Anne-Laure Thiel
email: thiel@math.unistra.fr

VIRTUAL BRAID GROUPS OF TYPE B AND WEAK CATEGORIFICATION

Keywords: braid group, virtual braid, categorification Mathematics Subject Classification (2000): 20F36, 57M27 (Primary); 13D, 18E30, 18G35 (Secondary)

We define virtual braid groups of type B and construct a morphism from such a group to the group of isomorphism classes of some invertible complexes of bimodules up to homotopy.

Introduction

Our aim in this paper is two-fold: we first define a virtual braid group of type B n and next construct a weak categorification of this group.

Virtual knots and braids have been introduced by Kauffman in [START_REF] Kauffman | Virtual knot theory[END_REF]; they can be represented by planar diagrams that are like usual link or braid diagrams with one extra type of crossings, called virtual crossings. Such crossings appear for instance when one projects a generic link in a thickened surface onto a plane (see [START_REF] Kamada | Abstract link diagrams and virtual knots[END_REF] or [START_REF] Kuperberg | What is a virtual link?[END_REF]).

The virtual braids with n strands form a group denoted VB n , which generalizes the usual Artin braid group B n . The group VB n has a presentation with 2(n -1) generators σ 1 , . . . , σ n-1 , ζ 1 , . . . , ζ n-1 , where σ 1 , . . . , σ n-1 satisfy the usual braid relations, ζ 1 , . . . , ζ n-1 satisfy the standard defining relations of the symmetric group S n , and the following "mixed relations" are satisfied

σ i ζ j = ζ j σ i , if |i -j| > 1,
and

σ i ζ i+1 ζ i = ζ i+1 ζ i σ i+1 , if 1 ≤ i ≤ n -2.
As is well known, the braid group B n can be generalized in the framework of Coxeter groups. Recall that given any Coxeter system (W, S) one defines a generalized braid group B W by taking the same generators s ∈ S and the same relations as for the Coxeter group W, except the relations s 2 = 1. When W is the symmetric group S n , i.e. of type A n-1 in the classification of Coxeter groups, then B W = B n .

A natural question is: can one similarly attach a generalized virtual group VB W to any Coxeter system (W, S)? The idea for defining such a group VB W would be, as in the type A case, to use two copies of the Date: 17.01.2011. 1 generating set S, and to require that the first copy satisfies the relations defining B W , the second one satisfies the relations defining W, and some mixed relations involving generators from the two copies of S are satisfied.

The problem is to come up with an appropriate and meaningful set of mixed relations. In this paper, we do not solve the problem in the general case, but we focus on the case of Coxeter groups of type B. We use a diagrammatic description of the generalized braid group B W of type B due to tom Dieck [tD94]; this description is in terms of symmetric braid diagrams. We define a generalized virtual braid group VB Bn of type B n by considering symmetric virtual braid diagrams up to some natural equivalence.

In a second part we "categorify" each newly-defined group VB Bn . Here we mean categorification in the weak sense of Rouquier, who actually proves a stronger version of this result in [START_REF] Rouquier | Categorification of sl2 and braid groups[END_REF]. More precisely, here to any word w in the generators of VB Bn we associate a bounded cochain complex F (w) of bimodules such that if two words w and w ′ represent the same element of VB Bn , then the corresponding cochain complexes F (w) and F (w ′ ) are homotopy equivalent. This leads to a morphism from the group VB Bn to the group of isomorphism classes of invertible complexes up to homotopy, which is not injective. We describe a part of its kernel. The bimodules used here have been introduced by Soergel, they have become important in the context of knot theory because they come up in the Khovanov-Rozansky link homology (see, e.g., [START_REF] Khovanov | Triply-graded link homology and Hochschild homology of Soergel bimodules[END_REF]). Our categorification extends Rouquier's weak categorification of generalized braid groups and our previous categorification [START_REF] Thiel | Categorification of the virtual braid groups[END_REF] of the virtual braid group VB n .

The paper is organized as follows. In Section 1 we recall the definition of the virtual braid groups VB n and of an invariant of virtual braids due to Manturov, which we see as a homomorphism of VB n into the automorphism group of a free group. In Section 2 we recall the definition of the generalized braid group of type B n and tom Dieck's graphical description in terms of symmetric braid diagrams.

