Antoine Ayache
email: antoine.ayache@math.univ-lille1.fr

Yassine Esmili
email: yassine.esmili@math.univ-lille1.fr

Wavelet analysis of a

Keywords: Chaotic multifractional process in an arbitrary Wiener chaos Wiener chaos, self-similar processes, modulus of continuity, wavelet bases, fractional processes AMS Subject Classification (2010): 60G17, 60G22

Introduction

Fractional Brownian motion (fBm) of an arbitrary Hurst parameter H ∈ (0, 1), denoted by {B H (t) : t ∈ R}, is defined, up to a multiplicative constant, as the unique (in distribution) Gaussian process with stationary increments which is globally self-similar of order H. Recall that a stochastic process {F (t) : t ∈ R} is said to be with stationary increments if, for any fixed point t 0 ∈ R, one has:

F (t 0 + u) -F (t 0 ) : u ∈ R law = F (u) -F (0) : u ∈ R ; (1.1)
and it is said to be globally self-similar of order H if, for each fixed positive real number ν, one has:

ν -H F (νt) : t ∈ R law = F (t) : t ∈ R . (1.
2)

The representation of fBm as a well-balanced moving average is given, for every t ∈ R, by the Wiener integral over R:

B H (t) = R |t -s| H-1/2 -|s| H-1/2 dB(s), (1.3)
with the convention that |t -s| 0 -|s| 0 = log |t -s| -log |s|. FBm was first introduced by Kolmogorov in 1940 as a way for generating Gaussian spirals in Hilbert spaces [START_REF] Kolmogorov | Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum[END_REF]. Later, in 1968, the well-known article [START_REF] Mandelbrot | Fractional Brownian Motions, Fractional Noises and Applications[END_REF] by Mandelbrot and Van Ness emphasised its importance as a model in several areas of application: hydrology, geology, finance, and so on. Since then many applied and theoretical aspects of this stochastic process have been extensively explored in the literature and, among many other things, its sample behavior has been well understood.

Despite its importance in modeling, fBm does not always succeed in giving a sufficiently reliable description of real-life signals. Indeed, fBm suffers from two main limitations:

(a) its Gaussian character, (b) local roughness of its sample paths remains everywhere the same; more precisely, their local and pointwise Hölder exponents are everywhere equal to the Hurst parameter H.

In order to overcome the limitation (b) of fBm, the so-called multifractional Brownian motion (mBm) was introduced, about twenty years ago, independently by Benassi, Jaffard and Roux [START_REF] Benassi | Elliptic gaussian random processes[END_REF] and by Lévy Vehel and Peltier [START_REF] Peltier | Multifractional Brownian Motion : Definition and Preliminary Results[END_REF]. The latter continuous Gaussian process with non-stationary increments can be obtained by substituting to the constant Hurst parameter H in (1.3) a deterministic continuous function H(•) depending on t and with values in the open interval (0, 1). Nowadays mBm has become a quite useful model in the fields of financial modeling and signal processing (see for instance [START_REF] Bianchi | Pathwise identification of the memory function of multifractional Brownian motion with application to finance[END_REF][START_REF] Bianchi | Multifractional properties of stock indices decomposed by filtering their pointwise Hölder regularity[END_REF][START_REF] Bianchi | Modeling stock prices by multifractional Brownian motion: an improved estimation of the pointwise regularity[END_REF]).

In order to overcome the limitations (a) and (b) together, extensions of mBm whose Hurst parameter is a stochastic process or more generally a sequence of stochastic processes were introduced in [START_REF] Ayache | Multifractional Processes with Random Exponent[END_REF][START_REF] Ayache | Wavelet construction of generalized multifractional processes[END_REF]. Other extensions of mBm to frames of heavy-tailed stable distributions were proposed in [START_REF] Stoev | Stochastic properties of the linear multifractional stable motion[END_REF][START_REF] Stoev | Path properties of the linear multifractional stable motion[END_REF][START_REF] Dozzi | Real harmonizable multifractional stable process and its local properties[END_REF]. More recently, [START_REF] Shevchenko | Local times for multifractional square Gaussian processes[END_REF] constructed a multifractional generalized Rosenblatt process belonging to the second order homogeneous Wiener chaos. In our present article, we construct a multifractional process, denoted by {Z(t) : t ∈ R}, which belongs to a homogeneous Wiener chaos of an arbitrary integer order d ≥ 2. The latter multifractional process is not a generalization of the Rosenblatt process but of a process {Y H (t) : t ∈ R} consisting in a very natural chaotic extension of the fBm in (1.3). Namely, it is defined, for all t ∈ R, through the multiple Wiener integral on R d :

Y H (t) = R d t * -x H-d 2 2 -x H-d 2 2 dB x 1 ...dB x d , (1.4)
where t * = (t, ..., t) ∈ R d and • 2 denotes the Euclidian norm over R d . A class of chaotic self-similar processes with stationary increments, which implicitly includes {Y H (t) : t ∈ R}, had been first introduced and investigated in [START_REF] Mori | The law of the iterated logarithm for self-similar processes represented by multiple wiener integrals[END_REF]. Long time later, {Y H (t) : t ∈ R} was explicitly introduced and studied in its own right in [START_REF] Arras | On a class of self-similar processes with stationary increments in higher order Wiener chaoses[END_REF] through wavelet methods inspired by the ones in [START_REF] Ayache | Multifractional Processes with Random Exponent[END_REF][START_REF] Ayache | Wavelet construction of generalized multifractional processes[END_REF].

Recall that a centred non-Gaussian square integrable real-valued random variable, on the underlying probability space (Ω, F, P), belongs to the homogeneous Wiener chaos of an arbitrary integer order d ≥ 2 when it can be represented by a multiple Wiener integral over R d . We always denote by I d (•) this stochastic integral, and use the classical convention that, for every

f ∈ L 2 (R d ), one has I d (f ) = I d ( f ); the function f being the symmetrization of f , defined, for all (t 1 , ..., t d ) ∈ R d , as f (t 1 , ..., t d ) = 1 d! σ∈S d f (t σ(1) , ..., f (t σ(d) ))
, where S d refers to the set of all permutations of {1, ..., d} (observe that d! is the cardinality of S d ). A very important property of multiple Wiener integrals, which somehow can be viewed as an isometry property, is that, for all function f ∈ L 2 (R d ), one has

E |I d (f )| 2 = d! f 2 L 2 (R d ) ≤ d! f 2 L 2 (R d ) . (1.5)
Before ending these very short recalls on multiple Wiener integrals, it is worth mentioning that two well-known books on them and related topics are [START_REF] Janson | Gaussian Hilbert Spaces[END_REF][START_REF] Nualart | The Malliavin calculus and related topics[END_REF].

Roughly speaking, we define Z = {Z(t) : t ∈ R}, the multifractional generalization of

Y H = {Y H (t) : t ∈ R}, as Z(t) = Y H(t) (t)
, for all t ∈ R, where H(•) is an arbitrary deterministic continuous function over R with values in the open interval (0, 1). More precisely, let us consider the chaotic stochastic field X = {X(u, v) : (u, v) ∈ R × (0, 1)}, such that, for every (u, v) ∈ R × (0, 1), one has

X(u, v) = R d u * -x v-d 2 2 -x v-d 2 2 dB x 1 ...dB x d . (1.6)
We mention in passing that, for each fixed v ∈ (0, 1), the stochastic processes

X(•, v) = {X(u, v) : u ∈ R} and Y v = {Y v (u) : u ∈ R} have the same law. The multifractional process Z = {Z(t) : t ∈ R} is defined, for all t ∈ R, as Z(t) = X(t, H(t)) . (1.7)
By expanding, for each fixed (t, H) ∈ R × (0, 1), the kernel function

x → t * -x H-d 2 2 - x H-d 2 2
in (1.4) into a Meyer wavelet basis of L 2 (R d ) (see e.g. [START_REF] Lemarié | Ondelettes et bases hilbertiennes[END_REF][START_REF] Meyer | Wavelets and operators[END_REF][START_REF] Daubechies | Ten Lectures on Wavelets[END_REF]), a random series representation for the chaotic fractional process {Y H (t) : t ∈ R} has been constructed in [START_REF] Arras | On a class of self-similar processes with stationary increments in higher order Wiener chaoses[END_REF], which also has shown that this series is almost surely absolutely convergent, for each fixed (t, H) ∈ R × (0, 1). The main goal of Section 2 of our article is to transpose these two results into the setting of the chaotic stochastic field {X(u, v) : (u, v) ∈ R × (0, 1)}, and more importantly to show that the random series representation of this field is almost surely uniformly convergent in (u, v) on each compact subset of R × (0, 1

Uniformly convergent random series representation

First, we need to introduce some additional notations. We denote by S(R d ) the Schwartz class, that is the space of the infinitely differentiable complex-valued functions over R d which,

as well as all their partial derivatives of any order, vanish at infinity faster than any power function.

Let E = {0, 1} d \{(0, ..., 0)}. A Meyer wavelet basis of L 2 (R d ) is an orthonormal (or Hilbertian) basis of L 2 (R d ) of the form: 2 jd 2 ψ ( ) (2 j x -k) : j ∈ Z, k ∈ Z d , ∈ E ; (2.1)
for the sake of convenience, one sets:

ψ ( ) j,k (x) = 2 jd 2 ψ ( ) (2 j x -k). (2.
2)

The 2 d -1 real-valued functions ψ ( ) , ∈ E, which generate the basis are called the d-variate

Meyer mother wavelets. They can be expressed as tensor products of ψ 0 and ψ 1 which respectively denote a 1-variate Meyer father and mother wavelet. More precisely, for each = ( 1 , . . . , d ) ∈ E and x = (x 1 , . . . , x d ) ∈ R d , one has that:

ψ ( ) (x) = d l=1 ψ l (x l ) . (2.3)
Let us emphasize that the 1-variate Meyer father and mother wavelet belong to S(R). Morever, their Fourier transforms F(ψ 0 ) and F(ψ 1 ) are infinitely differentiable compactly supported functions satisfying: supp

F(ψ 0 ) ⊂ [-4π 3 , 4π 3 ] , supp F(ψ 1 ) ⊂ ξ ∈ R : 2π 3 ≤ |ξ| ≤ 8π 3 ,
and

F(ψ 0 )(ξ) = 1, for all ξ ∈ [-2π 3 , 2π 3
]. Thus, in view of (2.3), the d-variate Meyer mother wavelets ψ ( ) , ∈ E, belong to S(R d ) and have infinitely differentiable compactly supported Fourier transforms which vanish in a neighborhood of 0. Using these nice properties of the d-variate Meyer mother wavelets, for each ∈ E, it can be shown (see [START_REF] Ayache | Multifractional Processes with Random Exponent[END_REF][START_REF] Ayache | Wavelet construction of generalized multifractional processes[END_REF] for instance) that the real-valued function Ψ defined, for all (u, v) ∈ R d × [0, 1], as

Ψ (u, v) = R d u -s v-d/2 2 ψ ( ) (s)ds ,
is infinitely differentiable on R d × (0, 1) and satisfies, as well as all its partial derivatives of any order, the following very useful localization property:

∀ (n, p, q) ∈ Z + ×Z d + ×Z + , sup α+ u 2 n (∂ p u ∂ q v Ψ )(u, v) : (u, v) ∈ R d ×(0, 1) < +∞ , (2.4)
where α is an arbitrary positive fixed real number.

Before giving the random series representation of the field X derived from the Meyer wavelet basis (2.1), let us state the following important lemma borrowed from [START_REF] Arras | On a class of self-similar processes with stationary increments in higher order Wiener chaoses[END_REF].

Lemma 2.1 For each (j, k, ) ∈ Z × Z d × E, let I d (ψ ( ) j,k
) be the multiple Wiener integral over R d of the wavelet function defined in (2.2). That is one has

I d (ψ ( ) j,k ) = R d ψ ( ) j,k (x)dB x 1 ...dB x d . (2.5)
Then, there exists an event Ω * of probability 1 and a finite positive random variable C d such that, for all ω ∈ Ω * and for each (j, k,

) ∈ Z × Z d × E, one has |I d (ψ ( ) j,k )(ω)| ≤ C d (ω) log(e + |j| + k 1 ) d 2 , (2.6)
where • 1 denotes the 1-norm over R d ; that is k 1 = d l=1 |k l |, the k l 's being the coordinates of k.

The following proposition provides the random series representation of the field X derived from the Meyer wavelet basis (2.1). The proposition has been obtained in [START_REF] Arras | On a class of self-similar processes with stationary increments in higher order Wiener chaoses[END_REF] with Y H in place of X (see (1.4) and (1.6)).

