Zabel | Raman Spectroscopy of Graphene and Bilayer under Biaxial Strain: Bubbles and Balloons[END_REF].

We carried out Hall effect measurements as a function of temperature to evaluate the electronic properties of the capped graphene. Figure 4 shows the sheet resistance, mobility and carrier density of the embedded graphene benchmarked to graphene on SiO2. Here both capped heterostructures i.e. (n,p) a-Si:H/Gr, present hole conduction with an average mobility of 1400 cm²/Vs and a sheet resistance as low as 420 Ω/sq for (p)-a-Si:H/Gr. For the latter the carrier density has increased to 1.3x10 13 cm -² compared to the 9.8x10 12 cm -² measured on SiO2 that is known to present hole accumulation. On the other hand, (n)-a-Si:H/Gr presents a lower mobility of 600 cm²/Vs with a sheet resistance of 1300 Ω/sq and a carrier density of 8x10 12 cm -². The lower mobility observed for the (n)-a-Si:H/Gr may be a result of structural defects induced by the mechanical strain. Clearly, the doping type of the as-deposited a-Si:H influences the electronic properties through field effect doping induced by the fixed charges at the interfaces notably for the p-type sample.

This doping has already been suggested by the Raman spectra of Figure 3; although the shift induced by the doping is hindered by the mechanical strain, the 2D mode intensity of the Gr/(p)-a-Si:H reduces and the full width half maximum increases. Using the measured carrier density we can calculate the position of the Fermi level EF relative to the Dirac point ED: ED-EF = -0.33 eV for (n)-a-Si:H/Gr and ED-EF = 0.65 eV for (p)-a-Si:H/Gr. Note that we do not observe any temperature dependence that is in contrast with the work on graphene capped with undoped a-Si:H by Gluba et al. that observed a strong temperature dependence of capped graphene assigned to phonon assisted scattering starting at 100 K [START_REF] Gluba | Embedded graphene for large-area silicon-based devices[END_REF]. In our configuration, the graphene sheet is embedded between SiO2 substrate and doped a-Si:H. These two materials are both impurity rich and therefore inherently hinder the electronic transport. It is therefore reasonable to consider the charge impurity scattering as the dominant mechanism. Nevertheless, the temperature dependence of the phonon scattering often observed on graphene on SiO2 is not absent but rather postponed beyond 300 K as discussed by Chen et al. for impurity rich samples by [START_REF] Chen | Charged-impurity scattering in graphene[END_REF][START_REF] Fratini | Substrate-limited electron dynamics in graphene[END_REF]. Nevertheless, despite this limitation, the magnitude of the mobility observed in the capped graphene remains very interesting for the largescale application of graphene. As shown in Figure 4, the graphene films capped with p-type a-Si:H have a higher mobility and carrier concentration, leading to a square resistance of 420 Ohm/square, which makes them suitable as an electrode for heterojunction solar cells and/or gate contact.

At last we have investigated the crystallization of a-Si:H on the Raman spectra using the 532 nm laser. The a-Si:H crystallizes at a power of 2.15 mW. Figure 5 presents the Raman spectra for both heterostructures before and after crystallization. We observe that the a-Si:H peaks at 480 cm -1 and 2000 cm -1 disappear whereas the c-Si peak appears around 520 cm -1 .

Figure 5: Raman spectra before and after crystallization of a) (p)-a-Si:H/Gr heterostructure and b) (n)-a-Si:H/Gr heterostructure. The as-deposited spectra present a shoulder at 480 cm -1 which is absent on the recrystallized spectra characterized by a single peak at ~ 520 cm -1 .

The G and 2D bands in the (p)-a-Si:H/Gr heterostructure exhibit an upshift of 19 and 14 cm -1 respectively. The (n)a-Si:H/Gr heterostructure, also presents upshifts for both G and 2D bands but of lower magnitude, 4 cm -1 for both G and 2D modes. This trend indicates two mechanisms: (i) the release of the compressive strain, (ii) a p-type doping.