We propose a definition of a generalized virtual braid group of type B n in Section 3. We show that each of its elements can be represented by a symmetric virtual braid diagram, but we do not prove the injectivity of this representation. Using Manturov's invariant, we show that certain relations do not hold in this group although they look natural.

Section 4 is devoted to the Soergel bimodules of type B n . In Section 5 we associate a cochain complex of bimodules to each generator of our virtual braid group of type B n , and we show that this leads to a categorification of this group in the weak sense of Rouquier.

Virtual braids

We first recall the definition of virtual braid groups and of Manturov's invariant for virtual braids.

1.1. The virtual braid groups. Let n be an integer ≥ 2. Following [START_REF] Manturov | Knot theory[END_REF], [START_REF] Vershinin | On homology of virtual braids and Burau representation[END_REF], we define the virtual braid group VB n as the group generated by σ 1 , . . . , σ n-1 and ζ 1 , . . . , ζ n-1 , and the following relations

σ i σ j = σ j σ i , if |i -j| > 1,
(1.1) The generator σ i can be represented by the usual braid diagram with a single positive crossing between the ith and the i + 1st strand; see Figure 1, whereas the generator ζ i is represented by the virtual braid diagram with a single virtual crossing between the ith and the i + 1st strand; see Figure 2.

σ i σ i+1 σ i = σ i+1 σ i σ i+1 , if 1 ≤ i ≤ n -2, (1.2) ζ i ζ j = ζ j ζ i , if |i -j| > 1, (1.3) ζ i ζ i+1 ζ i = ζ i+1 ζ i ζ i+1 , if 1 ≤ i ≤ n -2, (1.4) ζ 2 i = 1, if 1 ≤ i ≤ n -1, (1.5) σ i ζ j = ζ j σ i , if |i -j| > 1, (1.6) σ i ζ i+1 ζ i = ζ i+1 ζ i σ i+1 , if 1 ≤ i ≤ n -2. ( 1
1 i-1 i i+1 i+2 n Figure 1. The braid diagram σ i 1 i-1 i i+1 i+2 n Figure 2. The braid diagram ζ i
We use the following convention: if D (resp. D ′ ) is a virtual braid diagram representing an element β (resp. β ′ ) of VB n , then the product ββ ′ is represented by the diagram obtained by putting D ′ on top of D and gluing the lower endpoints of D ′ to the upper endpoints of D.

Relations (1.1), (1.3), (1.6) mean that we consider these planar diagrams up to planar isotopy preserving the crossings. Relation (1.2) illustrates the classical Reidemeister III move for ordinary braid diagrams.

Relations (1.4) and (1.5) mean that we consider the virtual braid diagrams up to the virtual Reidemeister II-III moves depicted in Figure 3. Relation (1.7) is a graphical transcription of the mixed Reidemeister move The braid group B n is obtained by dropping the generators ζ 1 , . . . , ζ n-1 . There is a natural homomorphism B n → VB n obtained by considering each braid diagram as a virtual braid diagram without virtual crossings. 1.2. Manturov's invariant. Manturov [START_REF] Manturov | On the recognition of virtual braids[END_REF] constructed an invariant of virtual braids, which he conjectured to be complete. Since we will be using it in Proposition 3.4, we have to recall its definition. We use the version that appeared in the review [START_REF] Izmestiev | Review of "On the recognition of virtual braids[END_REF].

In this version, Manturov's invariant can be seen as a group homomorphism f : VB n → Aut(F n+1 ), where F n+1 is the free group on n + 1 generators a 1 , . . . , a n , t. The homomorphism f is defined by the following formulas:

f (σ i )(a j ) =      a j+1 if j = i, a -1 j a j-1 a j if j = i + 1, a j otherwise, f (ζ i )(a j ) =      ta j+1 t -1 if j = i, t -1 a j-1 t if j = i + 1, a j otherwise,
and

f (σ i )(t) = f (ζ i )(t) = t for all i = 1, . . . , n -1.
Observe that this invariant specializes when t = 1 to an invariant constructed in [START_REF] Fenn | The braid-permutation group[END_REF] for welded braids; see also [START_REF] Vershinin | On homology of virtual braids and Burau representation[END_REF].

Braid groups of type B and symmetric braid diagrams

Let n be a positive integer. Consider the Coxeter group associated to the Dynkin diagram of type B n ; see [START_REF] Humphreys | Reflection groups and Coxeter groups, volume 29 of Cambridge Studies in Advanced Mathematics[END_REF].