Proposition 2.2 For each fixed (u, v, ω) ∈ R × (0, 1) × Ω * , one has j∈Z k∈Z d ∈E 2 -jv (Ψ (2 j u * -k, v) -Ψ (-k, v))I d (ψ ( ) j,k )(ω) < +∞ .
This means that the series of real numbers

j∈Z k∈Z d ∈E 2 -jv (Ψ (2 j u * -k, v) -Ψ (-k, v))I d (ψ ( ) j,k )(ω)
is absolutely convergent, and consequently that it converges to a finite limit not depending on the way the terms of the series are ordered. Moreover, for all fixed (u, v) ∈ R × (0, 1), one has, almost surely,

X(u, v) = j∈Z k∈Z d ∈E 2 -jv (Ψ (2 j u * -k, v) -Ψ (-k, v))I d (ψ ( ) j,k ) .
(2.7)

Remark 2.3 From now on and till the end of our article, the chaotic stochastic field X = {X(u, v) : (u, v) ∈ R × (0, 1)} will be systematically identified with its modification

j∈Z k∈Z d ∈E 2 -jv (Ψ (2 j u * -k, v) -Ψ (-k, v))I d (ψ ( ) j,k ) : (u, v) ∈ R × (0, 1) ,
which has just been introduced in Proposition 2.2. Also, we will always assume that X vanishes outside of Ω * , the event of probability 1 introduced in Lemma 2.1. The low and high frequency parts of X are the two chaotic stochastic fields, denoted respectively by

X lf = {X lf (u, v) : (u, v) ∈ R × (0, 1)} and X hf = {X hf (u, v) : (u, v) ∈ R × (0, 1)}, which
vanish outside of Ω * , and are defined, for every (u, v, ω) ∈ R × (0, 1) × Ω * , as:

X lf (u, v, ω) = -1 j=-∞ k∈Z d ∈E 2 -jv (Ψ (2 j u * -k, v) -Ψ (-k, v))I d (ψ ( ) j,k )(ω) (2.8)
and

X hf (u, v, ω) = +∞ j=0 k∈Z d ∈E 2 -jv (Ψ (2 j u * -k, v) -Ψ (-k, v))I d (ψ ( ) j,k )(ω). (2.9)
One clearly has, for all (u, v, ω) ∈ R × (0, 1) × Ω, that

X(u, v, ω) = X lf (u, v, ω) + X hf (u, v, ω). (2.10)
Recall that Ω is the underlying probability space.

Let us now state the main result of the present section.

Theorem 2.4 The random series in the right-hand side of (2.7) is, on the event Ω * of probability 1 , uniformly convergent in (u, v), on each compact subset of R × (0, 1).

The following lemma will play a major role in the proof of Theorem 2.4 and in other important proofs in our article.

Lemma 2.5 For all fixed (p, q) ∈ Z d + × Z + , there exists a positive finite random variable C p,q such that, for all (j, u, ω)

∈ Z × R d × Ω * , one has ∈E k∈Z d |I d (ψ ( ) j,k )(ω)| sup v∈(0,1) (∂ p u ∂ q v Ψ )(u -k, v) ≤ C p,q (ω)(log(e + |j| + u 1 )) d 2 . (2.

11)

As a straightforward consequence, for all (j, u, v, ω) ∈ Z × R d × (0, 1) × Ω * , the series

Φ j (u, v, ω) = ∈E k∈Z d I d (ψ ( ) j,k )(ω)Ψ (u -k, v) (2.12)
is absolutely convergent, and the real-valued function

Φ j (•, •, ω) : (u, v) → Φ j (u, v, ω) is well- defined and infinitely differentiable on R d × (0, 1). Moreover, for each (j, u, v, ω) ∈ Z × R d × (0, 1) × Ω * , one has (∂ p u ∂ q v Φ j )(u, v, ω) = ∈E k∈Z d I d (ψ ( ) j,k )(ω)(∂ p u ∂ q v Ψ )(u -k, v),
and the following inequality holds:

sup v∈(0,1) (∂ p u ∂ q v Φ j )(u, v, ω) ≤ C p,q (ω)(log(e + |j| + u 1 )) d 2 .
(2.13)

Proof of Lemma 2.5. It easily follows from (2.4) and from the finiteness of the set E that, for all (p, q) fixed in Z d + × Z + , there is a positive finite deterministic constant c p,q for which the following inequality holds for every (u, k,

) ∈ R d × Z d × E: sup v∈(0,1) (∂ p u ∂ q v Ψ )(u -k, v) ≤ c p,q ( √ d + 1 + u -k 2 ) 2d .
In the sequel, one sets u = ( u 1 , . . . , u d ), where u l denotes the integer part of the l-th coordinate u l of the vector u. Then, using Lemma 2.1, the change of variable m = k + u , the triangle inequality and the inequality

∀ (x, y) ∈ R 2 + , (log(e + x + y)) d 2 ≤ (log(e + x)) d 2 (log(e + y)) d 2 , (2.14)
one gets, for some positive finite constant C p,q (ω) and for all (j, u, ω) ∈ Z × R d × Ω * , that:

∈E k∈Z d |I d (ψ ( ) j,k )(ω)| sup v∈(0,1) (∂ p u ∂ q v Ψ )(u -k, v) ≤ C p,q (ω) k∈Z d (log(e + |j| + k 1 )) d 2 ( √ d + 1 + u -k 2 ) 2d ≤ C p,q (ω) m∈Z d (log(e + |j| + m + u 1 )) d 2 ( √ d + 1 + u -u -m 2 ) 2d ≤ C p,q (ω) m∈Z d (log(e + |j| + m 1 + u 1 )) d 2 ( √ d + 1 -u -u 2 + m 2 ) 2d ≤ C p,q (ω)(log(e + |j| + u 1 )) d 2 m∈Z d (log(e + m 1 )) d 2 (1 + m 2 ) 2d .
Thus, in view of the fact that

m∈Z d (log(e + m 1 )) d 2 (1 + m 2 ) 2d < +∞ , it turns out that (2.11) is satisfied.
Proof of Theorem 2.4. First, observe that, in view of (2.7) and (2.12), X(u, v, ω) can be expressed, for all (u, v, ω) ∈ R × (0, 1) × Ω * , as:

X(u, v, ω) = j∈Z A j (u, v, ω) , (2.15)
where, for each j ∈ Z, A j (•, •, ω) is the infinitely differentiable function on R × (0, 1) defined as:

∀ (u, v) ∈ R × (0, 1), A j (u, v, ω) = 2 -jv (Φ j (2 j u * , v, ω) -Φ j (0, v, ω)) . (2.16)
Thus, for proving the theorem, one has to show that the convergence of the series in (2.15) holds uniformly in (u, v) on each compact subset of R × (0, 1). To this end, it is enough to prove that, for all fixed positive real numbers ν and a < b < 1, one has

j∈Z sup (u,v)∈[-ν,ν]×[a,b] |A j (u, v, ω)| < +∞.
(2.17)

Let us sets

T 1 (ω) = +∞ j=0 sup (u,v)∈[-ν,ν]×[a,b] |A j (u, v, ω)| and T 2 (ω) = -1 j=-∞ sup (u,v)∈[-ν,ν]×[a,b] |A j (u, v, ω)|.
One can derive from (2.16) and (2.13) that

T 1 (ω) ≤ 2C 0,0 (ω) +∞ j=0 2 -ja (log(e + j + 2 j dν)) d 2 < +∞ . (2.18)
On another hand, using the mean value theorem and (2.13), one obtains, for every (j, u, v, ω) ∈

Z -× [ -ν, ν] × [a, b] × Ω * , that |A j (u, v, ω)| ≤ 2 -ja |Φ j (2 j u * , v, ω) -Φ j (0, v, ω)| ≤ 2 j(1-a) |u| d i=1 sup x∈[0∧2 j u,0∨2 j u] |∂ u i Φ j (x * , v, ω)| ≤ dνC 1,0 (ω)2 j(1-a) (log(e + |j| + 2 j dν)) d 2 .
Thus, one gets that

T 2 (ω) ≤ dνC 1,0 (ω) ∞ j=1 2 -j(1-a) (log(e + j + 2 -j dν)) d 2 < +∞ (2.19)
Finally, combining (2.18) and (2.19), it follows that (2.17) is satisfied.

Global behavior

First, we state the main results of the section and then we give their proofs.

Theorem 3.1 Let X lf and X hf be the low and high frequency parts of the field X which were introduced in Remark 2.3. The following two results hold, for all ω ∈ Ω * .

(i) The function

X lf (•, •, ω) : (u, v) → X lf (u, v, ω) is infinitly many times differentiable on R × (0, 1). (ii) For all fixed u ∈ R, the function X hf (u, •, ω) : v → X hf (u, v, ω
) is infinitly many times differentiable on (0, 1). Moreover, for each fixed q ∈ Z + , the function

(∂ q v X)(•, •, ω) : (u, v) → (∂ q v X)(u, v, ω) is continuous on R × (0, 1).
Corollary 3.2 For each (ω, q) ∈ Ω * × Z + , and for all non degenerate compact intervals J ⊂ R and H ⊂ (0, 1), one has:

sup (u,v 1 ,v 2 )∈J ×H 2 |(∂ q v X)(u, v 1 , ω) -(∂ q v X)(u, v 2 , ω)| |v 1 -v 2 | < +∞. (3.1)
Theorem 3.3 For each (ω, q) ∈ Ω * × Z + , and for all non degenerate compact intervals J ⊂ R and H ⊂ (0, 1), one has: Assuming that

sup (u 1 ,u 2 ,v 1 ,v 2 )∈J 2 ×H 2        |(∂ q v X)(u 1 , v 1 , ω) -(∂ q v X)(u 2 , v 2 , ω)| u 1 -u 2 v 1 ∨v 2 1 + log |u 1 -u 2 | q+ d 2 + v 1 -v 2        < +∞. ( 3
H(•) ∈ C γ L (L) for some γ L ∈ [H(L), 1), (3.4)
where C γ L (L) denotes the global space of Hölder on L of order γ L . Then, for all ω ∈ Ω * , one has:

sup (t 1 ,t 2 )∈L 2        |Z(t 1 , ω) -Z(t 2 , ω)| t 1 -t 2 H(L) 1 + log |t 1 -t 2 | d 2        < +∞ . (3.5)
Proof of Theorem 3.1. First, we point out that one knows from the proof of Theorem 2.4 that, for all ω ∈ Ω * , one has

X lf (u, v, ω) = -1 j=-∞ A j (u, v, ω) (3.6)
and

X hf (u, v, ω) = +∞ j=0 A j (u, v, ω) , (3.7)
where the series in (3.6) and (3.7) are uniformly convergent in (u, v) on each compact subset of R × (0, 1). Moreover, one knows that, for each j ∈ Z, the function A j (•, •, ω) is infinitely differentiable on R × (0, 1). Thus, in order to prove the theorem, it is enough to show that, for all (m, q, ω) ∈ Z + × Z + × Ω * , and for each positive real numbers ν and a < b, one has

-1 j=-∞ sup (u,v)∈[-ν,ν]×[a,b] (∂ m u ∂ q v A j )(u, v, ω) < +∞ and +∞ j=0 sup (u,v)∈[-ν,ν]×[a,b] (∂ q v A j )(u, v, ω) < +∞.
This can be done by following the main lines of the proof of (2.18) and (2.19).

Proof of Corollary 3.2. It follows from Theorem 3.1, that for all fixed (u, ω) ∈ R × Ω * , the function X(u, •, ω) is infinitely differentiable on (0, 1); and for all q ∈ Z + , the function

(∂ q v X)(•, •, ω) is continuous.
That is enough to prove that (3.1) holds.

Corollary 3.2 and the following lemma are the two main ingredients of the proof of Theorem 3.3.

Lemma 3.5 For each (ω, q) ∈ Ω * × Z + , and for all non degenerate compact intervals J ⊂ R and H ⊂ (0, 1), one has

sup (u 1 ,u 2 ,v)∈J 2 ×H        |(∂ q v X)(u 1 , v, ω) -(∂ q v X)(u 2 , v, ω)| u 1 -u 2 v 1 + log |u 1 -u 2 | q+ d 2        < +∞. (3.8)
Proof of Lemma 3.5. First, notice that Theorem 3.1 entails that the lemma holds when X in (3.8) is replaced by X lf . Thus, one only has to prove that the lemma is true when X in (3.8) is replaced by X hf . Using the continuity property of the function (

∂ q v X)(•, •, ω) (see Theorem 3.1), one has that sup (u 1 ,u 2 ,v)∈K        |(∂ q v X hf )(u 1 , v, ω) -(∂ q v X hf )(u 2 , v, ω)| u 1 -u 2 v 1 + log |u 1 -u 2 | q+ d 2        < +∞, (3.9)
where K is the compact subset of R 2 × (0, 1) defined as

K = (u 1 , u 2 , v) ∈ J 2 × H : |u 1 -u 2 | ≥ 2 -1 .
Thus, in order to derive the lemma, it is enough to prove that:

sup (u 1 ,u 2 ,v)∈K        |(∂ q v X hf )(u 1 , v, ω) -(∂ q v X hf )(u 2 , v, ω)| u 1 -u 2 v 1 + log |u 1 -u 2 | q+ d 2        < +∞, (3.10)
where K is the compact subset of R 2 × (0, 1) defined as

K = (u 1 , u 2 , v) ∈ J 2 × H : |u 1 -u 2 | ≤ 2 -1 .
We will show (3.10) for q = 0; the proof can be done in a rather similar way in the general case where q is an arbitrary nonegative integer. There is no restriction to assume that J = [-ν, ν]

and H = [a, b] ⊂ (0, 1), where ν and a < b are fixed positive real numbers. Let (u 1 , u 2 , v) ∈ K be arbitrary; there is no restriction to assume that u 1 = u 2 since (3.10) is clearly satisfied when u 1 = u 2 . Then, denote by j 0 be the biggest nonegative integer satisfying |u 1 -u 2 | ≤ 2 -j 0 .