The increase in the D band intensity implies an increase in the defect density assigned to the volume change of the Si thin film during crystallization. It is particularly pronounced for the (p)-a-Si:H/Gr heterostructure. Moreover, the 2D/G ratio also reduces both samples suggesting an increase in carrier density i.e. doping originating from the Gr/SiO2 and/or a-Si:H/Gr interfaces. The strength is nevertheless stronger for the (p)-a-Si:H/Gr heterostructure. It is known that (p)-a-Si:H conductivity increases after crystallization [START_REF] Hamma | Low-temperature growth of thick intrinsic and ultrathin phosphorous or boron-doped microcrystalline silicon films: Optimum crystalline fractions for solar cell applications Sol[END_REF] and therefore more charges are available for transferring to the graphene. These data show that embedded CVD graphene maintains its features and the transport properties can be modulated according to the nature and the phase of the deposited a-Si:H.

  XPS and UPS measurements for the study of the chemical composition and the work function (WF), respectively, have been performed using a PHI 5000 Versaprobe spectrometer (Physical Electronics) operating at a base pressure of 10 -9 mbar. For the XPS analysis, a focused monochromatic Al Kα X-ray source (hν = 1486.6 eV) was used. The work function was determined from the secondary electron cut-off in UPS experiments using a He I discharge lamp (hν = 21.2 eV). The position of the Fermi level was calibrated by measuring the Fermi edge of a sputtered gold sample. During the WF measurements, a bias of -8 V was applied to the sample. Pieces of the doped a-Si:H/Gr heterostructures have been patterned in Hall bars using optical lithography to investigate the transport properties of the encapsulated graphene, Figure 1. These measurements were performed in an ambient environment.

Figure 1 :

 1 Figure 1: Optical image of the graphene sample utilized in the experiments. The yellow region corresponds to the graphene capped with a-Si:H whereas the green region corresponds to graphene on SiO2. The light green region corresponds to a thinner thickness of a-Si:H that diffused under the hard mask during the deposition.

Figure 2 :

 2 Figure 2: Raman spectra of CVD graphene transferred onto n-and p-type a-Si:H. Indeed, XPS measurements (not shown) showed the presence of oxygen (O), fluorine (F), nitrogen (N) and chlorine(Cl) species on graphene. Oxygen is known as a p-type dopant whereas F and N are n-type dopants[START_REF] Cresti | Oxygen Surface Functionalization of Graphene Nanoribbons for Transport Gap Engineering[END_REF]. The concentrations of these species vary from sample to sample and they are not controllable. We have carried out transport measurements on graphene on top of the doped a-Si:H.

Figure 4 :

 4 Figure 4: Graphene Hall bars capped with a-Si:H (a) optical image (30x70 µm), (b) schematic of the Hall bars, temperature dependence of (c) sheet resistance, (d) Hall mobility and (e) charge carrier density of graphene capped with n or p-type a-Si:H. The dashed line represents data on SiO2.

  -a-Si:H /Gr / SiO 2 (n)-µ-Si (n) / Gr / SiO 2Raman Shift (cm -1 ) -a-Si:H /Gr /SiO 2 (p)-µ-si /Gr/SiO 2 Raman Shift (cm -1 )

ACKNOWLEDGMENT

The authors would like to thank the Ministère des Affaires Etrangères et Européennes (MAEE) and CNRS for their financial support through the STIC ASIA programme (3226/DGM/ATT/RECH). This work was also supported by a public grant overseen by the French National Research Agency (ANR) as part of the "Investissements d'Avenir" program (Labex NanoSaclay, reference: ANR-10-LABX-0035).

Conclusion

In this work, we have investigated the capping of CVD graphene film by n-and p-type doped a-Si:H. The deposition process of the a-Si:H has a limited impact on the structural integrity of graphene as observed by Raman spectroscopy and the graphene features are clearly visible on all heterosturctures. However the electronic properties are affected whether graphene is capped by n-or p-type a-Si:H. Post deposition Raman shifts indicate that the mechanical strain is dominating however the reduction of the I2D/IG ratio particularly for the (p)-a-Si:H/Gr heterostructure suggests a field effect doping as well. This observation has been evidenced by Hall effect measurements showing a substantial reduction of the sheet resistance. The transport is limited by the charge impurities from the SiO2 and the doped a-Si:H, the hole mobility reaches up to 1400 cm²/Vs without temperature dependence suggesting that phonon scattering is not dominant up to 300 K but would appear at higher temperature.

At last we have shown that the recrystallization of the doped a-Si:H film relaxes the mechanical strain and only the field effect doping remains. We have demonstrated that embedding CVD graphene under a-Si:H is a route for large scale graphene based solar cells or displays applications.