We denote the associated generalized braid group by B Bn . It has a presentation with n generators s 0 , s 1 , . . . , s n-1 , and three families of relations

s i s j = s j s i , if |i -j| > 1,
(2.1)

s i s i+1 s i = s i+1 s i s i+1 , if 1 ≤ i ≤ n -2, (2.2) s 0 s 1 s 0 s 1 = s 1 s 0 s 1 s 0 . (2.3)
In [tD94] tom Dieck gave a diagrammatic description of B Bn in terms of symmetric braid diagrams with 2n strands.

Fix a vertical line {0} × R in the plane R 2 . Consider the reflection in this vertical line. It acts on the underlying graph of a braid diagram. It lifts to a reflection on braid diagrams by imposing that the sign of the crossings is preserved. (Observe that if one identifies the plane R 2 with {0} × R 2 ⊂ R 3 , this reflection can be viewed as a rotation of angle π around the vertical axis {0} 2 × R.) A symmetric braid diagram with 2n strands is a planar braid diagram with 2n strands that is invariant under this reflection. To make things precise, we assume that the upper (resp. lower) endpoints of each symmetric braid diagram with 2n strands are the points {-n . . . , -1, 1, . . . , n} × {1} (resp. the points {-n . . . , -1, 1, . . . , n} × {0}). We consider the symmetric braid diagrams with 2n strands up to planar isotopy preserving the crossings, and up to the classical Reidemeister II-III moves, as depicted in Figure 5. We stress the fact that neither the isotopies nor the Reidemeister moves have to be preserved under the reflection.

Figure 5. Classical Reidemeister II-III moves

The equivalence classes of symmetric braid diagrams with 2n strands form a group, which tom Dieck [tD94] proved to be isomorphic to the generalized braid group B Bn of type B n . In this isomorphism, the generator s 0 is represented by the symmetric braid diagram with one positive crossing, as in Figure 6, and each remaining generator s 1 , . . . , s n-1 by the symmetric braid diagram with two symmetric positive crossings, as in Figure 7. From now on, we will identify each generator s i of B Bn with the corresponding symmetric braid. In terms of symmetric braid diagrams, we see that Relation (2.1) holds because the symmetric braid diagrams corresponding to each term of the relation are isotopic. Similarly, the equality in Relation (2.2) can be proven diagramatically using Reidemeister III move. As for Relation (2.3), it can Forgetting the symmetry condition yields an embedding of B Bn into the group B 2n of usual braids with 2n strands. Before we give a precise formula for this embedding, let us shift the indices of the 2n -1 generators σ i of B 2n by -n; in this way the indexing set for the generators becomes the set {-n + 1, . . . , n -1}. The re-indexed generators satisfy the same braid relations (1.1) and (1.2). We then define an embedding of B Bn into the group B 2n by sending s 0 to σ 0 , and each remaining s i to σ -i σ i = σ i σ -i (i = 1, . . . , n -1).

1 2 n -1 -2 -n
1 i-1 i i+1 i+2 n -1 -i+1 -i -i-1 -i-2 -n

Symmetric virtual braids

We now define a group VB Bn , which will be our generalized virtual braid group of type B n .

Definition 3.1. The group VB Bn has the following presentation: it is generated by 2n generators s 0 , s 1 , . . . , s n-1 and z 0 , z 1 , . . . , z n-1 , where s 0 , s 1 , . . ., s n-1 satisfy Relations (2.1), (2.2), (2.3), z 1 , . . . , z n-1 satisfy the relations

z i z j = z j z i , if |i -j| > 1,
(3.1)

z i z i+1 z i = z i+1 z i z i+1 , if 1 ≤ i ≤ n -2, (3.2) z 0 z 1 z 0 z 1 = z 1 z 0 z 1 z 0 , (3.3) z 2 i = 1, if 0 ≤ i ≤ n -1, ( 3
.4) and the following "mixed relations" are satisfied

s i z j = z j s i , if |i -j| > 1,
(3.5)