Observe that j 0 ≥ 1 and that one has:

2 -(j 0 +1) < |u 1 -u 2 | ≤ 2 -j 0 , (3.11)
which means that

j 0 = log |u 1 -u 2 | -1 log 2 . (3.12)
Notice that one knows from Lemma 2.5 and (2.16) that the function A 0 (•, •, ω) is infinitely differentiable on R × (0, 1), which implies that it satisfies (3.10). This allows to assume that the sum over j in (3.7) starts from j = 1 instead of j = 0. Thus, one has that

|X hf (u 1 , v, ω) -X hf (u 2 , v, ω)| ≤ S 1 (u 1 , u 2 , v, ω) + S 2 (u 1 , u 2 , v, ω), (3.13)
where

S 1 (u 1 , u 2 , v, ω) = j 0 j=1 |A j (u 1 , v, ω) -A j (u 2 , v, ω)| (3.14)
and

S 2 (u 1 , u 2 , v, ω) = +∞ j=j 0 +1 |A j (u 1 , v, ω) -A j (u 2 , v, ω)|. (3.15)
In order to derive appropriate upper bounds for S 1 (u 1 , u 2 , v, ω) and S 2 (u 1 , u 2 , v, ω), notice that there exists a deterministic positive finite constant c such that:

∀ x ≥ 1, log(e + x + 2 x dν) ≤ cx. (3.16)
Using (3.15), (2.16), the triangle inequality, Lemma 2.5, (3.16), the inequality

∀ (x, y) ∈ R 2 + , (1 + x + y) d 2 ≤ (1 + x) d 2 (1 + y) d 2 ,
(3.11) and (3.12), one gets:

S 2 (u 1 , u 2 , v, ω) ≤ 2 +∞ j=j 0 +1 2 -jv sup (u,v)∈[-ν,ν]×[a,b] |Φ j (2 j u * , v, ω)| ≤ C 2 (ω) +∞ j=j 0 +1
2 -jv log(e + j + 2 j dν)

d 2 ≤ C 2 (ω) +∞ j=j 0 +1 2 -jv j d 2 ≤ C 2 (ω)2 -(j 0 +1)v (1 + j 0 ) d 2 +∞ j=0 2 -ja (1 + j) d 2 ≤ C 2 (ω)2 -(j 0 +1)v (1 + j 0 ) d 2 ≤ C 2 (ω)|u 1 -u 2 | v 1 + log |u 1 -u 2 | d 2 , (3.17)
where C 2 is a positive finite random variable not depending on (u 1 , u 2 , v) derived from Lemma 2.5,

C 2 = C 2 c d/2 , C 2 = C 2 +∞ j=0 2 -ja (1 + j) d 2 < +∞ and C 2 = (log 2) -d 2 C
2 . On another hand, using (3.14), (2.16), the mean value theorem, the triangle inequality, Lemma 2.5, (3.16), (3.11) and (3.12), one gets:

S 1 (u 1 , u 2 , v, ω) ≤ j 0 j=1 2 j(1-v) |u 1 -u 2 | d i=1 sup (u,v)∈[-ν,ν]×[a,b] |∂ u i Φ j (2 j u * , v, ω)| ≤ C 1 (ω)|u 1 -u 2 | j 0 j=1 2 j(1-v) log(e + j + 2 j dν) d 2 ≤ C 1 (ω)|u 1 -u 2 | j 0 j=1 2 j(1-v) (j + 1) d 2 ≤ C 1 (ω)|u 1 -u 2 |(1 + j 0 ) d 2 2 j 0 (1-v) ≤ C 1 (ω)|u 1 -u 2 | v 1 + log |u 1 -u 2 | d 2 , (3.18)
where C 1 is a positive finite random variable not depending on (u 1 , u 2 , v) derived from Lemma 2.5,

C 1 = C 1 c d/2 , C 1 = 2 1-a (2 1-b -1) -1 C 1 and C 1 = (log 2) -d 2 C 1 .
Finally, puting together (3.13), (3.17) and (3.18), one obtains (3.10).

Proof of Theorem 3.3. For all (u 1 , u 2 , v 1 , v 2 ) ∈ J 2 × H 2 , one sets:

f (u 1 , u 2 , v 1 , v 2 ) = |(∂ q v X)(u 1 , v 1 , ω) -(∂ q v X)(u 2 , v 2 , ω)| u 1 -u 2 v 1 ∨v 2 1 + log |u 1 -u 2 | q+ d 2 + |v 1 -v 2 |
, with the convention that 0 0 = 0. Observe that one has:

f (u 1 , u 2 , v 1 , v 2 ) = f (u 2 , u 1 , v 2 , v 1 ).
Thus, one gets that:

sup (u 1 ,u 2 ,v 1 ,v 2 )∈J 2 ×H 2 f (u 1 , u 2 , v 1 , v 2 ) = sup (u 1 ,u 2 ,v 1 ,v 2 )∈J 2 ×H 2 f (u 1 , u 2 , v 1 ∨ v 2 , v 1 ∧ v 2 ). (3.19)
Moreover, using the triangle inequality, one obtains that:

sup (u 1 ,u 2 ,v 1 ,v 2 )∈J 2 ×H 2 f (u 1 , u 2 , v 1 ∨ v 2 , v 1 ∧ v 2 ) ≤ sup (u 1 ,u 2 ,v 1 ,v 2 )∈J 2 ×H 2        |(∂ q v X)(u 1 , v 1 ∨ v 2 , ω) -(∂ q v X)(u 2 , v 1 ∨ v 2 , ω)| u 1 -u 2 v 1 ∨v 2 1 + log |u 1 -u 2 | q+ d 2 + |v 1 -v 2 |        + sup (u 1 ,u 2 ,v 1 ,v 2 )∈J 2 ×H 2        |(∂ q v X)(u 2 , v 1 ∨ v 2 , ω) -(∂ q v X)(u 2 , v 1 ∧ v 2 , ω)| |u 1 -u 2 | v 1 ∨v 2 1 + log |u 1 -u 2 | q+ d 2 + |v 1 -v 2 |        ≤ sup (u 1 ,u 2 ,v 1 ,v 2 )∈J 2 ×H 2        |(∂ q v X)(u 1 , v 1 ∨ v 2 , ω) -(∂ q v X)(u 2 , v 1 ∨ v 2 , ω)| u 1 -u 2 v 1 ∨v 2 1 + log |u 1 -u 2 | q+ d 2        + sup (u 1 ,u 2 ,v 1 ,v 2 )∈J 2 ×H 2 |(∂ q v X)(u 2 , v 1 ∨ v 2 , ω) -(∂ q v X)(u 2 , v 1 ∧ v 2 , ω)| |v 1 -v 2 | ≤ sup (u 1 ,u 2 ,v)∈J 2 ×H        |(∂ q v X)(u 1 , v, ω) -(∂ q v X)(u 2 , v, ω)| |u 1 -u 2 | v 1 + log |u 1 -u 2 | q+ d 2        + sup (u,v 1 ,v 2 )∈J ×H 2 |(∂ q v X)(u, v 1 , ω) -(∂ q v X)(u, v 2 , ω)| |v 1 -v 2 | . (3.20)
Finally, putting together (3.19), (3.20), Corollary 3.2 and Lemma 3.5, one gets that sup

(u 1 ,u 2 ,v 1 ,v 2 )∈J 2 ×H 2 f (u 1 , u 2 , v 1 , v 2 ) < +∞,
which shows that (3.2) holds.

Proof of Corollary 3.4. Using (1.7) and Theorem 3.3, in the case where q = 0, J = L and H = [H(L), H(L)] (see (3.3)), one obtains, for all ω ∈ Ω * , that:

sup (t 1 ,t 2 )∈L 2        |Z(t 1 , ω) -Z(t 2 , ω)| |t 1 -t 2 | H(t 1 )∨H(t 2 ) 1 + log |t 1 -t 2 | d 2 + |H(t 1 ) -H(t 2 )|        < +∞. (3.21)
Then, (3.4) and (3.21) imply that (3.5) holds.

Local behavior

First, we state the main results of the section and then we give their proofs.

Theorem 4.1 Let u 0 ∈ R be an arbitrary fixed point. Then, one has, almost surely, for every q ∈ Z + and non degenerate compact interval H ⊂ (0, 1), that:

sup (u,v)∈[u 0 -1,u 0 +1]×H            |(∂ q v X)(u, v) -(∂ q v X)(u 0 , v)| |u -u 0 | v 1 + log |u -u 0 | q log e + log |u -u 0 | d 2            < +∞. (4.1)
Corollary 4.2 Let t 0 ∈ R be an arbitrary fixed point. Assume that there exists a constant

γ t 0 ∈ [H(t 0 ), 1) such that the continuous function H(•) satisfies sup t∈R |H(t) -H(t 0 )| |t -t 0 | γt 0 < +∞. (4.2)
Then, one has, almost surely:

sup t∈[t 0 -1,t 0 +1]            |Z(t) -Z(t 0 )| |t -t 0 | H(t 0 ) log e + log |t -t 0 | d 2            < +∞. ( 4.3)
The following theorem shows that the chaotic multifractional process {Z(t) : t ∈ R} has a local asymptotic self-similarity property rather similar to the one satisfied by the classical Gaussian multifractional Brownian motion (see [START_REF] Benassi | Identifying the multifractional function of a Gaussian process[END_REF][START_REF] Falconer | Tangent fields and the local structure of random fields[END_REF][START_REF] Falconer | The local structure of random processes[END_REF]).

Theorem 4.3 Let t 0 ∈ R be an arbitrary fixed point such that the condition (4.2) holds.

Then, the stochastic process {Z(t) : t ∈ R} is at t 0 , strongly locally asymptotically self-similar of order H(t 0 ) and the tangent process is X(s, H(t 0 )) : s ∈ R . More precisely, let (ν n ) n∈N be an arbitrary sequence of positive real numbers which converges to 0. For each n ∈ N, let T t 0 ,νn Z = (T t 0 ,νn Z)(s) : s ∈ R be the stochastic process with continuous paths, defined, for all s ∈ R, as

(T t 0 ,νn Z)(s) = Z(t 0 + ν n s) -Z(t 0 ) ν H(t 0 ) n . (4.4)
Then, when n goes to +∞, the probability measure induced on C(J ) by (T t 0 ,νn Z)(s) : s ∈ R converges to the one induced on C(J ) by X(s, H(t 0 )) : s ∈ R , where C(J ) denotes the usual Banach space of the real-valued continuous functions over an arbitrary non degenerate compact interval J of the real line equipped with the uniform norm.

|Z(t) -Z(t 0 )| |t -t 0 | H(t 0 )+η = +∞ ,
which means that the exponent H(t 0 ) in (4.3) is optimal. Moreover, when γ L in (3.3) belongs to H(L), 1 , then, using similar arguments, it can be shown that the exponent H(L) in (3.5) is optimal: one has, almost surely, sup

(t 1 ,t 2 )∈L 2 |Z(t 1 ) -Z(t 2 )| |t 1 -t 2 | H(L)+η = +∞ .
Theorem 4.5 Let δ ∈ (0, +∞) be arbitrary and fixed. One sets R + δ = {u ∈ R; |u| ≥ δ}. Then, for all ω ∈ Ω * and non degenerate compact interval H ⊂ (0, 1), one has:

sup (u,v)∈R + δ ×H            |X lf (u, v, ω)| |u| v log e + log |u| d 2            < +∞ (4.5)
and

sup (u,v)∈R + δ ×H        |X hf (u, v, ω)| log e + |u| d 2        < +∞. (4.6)
Notice that a straightforward consequence of (4.5), (4.6) and (2.10) is that:

sup (u,v)∈R + δ ×H            |X(u, v, ω)| |u| v log e + log |u| d 2            < +∞.
Corollary 4.6 Assume that the continuous function H(•) is with values in a compact interval included in (0, 1) (this means that inf t∈R H(t) > 0 and sup t∈R H(t) < 1). Then, for each fixed ω ∈ Ω * and δ > 0, one has:

sup |t|≥δ            |Z(t, ω)| |t| H(t) log e + log |t| d 2            < +∞. (4.7)
The following lemma will play a crucial role in the proof of Theorem 4.1.

Lemma 4.7 For all fixed integer j ≥ 1 and (u, θ) ∈ R d × [1, +∞), let D j (u, θ) be the finite non-empty set defined as:

D j (u, θ) = ( , k) ∈ E × Z d : u -2 -j k 1 ≤ d j θ 2 -j . (4.8)
Then, for each fixed (u, θ) ∈ R d × [1, +∞) there is a deterministic positive finite constant c * , only depending on (u, θ, d), such that one has, almost surely:

lim sup j→+∞      max ( ,k)∈D j (u,θ) I d (ψ ( ) j,k ) log(2 + j) d 2      ≤ c * . (4.9)
Proof of Lemma 4.7. The lemma can be derived from the Borel-Cantelli Lemma by showing that for some fixed well-chosen deterministic positive finite constant a ≥ 2, one has

+∞ j=1 P max ( ,k)∈D j (u,θ) |I d (ψ ( ) j,k )| > a √ d! log(2 + j) d 2 < +∞. (4.10)
It can easily be seen that, for all j ≥ 1, the following inequality holds:

P max ( ,k)∈D j (u,θ) |I d (ψ ( ) j,k )| > a √ d! log(2+j) d 2 ≤ ( ,k)∈D j (u,θ) P |I d (ψ ( ) j,k )| > a √ d! log(2+j) d 2 . (4.11)
Let us conveniently bound from above the probabilities in the right-hand side of (4.11).

Observe that (1.5) and the equality ψ

( ) j,k L 2 (R d ) = 1 imply, for all (j, k, ) ∈ Z × Z d × E, that E |I d (ψ ( ) j,k )| 2 = d! ψ( ) j,k 2 L 2 (R d ) ≤ d! ψ ( ) j,k 2 L 2 (R d ) = d! . (4.12)
Then (4.12) and Theorem 6.7 in [START_REF] Janson | Gaussian Hilbert Spaces[END_REF] entail that, for all real number α ≥ 2, one has

P |I d (ψ ( ) j,k )| > α √ d! ≤ P |I d (ψ ( ) j,k )| > α I d (ψ ( ) j,k ) L 2 (Ω) ≤ exp -κ d α 2 d , (4.13)
where κ d is a deterministic positive finite constant only depending on d. Thus, setting in

(4.13) α = a log(2 + j) d 2
, one gets that

P |I d (ψ ( ) j,k )| > a √ d! log(2 + j) d 2 ≤ exp -κ d a 2 d log(2 + j) = (2 + j) -κ d a 2 d . ( 4
( ,k)∈D j (u,θ) |I d (ψ ( ) j,k )| > a √ d! log(2 + j) d 2 ≤ c j dθ-κ d a 2 d . (4.16)
Thus, assuming that the constant a has been chosen big enough so that dθ -κ d a 2 d < -1, then it follows from (4.16) that (4.10) holds.

Proof of Theorem 4.1. Using the same arguments as in the proof of Lemma 3.5, it turns out that in order to derive the theorem it is enough to show that (4.1) holds when X in it is repalced by X hf , and one can assume that the sums over j in (3.7) and in (2.9) start from j = 1 instead of j = 0. Also, for the sake of simplicity one focuses on the case where q = 0.

The proof can be done in a rather similar way in the general case where q is an arbitrary nonegative integer.

Let us express the compact interval H as H = [a, b],

where the real numbers a and b are

such that 0 < a < b < 1. Let then (u, v) ∈ [u 0 -1, u 0 + 1] × [a, b] ⊂ R × (0, 1)

be arbitrary

and fixed. There is no restriction to assume that 0 < |u -u 0 | ≤ 2 -15 , since sample paths of X hf are almost surely continuous functions. One denotes by j 1 be the biggest nonegative integer which satisfies |u -u 0 | ≤ 2 -(j 1 -1) . Then, one has:

2 -j 1 < |u -u 0 | ≤ 2 -(j 1 -1) , (4.17)
which means that

j 1 = 1 + log |u -u 0 | -1 log 2 . ( 4.18)
One sets

j 2 = j 1 + d log j 1 2a log 2 . ( 4.19)
Observe that one has j 1 ≥ 16 and j 2 ≥ j 1 + 2. Moreover, for any j ∈ {j 1 + 1, ..., j 2 }, the following inequality holds:

j d a ≥ 2 j-j 1 +2 . (4.20)
Next, for all integer j ≥ 1, let D j (u 0 ) be the finite non-empty set defined as

D j (u 0 ) = D j (u * 0 , d a )
, where, as usual, u * 0 denotes the vector of R d whose coordinates are all equal to the real number u 0 , and D j (u * 0 , d a ) is defined through (4.8) with u = u * 0 and θ = d a . That is:

D j (u 0 ) = ( , k) ∈ E × Z d : u * 0 -2 -j k 1 ≤ d j d a 2 -j . (4.21)
Then, Lemma 4.7 entails that one has almost surely, for all integer j ≥ 1, max

( ,k)∈D j (u 0 ) |I d (ψ ( ) j,k )| ≤ C(log(2 + j)) d 2 , (4.22)
where C is a positive almost surely finite random variable not depending on j. Next, one denotes by D co j (u 0 ) the complement of D j (u 0 ) in E × Z d , that is:

D co j (u 0 ) = ( , k) ∈ E × Z d : u * 0 -2 -j k 1 > d j d a 2 -j . (4.23)
Let us mention in passing that

D co j (u 0 ) ⊂ d l=1 ( , k) ∈ E × Z d : |u 0 -2 -j k l | > j d a 2 -j , (4.24)
where k l is the l-th coordinate of k. One can derive from (2.9) (where the sum over j is assumed to start from j = 1 instead of j = 0), (4.21), (4.23) and the triangle inequality that

|X hf (u, v) -X hf (u 0 , v)| ≤ R 1 (u, u 0 , v) + R 2 (u, u 0 , v) + R 3 (u, u 0 , v) + R 4 (u, u 0 , v), (4.25)
where

R 1 (u, u 0 , v) = j 1 j=1 ( ,k)∈D j (u 0 ) 2 -jv |I d (ψ ( ) j,k )||Ψ (2 j u * -k, v) -Ψ (2 j u * 0 -k, v)|, (4.26)
R 2 (u, u 0 , v) = +∞ j=j 1 +1 ( ,k)∈D j (u 0 ) 2 -jv |I d (ψ ( ) j,k )||Ψ (2 j u * -k, v) -Ψ (2 j u * 0 -k, v)|, (4.27) R 3 (u, u 0 , v) = j 2 j=1 ( ,k)∈D co j (u 0 ) 2 -jv |I d (ψ ( ) j,k )||Ψ (2 j u * -k, v) -Ψ (2 j u * 0 -k, v)|, (4.28)
and

R 4 (u, u 0 , v) = +∞ j=j 2 +1 ( ,k)∈D co j (u 0 ) 2 -jv |I d (ψ ( ) j,k )||Ψ (2 j u * -k, v) -Ψ (2 j u * 0 -k, v)|. (4.29)
From now on, our goal is to derive an appropriate upper bound for each term in the righthand side of (4.25). In all the sequel, one assumes that L is an arbitrary large fixed positive integer. Therefore, one has

c := sup y∈R        k∈Z d log e + y * -k 2 d 2 1 + y * -k 2 L        < +∞. ( 4
R 2 (u, u 0 , v) ≤ K 2 +∞ j=j 1 +1 2 -jv log(2 + j) d 2 ≤ K 2 +∞ j=0 2 -(j+j 1 +1)v log(2 + j + j 1 + 1) d 2 ≤ K 2 2 -(j 1 +1)v log(e + j 1 + 1) d 2 +∞ j=0 2 -ja log(e + j) d 2 ≤ K 2 |u -u 0 | v log(e + | log |u -u 0 || d 2 , (4.31)
where K 2 and K 2 are two positive almost surely finite random variables not depending on (u, v).

Next, using (4.29), (2.11), the fact that |u| ≤ |u 0 | + 1, the inequality 2 j > j for all j ∈ Z + , 20

(2.14), (4.17), (4.18) and (4.19), one gets, on the event of probability 1 Ω * , that:

R 4 (u, u 0 , v) ≤ K 4 +∞ j=j 2 +1 2 -jv log e + (|u 0 | + d + 1)2 j d 2 ≤ K 4 2 -(j 2 +1)v (j 2 + 1) d 2 +∞ j=0 2 -ja log e + (|u 0 | + d + 1)2 j d 2 ≤ K 4 2 -(j 2 +1)v (j 2 + 1) d 2 = K 4 2 -j 1 v exp -(v log 2) 1 + d log j 1 2a log 2 + d log(j 2 + 1) 2 ≤ K 4 |u -u 0 | v exp d 2 log j 2 + 1 j 1 ≤ K 4 |u -u 0 | v , (4.32)
where K 4 , K 4 and K 4 are three positive finite random variables not depending on (u, v), and where

K 4 = K 4 2 + d 2a log 2 d 2 .
Next, observe that using the mean value theorem, it can be shown that, for all fixed ( , j, k) ∈ E × N × Z d , there exists a real number λ j,k (u, u 0 ) ∈ (0, 1) such that:

Ψ (2 j u * -k, v)-Ψ (2 j u * 0 -k, v) = 2 j (u-u 0 ) d n=1 (∂ yn Ψ ) 2 j u * 0 +λ j,k (u, u 0 )2 j (u * -u * 0 ) . ( 4
R 1 (u, u 0 , v) ≤ K 1 |u -u 0 | j 1 j=1 2 j(1-v) log(2 + j) d 2 ≤ K 1 |u -u 0 | log(2 + j 1 ) d 2 j 1 j=1 2 (j 1 -j+1)(1-v) ≤ K 1 |u -u 0 |2 (j 1 +1)(1-v) log(2 + j 1 ) d 2 +∞ j=1 2 -j(1-b) ≤ K 1 |u -u 0 | v log(2 + | log |u -u 0 ||) d 2 , (4.34)
where K 1 and K 1 are two positive almost surely finite random variables not depending on (u, v).

It only remains to obtain a convenient upper bound for R 3 (u, u 0 , v). Notice that, using the equivalence of all norms on R d , one deduces from (2.4) that:

∀ (n, p, q) ∈ N×Z d ×Z, sup α+ u 1 n |∂ p u ∂ q v Ψ (u, v)| : u ∈ R d , v ∈ [a, b] < +∞ , (4.35)
where α is an abritrary positive real number. Next, combining (4.28), (4.33) and (4.35), one obtains that

R 3 (u, u 0 , v) ≤ κ 3 |u-u 0 | j 2 j=1 ( ,k)∈D co j (u 0 ) 2 j(1-v) |I d (ψ ( ) j,k )| 2d + 1 + 2 j u * 0 -k + λ j,k (u, u 0 )2 j (u * -u * 0 ) 1 L (4.36)
where κ 3 denotes a positive finite and deterministic constant. Observe that using the triangle inequality, (4.17) and the fact that λ j,k (u, u 0 ) ∈ (0, 1) one has, for all j ∈ {1, ..., j 1 } and

( , k) ∈ D co j (u 0 ), that 2d + 1 + 2 j u * 0 -k + λ j,k (u, u 0 )2 j (u * -u * 0 ) 1 ≥ 2 j u * 0 -k 1 -λ j,k (u, u 0 ) d 2 j |u -u 0 | + 2d + 1 ≥ 1 + 2 j u * 0 -k 1 .
(4.37) Also observe that using the triangle inequality, the fact that λ j,k (u, u 0 ) ∈ (0, 1), (4.17), (4.23) and (4.20), one obtains, for all j ∈ {j 1 + 1, ..., j 2 } and all ( , k) ∈ D co j (u 0 ), that

2d + 1 + 2 j u * 0 -k + λ j,k (u, u 0 )2 j (u * -u * 0 ) 1 ≥ 2 j u * 0 -k 1 -λ j,k (u, u 0 ) d 2 j |u -u 0 | + 2d + 1 ≥ 1 2 2 j u * 0 -k 1 + 1 2 2 j u * 0 -k 1 -d 2 j-j 1 +1 + 2d + 1 ≥ 1 2 2 j u * 0 -k 1 + d 2 j d a -d 2 j-j 1 +1 + 2d + 1 ≥ 1 2 (1 + 2 j u * 0 -k 1 ). ( 4
R 3 (u, u 0 , v) ≤ 2κ 3 |u -u 0 | j 2 j=1 ( ,k)∈D co j (u 0 ) 2 j(1-v) |I d (ψ ( ) j,k )| 1 + 2 j u * 0 -k 1 L . (4.39)
Next, using (4.39), Lemma 2.1, the inequalities

log e + j + k 1 ≤ log e + j + 2 j u * 0 1 + 2 j u * 0 -k 1 ≤ log e + j + 2 j u * 0 1 d l=1 log e + |2 j u 0 -k l | , 1 + 2 j u * 0 -k 1 L ≥ d l=1 1 + |2 j u 0 -k l | L d ,
(4.24), and (4.30) with Z in place of Z d , it turns out that, on the event of probability 1 Ω * , one has

R 3 (u, u 0 , v) ≤ K 3 |u -u 0 | j 2 j=1 j d 2 2 j(1-v) k∈D co 1,j (u 0 )
log(e + |2 j u 0 -k|)

d 2 1 + |2 j u 0 -k| L d , (4.40)
where K 3 is a positive finite random variable not depending on (u, v), and where

D co 1,j (u 0 ) = k ∈ Z : |2 j u 0 -k| > j d a . (4.41)
Next, let us assume that η is an arbitrarily small fixed positive real number. using (4.41), and the fact that x → log(e + x) and x → x are increasing functions over R + , one gets that

k∈D co 1,j (u 0 )
log(e + |2 j u 0 -k|)

d 2 1 + |2 j u 0 -k| L d ≤ 2 +∞ j d a log(e + 1 + x) d 2 x L d dx ≤ κ 3 j -( L-d a -η) , (4.42)
where κ 3 is a positive finite deterministic constant not depending on j. Moreover the assumption that L is an arbitrarily large integer allows to assume that

L -d a -η - d 2 > d 2a > 0 . (4.43)
Thus, using the fact that v ∈ [a, b] ⊂ (0, 1), one obtains that

j 2 j=1 j -( L-d a -η-d 2 ) 2 j(1-v) ≤ j 2 /2 j=1 2 j(1-v) + j 2 /2 -( L-d a -η-d 2 ) j 2 j 2 /2 +1 2 j(1-v) ≤ 4 2 1-b -1 -1 2 j 2 (1-v)/2 + j 2 /2 -( L-d a -η-d 2 ) 2 j 2 (1-v) ≤ κ 3 j -( L-d a -η-d 2 ) 2 2 j 2 (1-v) , (4.44)
where the finite deterministic constant

κ 3 = 4 2 1-b -1 -1 2 ( L-d a -η-d 2 ) + sup n∈N 2 -n(1-b)/2 n ( L-d a -η-d 2 )
.