s i z i+1 z i = z i+1 z i s i+1 , if 1 ≤ i ≤ n -2, (3.6) s 0 z 1 z 0 z 1 = z 1 z 0 z 1 s 0 , (3.7) z 0 s 1 z 0 z 1 = z 1 z 0 s 1 z 0 , (3.8) s 0 z 1 s 0 z 1 = z 1 s 0 z 1 s 0 . (3.9)
By analogy with tom Dieck's graphical description, we want to represent elements of VB Bn by symmetric virtual braid diagrams with 2n strands. These are planar virtual braid diagrams with 2n strands, as defined in Section 1.1, that are symmetric under the reflection in the vertical line {0} × R. The reflection is supposed to preserve the virtual crossings as well as the positive (resp. the negative) crossings. We consider the symmetric virtual braid diagrams with 2n strands up to planar isotopy preserving the crossings, up to the classical Reidemeister II-III moves of Figure 5, up to the virtual Reidemeister II-III moves of Figure 3, and up to the mixed Reidemeister move of Figure 4.

We represent the generators s 0 , s 1 , . . . , s n-1 by the symmetric braid diagrams of Section 2. The generator z 0 is sent to the symmetric virtual braid diagram with a single virtual crossing, as in Figure 9. Each remaining generator z 1 , . . . , z n-1 is sent to a symmetric virtual braid diagram with two symmetric virtual crossings as in Figure 10.

1 2 n -1 -2 -n Figure 9. The symmetric virtual braid diagram z 0 1 i-1 i i+1 i+2 n -1 -i+1 -i -i-1 -i-2 -n Figure 10. The symmetric virtual braid diagram z i (i > 0)
Proposition 3.2. Considering symmetric braid diagrams as elements of VB 2n yields a well-defined group homomorphism j :

VB Bn → VB 2n defined by j(s 0 ) = σ 0 , j(z 0 ) = ζ 0 , (3.10) and j(s i ) = σ -i σ i = σ i σ -i , j(z i ) = ζ -i ζ i = ζ i ζ -i (3.11) for 1 ≤ i ≤ n -1.
Here again as in Section 2, we have shifted the index of the generators of the virtual braid group VB 2n by -n.

This shows that the relations defining the virtual braid group VB Bn of type B are adequate.

Remark 3.3. We do not claim that j : VB Bn → VB 2n is a monomorphism but we conjecture that it is. Should this hold, then we could claim that the defining relations are sufficient to define VB Bn as a subgroup of VB 2n .

Proof. Using the relations defining VB 2n (see Section 1.1), we now check that j(s 0 ), j(s 1 ), . . . , j(s n-1 ), j(z 0 ), j(z 1 ), . . ., j(z n-1 ) satisfy the defining relations of VB Bn .

Relations (2.1)-(2.3) and (3.1)-(3.6) are obviously satisfied.

A graphical proof of Relation (3.7) is given in Figure 11, where we use the virtual Reidemeister III move of Figure 3 We similarly obtain Relation (3.8) using four mixed Reidemeister moves, as in Figure 12.

σ i ζ i = ζ i σ i and ζ i σ i+1 σ i = σ i+1 σ i ζ i+1 . (b) In VB Bn we have z 0 s 1 z 0 s 1 = s 1 z 0 s 1 z 0 .
Proof. (a) For each relation we will use Manturov's invariant f (see Section 1.2) to show that both sides of the relation do not have the same invariant. This will imply the desired inequality in VB n . For the leftmost relation, we have

f (ζ i σ i ) (a i ) = t -1 a i t and f (σ i ζ i ) (a i ) = ta -1 i+1 a i a i+1 t -1 , which are different.
For the righmost relation, we have

f (ζ i σ i+1 σ i ) :            t → t, a i → a i+2 , a i+1 → a -1 i+2 ta i+1 t -1 a i+2 , a i+2 → a -1 i+2 t -1 a i ta i+2 , a j → a j if j = i, i + 1, i + 2 and f (σ i+1 σ i ζ i+1 ) :            t → t, a i → a i+2 , a i+1 → ta -1 i+2 a i+1 a i+2 t -1 , a i+2 → t -1 a -1 i+2 a i a i+2 t, a j → a j if j = i, i + 1, i + 2. So f (ζ i σ i+1 σ i ) and f (σ i+1 σ i ζ i+1 ) are not equal in Aut (F n+1 ).
(b) We want to compare the two elements z 0 s 1 z 0 s 1 and s 1 z 0 s 1 z 0 of VB Bn . Their image in VB 2n under the homomorphism j are respectively the elements

ζ 0 σ -1 σ 1 ζ 0 σ -1 σ 1 and σ -1 σ 1 ζ 0 σ -1 σ 1 ζ 0 .
To them, we apply Manturov's invariant, the group homomorphism f : VB 2n → Aut(F 2n+1 ), where F 2n+1 is the free group on the generators a -n+1 , . . . , a -1 , a 0 , a 1 , . . . , a n , t, which is given by the same formulas as in Section 1.2.