Moreover, one can derive from (4.18), (4.19) and (4.43) that j

-( L-d a -η-d 2 ) 2 2 j 2 (1-v) ≤ 4|u -u 0 | v-1 j -( L-d a -η-d 2 ) 2 2 d log(j 1 )/(2a log 2) ≤ 4|u -u 0 | v-1 j -( L-d a -η-d 2 -d 2a ) 2 ≤ 4|u -u 0 | v-1 . ( 4
R 3 (u, u 0 , v) ≤ K 3 |u -u 0 | v , ( 4
|Z(t) -Z(t 0 )| |t -t 0 | H(t 0 ) log(e + log |t -t 0 | ) d 2 ≤ U 1 (t 0 ) + U 2 (t 0 ), (4.47)
where

U 1 (t 0 ) = sup t∈[t 0 -1,t 0 +1] |X(t, H(t)) -X(t, H(t 0 ))| |t -t 0 | H(t 0 ) log(e + log |t -t 0 | ) d 2 (4.48) and U 2 (t 0 ) = sup t∈[t 0 -1,t 0 +1] |X(t, H(t 0 )) -X(t 0 , H(t 0 ))| |t -t 0 | H(t 0 ) log(e + log |t -t 0 | ) d 2
.

(4.49)

Next, observe that it follows from the assumption that sup

t∈[t 0 -1,t 0 +1] |H(t) -H(t 0 )| |t -t 0 | H(t 0 ) log(e + log |t -t 0 | ) d 2 < +∞ . (4.50)
On the other hand, denoting by H the compact interval included in (0, 1) defined as Proof of Theorem 4.3. It easily follows from (4.4) and (1.7) that, for every n ∈ N, the stochastic process T t 0 ,νn Z = (T t 0 ,νn Z)(s) : s ∈ R can be expressed as the sum of the two stochastic processes T 1 t 0 ,νn X = (T 1 t 0 ,νn X)(s) : s ∈ R and T 2 t 0 ,νn X = (T 2 t 0 ,νn X)(s) : s ∈ R , defined, for all s ∈ R, as: = 0, one gets that

H = H([t 0 -1, t 0 + 1]) = H(t) : t ∈ [t 0 -1, t 0 + 1] , one clearly has that sup t∈[t 0 -1,t 0 +1] |X(t, H(t)) -X(t, H(t 0 ))| |H(t) -H(t 0 )| ≤ sup (u,v 1 ,v 2 )∈[t 0 -1,t 0 +1]×H 2 |X(u, v 1 ) -X(u, v 2 )| |v 1 -v 2 | . ( 4
U 2 (t 0 ) ≤ sup (u,v)∈[t 0 -1,t 0 +1]×H |X(u, v) -X(u 0 , v)| |u -u 0 | v log(e + log |u -u 0 | ) d 2 < +∞. ( 4
(T 1 t 0 ,νn X)(s) = X t 0 + ν n s, H(t 0 ) -X t 0 , H(t 0 ) ν H(t 0 ) n (4.54) and (T 2 t 0 ,νn X)(s) = X t 0 + ν n s, H(t 0 + ν n s) -X t 0 + ν n s, H(t 0 ) ν H(t 0 ) n . ( 4
(T 1 t 0 ,νn X)(s) : s ∈ R law = X(s, H(t 0 )) : s ∈ R .
This equatility in the sense of finite-dimensional distributions and the fact that (T 1 t 0 ,νn X)(s) : s ∈ R and X(s, H(t 0 )) : s ∈ R have continuous paths imply that these two processes induce the same probability distribution on the space of continuous functions C(J ). Thus, in order to derive the theorem, it is enough to show that T 2 t 0 ,νn X, viewed as a random variable with values in the space C(J ), converges to 0 in this space, when n goes to +∞. That is

lim n→+∞ sup s∈J (T 2 t 0 ,νn X)(s) a.s. = 0 . (4.56)
There is non rectriction to assume that J = [-M, M ] for some fixed positive real number M , and that ν n ∈ (0, 1], for every n ∈ N. Let then I and H be the compact intervals defined as I = [t 0 -M, t 0 + M ] and H = H(I) = H(t) : t ∈ I . It follows from Corollary 3.2 that the positive random variable A defined as Proof of Theorem 4.5 In view of the fact that on the event of probability 1 Ω * the fields X lf and X hf are with continuous paths, one can assume without any restriction that δ = 2.

A = sup (u,v 1 ,v 2 )∈I×H 2 |X(u, v 1 ) -X(u, v 2 )| |v 1 -v 2 | ( 4
(T 2 t 0 ,νn X)(s) ≤ ν -H(t 0 ) n A sup s∈J H(t 0 + ν n s) -H(t 0 ) . ( 4
Let H = [a, b] ⊂ (0, 1) be an arbitrary compact interval and let u be an arbitrary real number such that |u| ≥ 2. One denotes by j 3 the biggest positive integer satisfying: |u| ≥ 2 j 3 . Then, one gets that: First, one shows that (4.5) holds. Recall that, for all (v, ω) ∈ (0, 1) × Ω * , one has:

2 j 3 ≤
X lf (u, v, ω) = +∞ j=1
2 jv (Φ -j (2 -j u * , v, ω) -Φ -j (0, v, ω)) , (4.61)

where Φ -j (•, •, ω) is the infinitely differentiable function on R d × (0, 1), introduced in (2.12).

Also, recall that the series in (4.61) is uniformly convergent in (u, v) on each compact subset of R × (0, 1). Using the mean value theorem, one gets, for all (j, v) ∈ N × H, that:

|Φ -j (

Remark 4 . 4

 44 One can derive from Theorem 4.3 and zero-one law that, for any fixed arbitrarily small positive real number η, one has, almost surely, sup t∈[t 0 -1,t 0 +1]

  .53) Finally, putting together (4.52), (4.53) and (4.47), one obtains (4.3).

  .58) Finally, combining (4.2) and (4.58) one obtains (4.56).

2 , 2 ≤ 2 ≤ 2 ≤ 2 < 2 . 3 j=1 2 3 j=1 2 2 = 3 j=1 2 - 2 ≤ C d (ω)|u| v j 3 j=1 2 - 2 ≤ 2 ≤ 2 ,→ z d 2 , 2 - 2 ≤ 2 - 2 - 2 d 2 2 -

 222222323223223222222222222 2 -j u * , v, ω) -Φ -j (0, v, ω)| ≤ 2 -j |u| d n=1 sup (y,v)∈[0∧2 -j u,0∨2 -j u]×H |∂ yn Φ -j (y * , v, ω)|.Then, Lemma 2.5 entails that:|Φ -j (2 -j u * , v, ω) -Φ -j (0, v, ω)| ≤ C d (ω)2 -j |u|(log(e + j + 2 -j |u|)) d (4.62)where C d is a positive finite random variable not depending on u. Thus, one can derive from (4.62), (2.14), (4.59) and (4.60) that:+∞ j=j 3 +1 2 jv |Φ -j (2 -j u * , v, ω) -Φ -j (0, v, ω)| ≤ C d (ω)|u| +∞ j=j 3 +1 2 -j(1-v) (log(e + 1 + j)) d C d (ω)|u|2 -(j 3 +1)(1-v) +∞ l=0 2 -l(1-v) (log(e + 2 + l + j 3 )) d C d (ω)2 j 3 v (log(e + j 3 )) d C d (ω)|u| v (log(e + log |u| ) +∞ , and C d = C d (log(e+ 1 log 2 ))d On the other hand, Lemma 2.5, (4.59),(2.14), and (4.60) imply that:j jv |Φ -j (2 -j u * , v, ω) -Φ -j (0, v, ω)| ≤ Cd (ω) j jv (log(e + j + 2 -j |u|)) d Cd (ω)2 (j 3 +1)v j jv (log(e + (j 3 + 1 -j) + 2 -(j 3 +1-j) |u|)) d jv (log(e + j 3 + 2 j )) d C d (ω)|u| v (log(e + j 3 )) d C d (ω)|u| v (log(e + log |u| ) d (4.64)where Cd is a positive finite random variable not depending on u, C d Finally, (4.63) and (4.64) entail that (4.5) holds. Now, let us show that (4.6) is satisfied. Recall that, for all (v, ω) ∈ (0, 1) × Ω * , one has:X hf (u, v, ω) = +∞ j=0 2 -jv (Φ j (2 j u * , v, ω) -Φ j (0, v, ω)) ,(4.65)where Φ j (•, •, ω) is the infinitely differentiable function on R d × (0, 1), introduced in (2.12).Also, recall that the series in (4.65) is uniformly convergent in (u, v) on each compact subset of R × (0, 1). Next, let us mention that thanks to the convexity property of the function z one has the following inequaty:∀ (x, y) ∈ R 2 + , (x + y) mentions that the inequality ∀ (x, y, z) ∈ R 3 + , log(e + x +yz) ≤ log(e + x + y) + log(e + z) (4.67) holds, since (e + x + yz) ≤ (e + x + y)(e + z) and the logarithm is an increasing function. Using (4.65), the triangle inequality, Lemma 2.5, (4.67) and (4.66), one obtains: |X hf (u, v, ω)| ≤ Ĉd (ω) +∞ j=0 ja log(e + j + 2 j |u|) d Ĉd (ω) +∞ j=0 ja log(e + |u|) + log(e + j + 2 j ) ja log(e + |u|) d 2 + log(e + j + 2 j ) a positive finite random variable not depending on u, Ĉ d = ja log(e + j + 2 j ) d 2 < +∞ . It easily results from (4.68) that (4.6) holds. Proof of Corollary 4.6. The corollary is a straightforward consequence of Theorem 4.5 and (1.7).

  ). Then, thanks to this nice representation, global and local behavior of {X(u, v) : (u, v) ∈ R × (0, 1)} and {Z(t) : t ∈ R} are studied in Sections 3 and 4 of our article, by using a wavelet methodology which is, to a certain extent, inspired by the one introduced in[START_REF] Ayache | Multifractional Processes with Random Exponent[END_REF][START_REF] Ayache | Wavelet construction of generalized multifractional processes[END_REF].

  .46) where K 3 is a positive finite random variable not depending on (u, v).

	Finally, combining (4.25), (4.31), (4.32), (4.34) and (4.46), one obtains the theorem.
	Proof of Corollary 4.2. Using (1.7) and the triangle inequality, one gets that
	sup
	t∈[t 0 -1,t 0 +1]

  |u| < 2 j 3 +1 ,

					(4.59)
	which means that	j 3 =	log |u| log 2	.	(4.60)

Acknowledgements

The authors are very grateful to the anonymous referee for his valuable comments which have led to improvements of the article. This work has been partially supported by the Labex CEMPI (ANR-11-LABX-0007-01) and the GDR 3475 (Analyse Multifractale).


Owen K Appadoo
email: owen.appadoo@uca.fr

Hugo Rositi
email: hugo.rositi@uca.fr

Sylvie Valarier
email: sylvie.valarier@ch-lepuy.fr

Marie-Claire Ombret
email: marieclaire.ombret@ch-lepuy.fr

Émilie Gadéa

Christine Barret-Grimault
email: christine.barret@ch-lepuy.fr

Christophe Lohou
email: christophe.lohou@uca.fr

IMPLEMENTATION OF COMMUNICATION MEDIA AROUND A MIXED REALITY EXPERIENCE WITH HOLOLENS HEADSET, AS PART OF A DIGITALIZATION OF A NUTRITION WORKSHOP MULTIMEDIA COMMUNICATION ARTICLE

Keywords: Visual Communication, Multimedia, Mixed Reality, HoloLens, Nutrition

The release of Microsoft's HoloLens headset addresses new types of issues that would have been difficult to design without such a hardware. This semi-transparent visor headset allows the user who wears it to view the projection of 3D virtual objects placed in its real environment. The user can also interact with these 3D objects, which can interact with each other. The framework of this new technology is called mixed reality. We had the opportunity to numerically transform a conventional human nutrition workshop for patients waiting for bariatric surgery by developing a software called HOLO_NUTRI using the HoloLens headset. Unlike our experience of user and

Context 1.Introduction

One of the patient therapeutic education workshops of the Hospital Center Emile Roux nutrition department in Le Puy-en-Velay, France, is to make patients in need of bariatric surgery (reducing the stomach volume) aware of the changes in their diet after surgery. Patients have to compose a standard menu after surgery using printed cards on a table (food/associated quantity), then the nutrition team analyzes their choices and delivers qualitative and quantitative messages. We call this workshop CONV_WORKSHOP.

A team of computer scientists collaborated with the nutrition team to digitally transform their menu composition workshop by using the HoloLens headset, recently introduced by Microsoft. We call DIG_WORKSHOP this digital workshop version. These scientists then developed a computer application, called HOLO_NUTRI, to allow patients to compose their menu while using the headset. Synthetic 3D elements (such as a cube) are superimposed on the semi-transparent visor without occulting the vision through the headset. Unlike a direct approach to augmented reality, the user can interact with these synthetic objects, they can also interact with each other and with the environment, this new technological framework is called mixed reality.

In addition to the computer design for the development of HOLO_NUTRI operating various software and libraries, a specific development is required to provide the most appropriate user experience, for example, an arrangement of manipulable 3D objects at a distance from the user (aspects developed in the article [START_REF] Rositi | Presentation of a mixed reality software with a HoloLens headset for a nutrition workshop[END_REF]). The obligation to propose an appropriate communication for a good comprehension of both the material and the software HOLO_NUTRI quickly appeared to us, so that the patients could use this software and understand the exploitation of the headset (superimposition of 3D objects, gestures to perform, etc.) in less than 1h30.

In this article, we describe how we digitalized such a workshop using both holographic headsets and our produced software HOLO_NUTRI. More specifically, we justify all the steps of communication media necessary for the success of this project. These communication elements are offered in three different forms: paper support (leaflet), multimedia support (video sequences), software support (learning exercise). More precisely, four leaflets, four video sequences and a "learning exercise" software, named LEARN_EX, have been developed for this purpose. We think that this experiment could be adopted for any communication using this technology, in our case it represents about a quarter of the overall preparation time of the project, the remaining concerns the encoding of our software, so it is an important element to take into account.

The article is structured in the following way: in section 1.2, we first describe the progress of the conventional CONV_WORKSHOP workshop in order to understand later the complexity of its digitization exploiting the mixed reality. Then we explain the objectives to be proposed in the context of this digitization (section 1.3.1), and justify the choice of mixed reality (section 1.3.2) to achieve them in an optimal way. We briefly describe the features of the HOLO_NUTRI application (section 1.4.1), and the correct way to dispose the headset (section 1.4.2). Then, we detail the steps of the numerical workshop DIG_WORKSHOP (section 2), in order to understand the relevance of the various elements of communication. Section 3 presents the global communication process for setting up the workshop DIG_WORKSHOP; this same process is more detailed in section 4. In this same section, we will focus on the development of video sequences for the presentation of the headset, the actions to be performed and associated gestures (section 4.1), the presentation sequence of the LEARN_EX learning exercise (section 4.2), and finally the design of an information leaflet used in HOLO_NUTRI (section 4.3). Section 5 will summarize the assessments made by the thirty patients concerning the communication. We conclude in section 6.

Progress of the conventional nutrition workshop

The nutrition team composed of endocrinologists, digestive surgeon, nurse coordinator psychologist and dietician receives patients who will undergo a bariatric surgery. This surgery aims at a weight loss of the patient. For that, two surgical solutions are considered: either the reduction of the volume of the stomach "Sleeve Surgery" (Figure 1(a)), or to shunt the normal way of progression of the foods "Bypass Surgery" (Figure 1(b)). The proposed workshop CONV_WORKSHOP aims to make the patients understand (on average 6 by group) the difference of diet before and after surgical intervention. The workshop lasts one hour and half.

The workshop is organized according to the 9 steps described below (summarized in Table 1): Introduction of the session (step 1); debate around the question «Feeding ourselves, eating, when we have been under a bariatric surgery, what does it bring to mind ?» (step 2); initiation of the workshop to discuss with patients about their daily diet to adopt after intervention (step 3); patients are then invited to compose their menu by choosing cards representing different foods according different quantities, the images come from the SU.VI.MAX study [START_REF] Hercberg | The SU.VI.MAX Study: A Randomized, Placebo-Controlled Trial of the Health Effects of Antioxidant Vitamins and Minerals[END_REF][START_REF] Hercberg | Portions alimentaires : Manuel photos pour l'estimation des quantités[END_REF] (step 4); collection by the clinical team of the menus developed by patients (step 5); collegial discussion of the choice made by each patient (step 6); the dietician then gives each patient some advices based on the foods and quantities she or he has chosen (step 7); dissemination of quantitative and qualitative messages (step 8). The qualitative messages relate on the need to vary the number of dishes, the duration of the meal, the chewing time; quantitative messages are about reducing the amount of food compared to a meal before surgery. Finally, patients complete forms to evaluate the session (step 9). We stress on the importance of oral communication throughout this workshop. The gestural interaction of the patients, on the other hand, is done only when they take the cards representing the food.

Our incentive : a digitalization of the workshop by mixed reality 1.3.1 Workshop digitalization

We have proposed a software version (digitalization) of the conventionnal workshop CONV_WORKSHOP, previously described in section 1.2. This new workshop DIG_WORKSHOP aims to propose:

• the digital version of menu composition using a computer application (HOLO_NUTRI), • a more detailed analysis of the developed menu (several quantities associated with food, simulation of the meal time, taking into account the chewing time, etc.), • a similar approach, in the computer application, of the patient's gesture when she or he grabs the card representing the food and the associated quantity in CONV_WORKSHOP.

The first two items above can be done using a conventional computer and software development. On the other hand, we have sought the most appropriate solution as regards to the implementation of the gesture to catch the numerical model corresponding to the printed card of the food, or even to reinforce this learning. To do this, we then opted for a solution setting up a mixed reality framework that we define below.

Mixed Reality

Microsoft has recently released the HoloLens headset [START_REF]Microsoft HoloLens[END_REF]. In addition to scanning the real environment around the user, this headset allows to display virtual 3D objects (called holograms) superimposed on the visualization of the real environment, and to interact with these same objects thanks to different sensors headset (Figure 2).

Therefore, it is the physical aspect (collisions, etc.) which is put forward for the exploitation of the mixed reality. The user can interact with his environment using the GGV triplet (Gaze, Gesture, Voice) [START_REF]Gaze, Gesture and Voice[END_REF], by triggering actions in the computer application. Indeed, the integrated gyroscope computes the movements of the head -Gaze, it should be noted that it is the orientation of the head which is taken into account and defines this notion of gaze, the eyes are not tracked; the depth cameras of the headset detects certain gestures of the user (hand fingers tight, open hand, etc.) -Gesture; a microphone can hear the words or sounds of the user -Voice.

More and more applications using HoloLens within an industrial framework (automobile, architecture, engineering, medical) are proposed, to give, for example, information on a product during industrial maintenance or bring anatomical information during surgical operations [START_REF] Lohou | Preliminary experiment of the interactive registration of a trocar for thoracoscopy with HoloLens headset[END_REF][START_REF] Lohou | Mixed reality experiment for hemodialysis treatment[END_REF]. In the next section, we describe our mixed reality software.

1.4 Our solution

Description of HOLO_NUTRI software

The HOLO_NUTRI software is composed of two stages.

The first stage shows the user the type of intervention she or he will undergo (either using buttons in the application (Figure 3(a)), or by scanning a leaflet with the HoloLens (Figure 3(b)), knowing that there is a leaflet for each type of intervention. For the implementation of the communication that will be described later (sections 3 and 4), we use Photoshop and Illustrator from Adobe Creative Suite [START_REF]Adobe creative suite[END_REF]. Then a virtual stomach is displayed on which the user will interact (for example, for the Sleeve operation, the user will select a portion of the stomach and remove it (Figure 4). This analogous corresponding part was not present in the conventional workshop CONV_WORKSHOP. Once the action is done, the second stage starts.

The second stage shows a virtual self-service restaurant containing thirty foods. By the amount of information and so that the patient can interact easily, this self-service restaurant is proposed as a half-cylinder model with three different quantities (small, medium, large) of a given food in the same column (Figure 5). This seems to be the optimal solution to present a maximum of elements with which the user wearing the headset can interact. In other words, the environment has been specifically designed in relation to the use of such headsets.

These foods are illustrated as 2D images from the SUVIMAX study [START_REF] Hercberg | The SU.VI.MAX Study: A Randomized, Placebo-Controlled Trial of the Health Effects of Antioxidant Vitamins and Minerals[END_REF][START_REF] Hercberg | Portions alimentaires : Manuel photos pour l'estimation des quantités[END_REF]. The user then chooses her or his food by looking at it (Gaze of the HoloLens) then grasps it by pinching her or his fingers and then releasing them (Gesture of the HoloLens). The food appears in front of a 3D jaw in the shape of a cube (Figure 5). The user grasps the food again and brings it to her or his own mouth, an animation (cycle of opening and closing of the jaw) is then launched so that the user becomes aware of the chewing time. The user repeats this process as long as she or he thinks it is necessary to add food to compose her or his menu. When the menu composition is achieved, a first assessment is proposed concerning the number of meals and the corresponding quantity. Then the quantitative and qualitative messages, the same as those of CONV_WORKSHOP, are displayed. The user can then repeat this training a second time. During this second iteration, additional indicators (number of portions ingested, etc.) can help the patient to better compose her or his menu.

Setting up the digital workshop DIG_WORKSHOP

This workshop was performed six times with three new patients each time who had to use HOLO_NUTRI at the same time. Each patient wore a HoloLens headset. Since the headset is semi-transparent, patients could then keep in touch with the coordinating nurse. The solution is quite complex to set up:

• this headset must be presented to patients who, for a few, have poor skills in computer science,

• the gestures to the HoloLens can be quite complicated to perform,

• the connection between the computer and the HoloLens can cause problems.

Unlike CONV_WORKSHOP, a team of computer scientists is required to assist patients to use this headset and therefore this application. We then chose to display a visual feedback of what the user sees by connecting the headset to a laptop: each patient is then assisted by a computer specialist who can help her or him through the visual feedback on the computer screen.

Nevertheless the guidance of a patient by a computer scientist is not enough for the effective exploitation of the application. The latter can be described as "pure mixed reality" in the sense that it is not proposed to use only the traditional computer / screen / keyboard / mouse set; indeed, the objectives described in section 1.3.1 would not be reached and in particular the gesture made by the patient to grab the food. Users may have difficulty using mixed reality, unlike a more traditional computer program. In order to guarantee a good flow, this application requires a very specific communication so that patients are as autonomous and efficient as possible.

We then describe the communication we have put in place for this workshop; it relies on the use of multimedia supports briefly described in section 3 and more precisely in section 4. We will, first of all, describe the different stages of the workshop DIG_WORKSHOP in section 2 so as to understand the proposed communication.

Description of the DIG_WORKSHOP

The workshop DIG_WORKSHOP consists of 9 steps (see Table 2): Introduction of the session (step 1), identical to step 1 of CONV_WORKSHOP; debate around the question "feeding ourself, eating, when we have been under a bariatric surgery, what does it bring to mind? (Step 2), identical to step 2 of CONV_WORKSHOP; presentation of the group of computer scientists (step 3); presentation of the material and its use (step 4) through a video. We have developed a software LEARN_EX to learn the basics of the HoloLens that are essential to the use of HOLO_NUTRI.

We then give a user manual on this learning exercise LEARN_EX (step 5), exercice that patients must complete. Then, we begin the workshop to discuss with patients about their daily diet after intervention (step 6), identical to step 5 of CONV_WORKSHOP; patients are invited to compose their menu within HOLO_NUTRI software (step 7); the dietician then gives each patient advices based on the foods and quantities she or he has chosen (step 8); patients complete forms to evaluate the session (step 9), the same step in step 9 of CONV_WORKSHOP.

3 Preliminary thinking on the setting of a communication dedicated to the digital workshop DIG_WORKSHOP

From a logistical point of view, the constraints we face for this kind of workshop are multiple (small room space, several patients at the same time, time limit of 1:30 hour for a 3-person group who must stay seated for the entire session). The official tutorial offered by Microsoft aims at computer experienced users. Here, we have to present the headset as simply and efficiently as possible in order to this one to be used in the most optimal way no matter the skills the user could have in computer science. It was also unnecessary to show all features of this headset as they are not all used in the HOLO_NUTRI application (e.g. voice control). In other words, we could not use original demonstrations as proposed by the headset construtor, we had to develop a dedicated communication designed both for patients and for the purpose of our application.

Global communication process

We provide all communication actions designed specifically for DIG_WORKSHOP by referring to Table 2:

• Step 1. Presentation by the usual team (clinicians) of the workshop objectives (awareness on feeding modification after the surgery), all workshop steps presentation and free consent form signature, The communication is described in more details for each of those steps in the next section.