A simple computation shows that

f (ζ 0 σ -1 σ 1 ζ 0 σ -1 σ 1 )(a 2 ) = a -1 2 t -1 a -1 0 ta 2 a -1 1 t -1 a -1 ta 1 a -1 2 t -1 a 0 ta 2 and f (σ -1 σ 1 ζ 0 σ -1 σ 1 ζ 0 )(a 2 ) = a -1 2 a -1 1 a 2 t -1 a -1 0 a -1 a 0 ta -1 2 a 1 a 2 , which are different.
Remarks 3.5.

• The inequalities of Proposition 3.4 (a) imply that, in VB Bn we have

s i z i = z i s i if 0 ≤ i ≤ n -1 and z i s i+1 s i = s i+1 s i z i+1 if 1 ≤ i ≤ n -1.
• The second inequality of Proposition 3.4 (a) means that the mixed Reidemeister move of Figure 14, involving one virtual crossing and two positive crossings, is not allowed. Specializing at t = 1 one observes that

f (ζ i σ i+1 σ i ) and f (σ i+1 σ i ζ i+1 ) become equal in Aut (F n ). Adding the re- lation ζ i σ i+1 σ i = σ i+1 σ i ζ i+1
to the presentation of VB n , one precisely obtain a presentation of the group of welded braids with n strands defined in [START_REF] Fenn | The braid-permutation group[END_REF]. Generalities. We consider some bimodules which have been introduced by Soergel [START_REF] Soergel | The combinatorics of Harish-Chandra bimodules[END_REF], [START_REF] Soergel | Gradings on representation categories[END_REF], [START_REF] Soergel | Kazhdan-Lusztig-Polynome und unzerlegbare Bimoduln über Polynomringen[END_REF] in the general context of Coxeter groups.

Let (W, S) be a Coxeter system with generating set S = {s 0 , . . . , s n-1 } of cardinality n. As usual, we denote the order of st, where s, t ∈ S, by m(s, t) and we set s i , s j = -cos π m(s i , s j ) ∈ R .

Let R = R[X 0 , . . . , X n-1 ] be a real polynomial algebra in n indeterminates X 0 , . . . , X n-1 .

For i = 0, 1, . . . , n -1 we define an algebra automorphism α i of R by

α i (X j ) = X j -2 s i , s j X i
for all j = 0, 1, . . . , n -1. It is easy to check that α i is an involution and that there is a unique action of W on R by algebra automorphisms such that

s i (P ) = α i (P )
for all i = 0, 1, . . . , n -1 and P ∈ R.

For any w ∈ W we consider the following objects: (a) the subalgebra R w of elements of R fixed by w;

(b) the R-bimodule B w = R ⊗ R w R;
(c) the R-bimodule R w , which coincides with R as a left R-module whereas a ∈ R acts on R w on the right by multiplication by w(a).

Remarks 4.1.

• To avoid confusion, let us point out that:

-among these bimodules, the only ones which are commonly called Soergel bimodules are the bimodules B s for s ∈ S, -for an arbitrary w ∈ W, Soergel uses the notation B w for different bimodules than us (some indecomposables) that we will not consider here.

• The bimodules R w already appeared in Soergel's work and they turned out to be important in the study of Soergel bimodules.

We introduce a grading on the algebra R by setting deg(X k ) = 2 for all k = 0, . . . , n -1. It induces a grading on the bimodules we consider. We will indicate a shift of the grading by curly brackets: if M = i∈Z M i is a Z-graded bimodule and p an integer, then the Z-graded bimodule M {p} is defined by M {p} i = M i-p for all i ∈ Z.