Clinical team action Clinical team + patients action

Patients action

Computer scientists action

Computer scientists team + patients action Application action

Description of the content of the global communication

In this section, we detail the different steps proposed in the previous section, while putting forward the differences of communication between the conventional session CONV_WORKSHOP and the digital session DIG_WORKSHOP resulting from the exploitation of the mixed reality by patients. In addition to reviewing these different steps, we justify the choices of communication or used media as well as their design to be usable both in the digital workshop DIG_WORKSHOP and in practical by a HoloLens headset. More particularly, we will thus detail the content of the learning exercise (content) designed to understand the gestures to handle HoloLens headset (section 4.2.1), the video sequences (sections 4.2.2 and 4.3.1) and also the design of the leaflets (content and conception) (section 4.2.3 and 4.3.4). The reflections are certainly constrained within the framework of our workshop, but could be useful to other people in a completely different framework of exploitation of mixed reality. • The three patients are installed around an oval table, a laptop is installed in front of each patient. A computer scientist is located on the side of each patient and can then see the visual feedback of what the patient sees in the headset, on the laptop screen (Figure 6).

4.1.3

Step 4. Presentation of the hardware

• Step 4.1.
Oral presentation of the HoloLens headset. A computer scientist shows the headset to the patients so that they realize the weight and the bulk of the hardware. Some notions of mixed reality are introduced: possibility to see the real environment and 3D objects on the transparent visor of the headset, taking into account the space around the patient for the display and manipulation of virtual objects. This step also aims to reassure patients who are not necessarily used to computers and even less devices like these headsets. • Step 4.2. Presentation of the video sequence SEQ_HEADSET about the headset and its carrying. It is simultaneously commented by computer scientists. The duration of this sequence is 52 seconds. Description of the video sequence SEQ_HEADSET: an operator shows the headset from different views (front, rear, side); then the operator positions the headset on his head and shows that he can sweep his environment by turning his head. During the sequence, we indicate that it is the orientation of the head that is taken into account by the HoloLens and that the headset does not allow the follow-up of the eyes (known as eye-tracking; note that HoloLens in its second version, not yet available during our experiment, takes into account eye-tracking).

• Step 4.3. Presentation of the video sequence SEQ_GESTURES on gestures recognized by the HoloLens, specifically only the gestures used in the LEARN_EX learning exercise and in the HOLO_NUTRI application are presented, these gestures are identical for these two applications. The duration of this sequence is 57 seconds.

The interaction system based on the gesture recognition is introduced. Different gestures are natively recognized by the hardware to interact, that is to say a precise gesture (or a combination of gestures) is associated with a specific action. For example, pinching the thumb and forefinger with the rest of the hand closed triggers the "selection action", releasing the thumb and index finger after the selection results in the "validation action", holding the inch and index finger triggers the "move action". In this sequence SEQ_GESTURES, we describe these three gestures and highlight on the good positioning of the hand to perform these gestures (Figure 4). We also show that it is possible to do wrong gestures (for example, selection and pinching errors) always in order to reassure patients and not to rush during the experiment. We deem that the part presenting the bad handling to make the gestures (too fast speed of pinching, bad orientation of the hand) is mandatory. During pauses occurring while the video is playing, we also highlighted the presence of cameras in the headset which detect hands: this technical justification implies that gestures must be made correctly (right angle between the index finger and thumb to select, pinch speed, etc.) and should be done according to the orientation of the head. The announcement of this technical constraint made it easier for patients to correct themselves, and thus to perform the session more efficiently.

• Step 4.4. Distribution of the leaflet LEAFLET_HEADSET_GESTURES

The team of computer scientists distributes LEAFLET_HEADSET_GESTURES leaflet (Figure 7) containing the information of the two video sequences (SEQ_HEADSET and SEQ_GESTURES), namely the setting of the port of the headset (step 4.2) as well as a summary of the basic gestures to be carried out (step 4.3): orient your head, point your hand towards the headset camera, spread your thumb and forefinger. Moreover, thanks to the transparent visor of the HoloLens, the patient can at any time visualize the most important steps of the application included on this leaflet, provided that the occupation of the information projected on the visor leaves enough of space in the user's field of view.

Step 5. Learning exercise LEARN_EX

To successfully use the application HOLO_NUTRI, it is mandatory that patients gestures are performed correctly with the HoloLens. It quickly became necessary to offer a learning exercise to patients, and as noted above, the most suitable for the application HOLO_NUTRI (and not to use the official tutorial). We first describe the LEARN_EX learning exercise and then the associated communication.

Description of the learning exercise LEARN_EX

The learning exercise LEARN_EX (Figure 8) is composed of the following three stages:

• Stage 1: Menu Validation Action (Figure 8(a)) > This step teaches patients to look at a button (whose default color is gray). When the gaze is put on the button after a few seconds, the button turns green. The patient can then select it by performing the "pinching" gesture, then the "loosening" gesture. Once the gesture is done correctly, the button displays the text to the next step and turns blue.

• Stage 2: 3D Object Selection Action (Figure 8(b)) > In this step, patients learn to look at a 3D object, a gray cube. Like stage 1, the cube turns green after a few seconds when the gaze is held on the button. The "pinching" gesture allows the user to select the cube, the "loosening" gesture to deselect the cube and to display a next button to move forward, on that button you have to repeat the same gestures as during stage 1 to go through in the next stage.

• Stage 3: Move Action (Figure 8(c)) > This last stage teaches patients to look at the cube and with the only "pinching" gesture, i.e. keeping the thumb and index finger tight, thus the cube becomes blue. It is then possible to shift it by the gesture "movement of the arm". The displacement stops during the "loosening" gesture. Then the learning exercise stops.

The objective of LEARN_EX is that the user becomes familiar with this new type of manipulation. Unlike the official tutorial provided with the headset (heavy digital scenes), we propose an application manipulating only a 2D object (menu button), a 3D object (cube), the 3 actions (validate, select, displace) and the associated gestures or gesture combinations (pinching, loosening and moving the arm). This allows patients to be as effective as possible on this new technology for our purposes.

Therefore the user becomes more familiar, on one hand, with 3D objects (holograms), her or his environment and this new augmented visualization both, and on the other hand, with their selection and their displacement in the 3D space.

The positioning of the gaze and the 3 actions "validation", "selection" and "displacement", by the gestures "pinching", "loosening" and "movement of the arm" are also the only actions and gestures necessary to perform the application HOLO_NUTRI. We also took on the same visual appearance on the graphical elements of the interface for the two "validation" and "selection" actions between the LEARN_EX learning exercise and the HOLO_NUTRI application.

The communication for LEARN_EX is composed of two supports. The first one, described in section 4.2.2, is the SEQ_LEARN_EX video sequence explaining the exercise (step 5.1). The second one, described in section 4.2.3, is the leaflet LEAFLET_LEARN_EX. The latter is given out to patients during the step 5.2. We now detail these two supports. The duration of the video sequence (Figure 8) is of 1 minute and 13 seconds. We chose to show the whole sequence to patients so that they do not have to put on and take off the headset between each stage of the exercise LEARN_EX. In addition, it would have been difficult for a user to view this video clip played thoroughly on a computer screen, through the visor of the headset.

During the viewing of the sequence, we emphasize the importance of the gaze when selecting an object in the environment. Take the example of a cube (Figure 9(a)). When we look at the cube, a cursor occurs with the form of a disc (reticle) (Figure 9(b)). We recall that the movement of the eyes is not taken into account by this headset; for the sake of simplicity, we call "put the gaze on" the "orientation of the head" action. Thus, any interaction with the environment is carried out in two stages: the user must first pose and maintain his gaze on a hologram (3D object or button to be selected), then perform the expected gesture while maintaining the gaze on the object. When the gaze is putted on the object of interest, then the reticle is transformed into a hand shape after a few seconds (Figure 9(c)) if the camera has detected that the hand was ready to perform one of the expected actions on this object (corresponding to an action programmed in the application); it is then possible to interact (Figure 9(d)) with this object (selection, validation, displacement).

Note that the video seems to be the proper support for patients to understand the notions of reticle, its transformation in a hand shape (indicating that a gesture is ready to be detected on the object). We realized that the biggest difficulties with the learning exercise were:

• to focus on the object of interest, more specifically patients must understand that it is necessary to turn the head and not the eyes,

• to position one hand in front of the cameras of the headset, according to the orientation of the head,

• to spread the index finger and the thumb to make the "pinching" gesture.

Step 5.2 : Description of the leaflet LEAFLET_LEARN_EX of the learning exercise

The leaflet LEAFLET_LEARN_EX (Figure 10) recalls the gestures associated with the three actions: selection, validation and displacement (step 5.1).

4.3

Step 7 : Nutrition session HOLO_NUTRI

Step 7.1 : Presentation of the video sequence SEQ_HOLO_NUTRI describing HOLO_NUTRI application

We project the video sequence SEQ_HOLO_NUTRI, lasting 2 minutes and 58 seconds, presenting the contents arranged in the environment of the user. In the video, we emphasized the following points:

• the user must turn the head to scan the cylindrical virtual restaurant as a whole since the number of foods occupies a large visual space (Figure 2, Figure 5 and Figure 11).

• we inform users that different quantities of the same food are organized by rows (Figure 2, Figure 5 and Figure 11).

• we explain to the user that he must choose a food for a desired quantity. Once it is chosen, the food first appears in front of the virtual jaw, then the user must bring this food to her or his mouth. The gesture was not proposed in one go "bring the food to the mouth" but was broken down into two parts: to grab the food then to swallow it. Then a cycle of animation of the jaw starts preventing the user from taking another food until a predefined time (chewing time) has elapsed. This time is displayed in a timer,

• we recall that there are two iterations and that in the second one, additional indicators are shown (stomach in transparency, color of the stomach according to the number of dishes and portions as well as indicator on the number of dishes and portions),

• a summary is displayed with qualitative and quantitative messages corresponding to the menu composition.

Step 7.2 : Distribution of the leaflet LEAFLET_HOLO_NUTRI

We chose to have an introductory leaflet for each patient (step 7.2) from the beginning of the session in connection with the type of intervention she or he will undergo, as explained above. This leaflet contains both textual elements and an image representing one of the two types of interventions (Figure 14).

Interest of the leaflet for the HOLO_NUTRI application

The computer scientist assist the patient to launch the HOLO_NUTRI application in the headset. The patient chooses the leaflet corresponding to her or his intervention, then puts the headset. The image of the stomach is scanned by the headset when the user looks it. A virtual 3D stomach appears above the leaflet, stomach having undergone the type of intervention determined by the leaflet (Bypass or Sleeve), as shown in Figure 3. Then the user is asked to select a part of the stomach and to displace it (step 7.4.1), as shown in Figure 4, the goal is that the patient realizes the shrinking of the stomach's volume. Once the part of the stomach is removed, a new scene is launched in which the user can compose the menu; these are the two iterations of step 7.4.2.

Description of the leaflet LEAFLET_HOLO_NUTRI

Unlike the other two leaflets LEAFLET_HEADSET_GESTURES and LEAFLET_LEARN_EX that rather act as memory aid and reassure the patient, the leaflet LEAFLET_HOLO_NUTRI is scanned by the HoloLens headset: an adequate design had then to be proposed.

In this section, we describe the content of the leaflet and its layout. Then, we insist on the constraints related to HoloLens for the design of graphic elements of the leaflet (size and contrast of the image to be scanned). The two leaflets for chirurgical procedures, Bypass and Sleeve, are also shown.

Graphical content of the leaflet The content of the leaflet (shown in Figure 12) are : Disk 1. Title: type of intervention for a given pathology, Disk 2. Description of the pathology, Disk 3. 3D organ: this image is a marker registered in the application database. When this marker is scanned by the HoloLens, a 3D stomach hologram appears allowing the simulation to start (step 7.4.1), Disk 4. Brief explanation of eating habits, Disk 5. Brief explanation of the surgical procedure.

Constraints related to HoloLens We did several tests for marker detection by HoloLens in step 7.4.1, using the Vuforia library [START_REF]Vuforia[END_REF], directly integrated with the latest versions of Unity framework [START_REF]Unity[END_REF], that we use to develop application HOLO_NUTRI. The first marker was in line with the visual identity in flat design (Figure 13(a)) of the template leaflet described in the previous paragraph (Figure 12), but the image was not recognized. We then proposed a marker with a volume feature (Figure 13(b)), the marker was not recognized either. The detection finally works only by contrasting the previous marker (Figure 13(c)).

Leaflets for both types of interventions

The leaflet must be rigid enough both to be easily held in hand and durable to repeat the experience several times. Therefore we chose to laminate this leaflet. Laminating makes the leaflet brighter but does not alter the HoloLens' recognition of the leaflet. It should be noted that in case of unfavorable lighting conditions that do not allow the image to be scanned, the simulation (step 7.4.1) can also be launched by two menu buttons (Figure 3(a)). The two resulting leaflets are shown in Figure 14.

Results

We first report the duration of the video sequences and explanations projected and given to patients, then analyze the patient's assessment of different communication elements.

Time information

The sequences SEQ_HEADSET, SEQ_GESTURES, SEQ_LEARN_EX and SEQ_HOLO_NUTRI last respectively: 0:52 minute, 0:57 minute, 1:13 minute and 2:58 minutes. Ten sessions occurred at the hospital. The average watch time of the sequences SEQ_HEADSET and SEQ_GESTURES paused several times to give additional explanations (see section 4.1) is 5 minutes (standard deviation 0:50 minutes). The average duration of viewing of the SEQ_LEARN_EX sequence and the completion of the LEARN_EX learning exercise is 4 minutes (standard deviation 2:38 minutes). The average duration of viewing of the sequence SEQ_HOLO_NUTRI is 4 minutes (standard deviation 00:42 minutes). To give a comparison order, the realization of the two iterations of HOLO_NUTRI lasted on average 18 minutes (standard deviation 5:54 minutes) [START_REF] Rositi | Presentation of a mixed reality software with a HoloLens headset for a nutrition workshop[END_REF].

Detail of the feasibility study following LEARN_EX

Thirty patients participated at the sessions. The patients' feedback following the completion of the LEARN_EX learning exercise is given in Tables 3 and4.

Global analysis

See Table 3. (Question Q1) Twenty-two patients (73.3%) thought the exercise LEARN_EX was easy to use after watching the video SEQ_LEARN_EX, eight patients (26.7%) felt difficulty to understand the mixed reality environment. (Question Q2) twenty-eight patients (93.3%) found the communication efficient regarding the actions to be carried out either in the video sequence SEQ_LEARN_EX or in the exercise of learning LEARN_EX. Only two patients (6.7%) have experienced technical difficulties. (Question Q3) nineteen patients (63.3%) felt comfortable during the learning exercise, nine patients (30%) did not feel really comfortable, two patients (6.7%) not at all. For example, one patient showed signs of nervousness because of difficulty to grab the food (too short distance between the hand and the headset).

Analysis by patient

See Table 4. We can notice that among the nine patients (30%) who were not really comfortable using the learning exercise (orange cell, question Q3), six of them (orange cell, question Q1) thought the exercise was not easy to use. One of the two patients (patient id#2) who expressed not at all comfortable using the exercise (Q3) thought the exercise was easy to use. Eleven patients (36.7%) were not really or at all comfortable and eight of them have answered that the initiation to the technical gestures seemed to them rather clear (Q2) and for one of them it was quite clear (green cells, question Q3).

Nineteen patients (63.3%) who were more or somewhat comfortable using the learning exercise (question Q3), eighteen of them found the learning exercise rather easy to use after viewing the sequence (question Q1), and that the initiation to the technical gestures seemed to them clear (question Q2), only one patient (patient id#6, purple cell) thought the application was not easy to use and it is the only disconnect between communication and practice.

The answers are rather positive concerning the communication (questions Q1 and Q2), they are less positive concerning the roll-out of LEARN_EX (question Q3). Additional results concerning the application HOLO_NUTRI, and not only the communication set up as in this section, were also collected [START_REF] Rositi | Presentation of a mixed reality software with a HoloLens headset for a nutrition workshop[END_REF]; the results about this project are based on the NASA-TLX, SUS and on clinical feasibility study CARACO [START_REF] Barret-Grimault | Innovons en ETP grâce au casque HoloLens en chirurgie bariatrique[END_REF].

Conclusion

Due to its complexity of handling and its novelty, adapted communication was necessary to present the HOLO_NUTRI hardware and software to thirty patients. The latter found the use of this material and the application quite simple thanks to the communication deployed throughout the workshop. Global communication was important and essential. It has been lightened enough not to overwhelm patients with technological concepts.

Despite our experience as computer scientists and communication experts, we did not anticipate as much work was necessary on this part of communication. Finally it represents a quarter of the total work done in this experiment; it would have required much less work for a computer application in a conventional setting but the latter would not have allowed a similar learning in the gesture of patients and therefore the desired impact of this new workshop.

The experience was very much appreciated by both the patients and the teams, it would not have been the case if the communication had not been prepared, thought out and carefully presented during the experiment. If we had to deal with a new type of intervention on patients with this type of material in a mixed reality setting, we think that these elements (leaflets and videos) are sufficient. More generally, this feedback of communication experience may be of interest to any developer of mixed reality applications.

Acknowledgments

• Grant: PEPS INSIS CNRS " Engineering Sciences for Health to support translational projects ", 2017, AVACM Project (Augmented Visual Assistance during Medical Consultations),

• Patients of the Bariatric Surgery / Human Nutrition departments, Le Puy-en-Velay, Hospital Center,

• Jean-Raymond Casimir, Jonathan Etienne, students of the bachelor of Development of Interactive 3D Graphics Applications option, Computer Graphics Department, IUT of Le Puy-en-Velay, for the development of HOLO_NUTRI,

• Arthur Jacquin, student of Diploma of Computer Graphics Department, IUT of Le Puy-en-Velay, for the development of HOLO_NUTRI,

• Marianne Bonneton, Robin Morel, Romain Vergnaud, students of Diploma of Multimedia Department, IUT of Le Puy-en-Velay for the research on visual design of leaflets.

Figure 1 :

 1 Figure 1: Both types of interventions considered : (a) Sleeve Surgery -the dark part is removed, (b) Bypass Surgerythe dark part represents the middle part of the small intestine which is a shortcut for food flow. Figure from [1].

7 .Figure 2 :

 72 Figure 2: Photomontage illustrating the visualization of the headset wearer in his environment.

Figure 3 :

 3 Figure 3: (a) Sleeve / Bypass Surgery toolbars. (b) Scanned leaflet (Sleeve intervention). A virtual 3D stomach occurs in front of the leaflet when the latter is scanned by the headset.

Figure 4 :

 4 Figure 4: User interaction with the virtual stomach. Frame retrieved from SEQ_GESTURES sequence (Section 4.1.3).

Figure 5 :

 5 Figure 5: Self-service restaurant shown as a half cylinder model.

Figure 6 :

 6 Figure 6: Visual feedback (seen by a patient through the headset (left), shown on the laptop screen (right)).

4. 1

 1 Description of steps 1 to 4 4.1.1 Step 1. Session introduction • Steps 1.1 and 1.2. Presentation of the protocol by the clinical team and signature of the consent forms by the patients (participation in the study, image rights), • Step 1.3 Presentation of the group of clinician and patients, • Step 1.4 Presentation of the objectives of the session. 4.1.2 Step 3. Presentation of the team of computer scientists.

Figure 7 :

 7 Figure 7: LEAFLET_HEADSET_GESTURES leaflet.

Figure 8 :

 8 Figure 8: The three stages of the learning exercise LEARN_EX: actions (a) Menu validation, (b) 3D object selection, (c) 3D object displacement. Frames extracted from our video sequence SEQ_LEARN_EX.

Figure 9 :

 9 Figure 9: Display and cursor shape: (a) cube, (b) reticle, (c) hand shape, (d) move action.

Figure 10 :

 10 Figure 10: Leaflet LEAFLET_LEARN_EX summarizing the learning exercise LEARN_EX.

Figure 11 :

 11 Figure 11: Screenshot of a part of the virtual self-service restaurant. Zoom-in boxed shows three different foods with three different quantities. Frame extracted from SEQ_HOLO_NUTRI.

Figure 12 :

 12 Figure 12: Template leaflet for a chirurgical procedure.

Figure 13 :

 13 Figure 13: Marker tests for better detection by HoloLens: (a) flat design rendering, (b) initial 3D rendering, (c) 3D contrast rendering.

Figure 14 :

 14 Figure 14: Leaflets LEAFLET_HOLO_NUTRI scanned by HOLO_NUTRI (step 7.4.1): (a) Bypass, (b) Sleeve.

Table 1 :

 1 Description of both steps and actions during conventional workshop CONV_WORKSHOP. Actors of all actions are identified by specific colors detailed at the end of this table.

	1. Session introduction
	1.1. Presentation of the protocol and signature of the consent forms
	1.2. Signature by patient
	1.3. Group presentation (clinicians / patients)
	1.4. Presentation of the objectives of the session
	2. Debate around the question «Feeding ourselves, eating, when we have been
	under a bariatric surgery, what does it bring to mind?»
	3. Beginning of the workshop: Initial question "after the surgery, how
	to eat daily? "
	4. Carrying out the workshop experience with cards
	4.1. Presentation of the workshop with cards
	4.2. Food choices via cards
	4.3. Register quantities in a table
	5. Discussion around the choices of each patient with the whole group
	(collecting information of each patient)
	6. Highlighting important messages (in relation to the choices made
	by each patient)

Table 2 :

 2 Description of both steps and actions during digital workshop DIG_WORKSHOP. Actors of all actions are identified by specific colors detailed at the end of this table.

	1. Session introduction
	1.1. Experimental protocol presentation and free consent form signature
	1.2. Signature of the free consent form by each patient
	1.3. Group presentation (clinicians/patients)
	1.4. Session objectives presentation
	2. Debate around the question «Feeding ourselves, eating, when we have been
	under a bariatric surgery, what does it bring to mind?»
	3. Group presentation (computer scientists/patients)
	4. Hardware presentation
	4.1. HoloLens presentation (oral + hardware)
	4.2. Headset presentation and its carrying (video sequence SEQ_HEADSET + oral)
	4.3. Gesture presentation (video sequence SEQ_GESTURES + oral)
	4.4. Reminder leaflet delivery (leaflet LEAFLET_HEADSET_GESTURES)
	5. Learning exercise
	5.1. Learning exercise presentation (video sequence SEQ_LEARN_EX + oral)
	5.2. Reminder leaflet delivery (1 LEAFLET_LEARN_EX per person)
	5.3. Headset setup (computer scientists/patients)
	5.4. Launching of learning exercise LEARN_EX
	5.5. Headset removal
	6. Workshop trigger
	6.1. Starting question «After the surgery, how to eat daily?»
	7. HOLO_NUTRI application
	7.1. Presentation of HOLO_NUTRI application (video sequence SEQ_HOLO
	_NUTRI + oral)
	7.2. Distribution of paper reminder on the application (1 leaflet LEAFLET_HOLO
	_NUTRI per person according to the type of surgical operation)
	7.3. Headset setup (computer scientists/patients)
	7.4. Launching of HOLO_NUTRI
	7.4.1. Surgical operation simulation (using leaflet or the menu buttons)
	7.4.2. First iteration (simple interface) and second iteration (same interface
	with more visual information and with continuous analysis of the food choice)
	7.4.2.1. Food choice via 2D images in the virtual self-service restaurant
	7.4.2.2. Recap panel
	7.4.2.3. Analyse of the patient choice
	7.4.2.4. Informative messages
	7.5. Headset removal
	8. Discussion on each patient choice in an individual manner
	9. Forms (knowledge evaluation + satisfaction)

Table 3 :

 3 Patients' feedback following the LEARN_EX learning exercise (30 patients) -summary

			Not at	Not re-	Yes,	Yes	NA
			all	ally	rather	abso-
						lutely
	Q1 After watch-	Number of	0	8	16	6	0
	ing the video, do	persons
	you think this app
	is easy to use ?
		% (on 30)	0%	26.7%	53.3%	20.0%	0%
	Q2 Did the in-	Number of	0	2	17	11	0
	troduction to the	persons
	technical gestures
	of the application
	seems clear to you
	?
		% (on 30)	0%	6.7%	56.7%	36.6%	0%
	Q3 Are you com-	Number of	2	9	12	7	0
	fortable using this	persons
	application ?
		% (on 30)	6.7%	30.0%	40.0%	23.3%	0%

Table 4 :

 4 Patients' feedback following the LEARN_EX learning exercise (30 patients) -detail per patient

		Q1	Q2	Q3
	Patient ID.	After watching	Did the introduc-	Are you comfort-
		the video, do you	tion to the techni-	able using this ap-
		think this app is	cal gestures of the	plication ?
		easy to use ?	application seems
			clear to you ?
	1	yes absolutely	yes absolutely	yes absolutely
	2	yes absolutely	yes, rather	not at all
	3	not really	yes, rather	not really
	4	yes, rather	yes, rather	not really
	5	yes absolutely	yes, rather	yes, rather
	6	not really	yes, rather	yes, rather
	7	yes, rather	yes, rather	yes, rather
	8	yes, rather	yes, rather	yes, rather
	9	yes, rather	yes absolutely	yes, rather
	10	yes, rather	yes absolutely	yes absolutely
	11	not really	yes, rather	not at all
	12	yes, rather	yes absolutely	yes absolutely
	13	yes, rather	yes absolutely	yes, rather
	14	yes, rather	yes absolutely	yes, rather
	15	yes absolutely	yes absolutely	yes absolutely
	16	yes, rather	yes, rather	yes, rather
	17	yes, rather	yes, rather	yes, rather
	18	yes, rather	yes, rather	yes, rather
	19	yes, rather	yes, rather	not really
	20	not really	not really	not really
	21	not really	yes, rather	not really
	22	yes, rather	yes, rather	yes, rather
	23	not really	yes, rather	not really
	24	not really	yes absolutely	not really
	25	not really	not really	not really
	26	yes, rather	yes, rather	not really
	27	yes, rather	yes absolutely	yes absolutely
	28	yes absolutely	yes absolutely	yes absolutely
	29	yes, rather	yes, rather	yes, rather
	30	yes absolutely	yes absolutely	yes absolutely
	8 Bibliography/References