In the sequel we will systematically identify any bimodule

R w ⊗ R B w ′ (resp. B w ⊗ R B w ′ ) with R w ⊗ R w ′ R (resp. R ⊗ R w R ⊗ R w ′ R) since R w ⊗ R B w ′ = R w ⊗ R R ⊗ R w ′ R ∼ = R w ⊗ R w ′ R and B w ⊗ R B w ′ = R ⊗ R w R ⊗ R R ⊗ R w ′ R ∼ = R ⊗ R w R ⊗ R w ′ R.
The following lemma is straightforward. It will be used repeatedly in Section 5.2. Lemma 4.2. For all w, w ′ ∈ W, there are isomorphisms of R-bimodules

R w ⊗ R R w ′ ∼ = -→ R ww ′
R w ⊗ R B w ′ ∼ = -→ B ww ′ w -1 ⊗ R R w sending a ⊗ b to a ⊗ w(b) (a, b ∈ R).

4.2.

The type B n . When the Coxeter system (W, S) is of type B n , then

s i , s i = -cos (π) = 1 if i = 0, . . . , n -1, s i , s j = -cos (π/2) = 0 if |i -j| > 1, s i , s i+1 = -cos (π/3) = -1/2 if i = 1, . . . , n -2, s 0 , s 1 = -cos (π/4) = - √ 2/2.
See [START_REF] Humphreys | Reflection groups and Coxeter groups, volume 29 of Cambridge Studies in Advanced Mathematics[END_REF]. It follows that the automorphisms α 0 , α 1 , . . . , α n-1 act on the polynomial algebra R = R[X 0 , . . . , X n-1 ] by the formulas

α 0 :    X 0 → -X 0 , X 1 → X 1 + √ 2 X 0 , X i → X i if i > 1, α 1 :        X 0 → X 0 + √ 2 X 1 , X 1 → -X 1 , X 2 → X 1 + X 2 , X i → X i if i > 2. For 1 < j < n, we have α j :    X j → -X j , X i → X i + X j if i = j -1, j + 1, X i → X i if i = j -1, j, j + 1.
For future use, we shall need an explicit set of algebraically independent homogeneous generators for certain subalgebras R w of invariants. Using Chevalley's theorem (see [START_REF] Humphreys | Reflection groups and Coxeter groups, volume 29 of Cambridge Studies in Advanced Mathematics[END_REF]Chap. 3] or [START_REF] Tauvel | Lie algebras and algebraic groups[END_REF]Chap. 31] for more details), one can verify that

R s 0 = R[ √ 2X 0 + 2X 1 , X 2 0 , X 2 , . . . , X n-1 ] , R s 1 s 0 s 1 = R[X 0 , X 1 + X 2 , X 1 ( √ 2X 0 + X 1 ), X 3 , . . . , X n-1 ] , R s 1 = R[2X 0 + √ 2X 1 , X 1 + 2X 2 , X 2 1 , X 3 , . . . , X n-1 ] and R s 0 s 1 s 0 = R[X 0 + √ 2X 2 , X 1 , X 0 (X 0 + √ 2X 1 ), X 3 , . . . , X n-1 ] .
5. Weak categorification of the virtual braid group of type B n 5.1. Rouquier's weak categorification of generalized braid groups.

We explain Rouquier's construction following [START_REF] Rouquier | Categorification of sl2 and braid groups[END_REF]. It works for any generalized braid group B W associated to a finite Coxeter group W.

To each braid generator s i ∈ B W we assign the cochain complex F (s i ) of graded R-bimodules

F (s i ) : 0 -→ R{2} rb i --→ B s i -→ 0 , (5.1)
where B s i sits in cohomological degree 0. The degree-preserving R-bimodule morphism rb i sends any a ∈ R to aX i ⊗ 1 + a ⊗ X i ∈ B s i . To the inverse s -1 i of s i we assign the cochain complex F (s -1 i ) of graded R-bimodules

F (s -1 i ) : 0 -→ B s i {-2} br i --→ R{-2} -→ 0 , (5.2)
where B s i {-2} sits in cohomological degree 0 and the degree-preserving Rbimodule morphism br i is given by multiplication; in other words, it sends a ⊗ b ∈ B s i to ab ∈ R.

To the unit element 1 ∈ B W we assign the trivial complex of graded R-bimodules

F (1) : 0 -→ R -→ 0 , (5.3)
where R sits in cohomological degree 0. The complex F (1) is obviously a unit for the tensor product of complexes. Finally to any word w = s ε 1 i 1 . . . s ε k i k where ε 1 , . . . , ε k = ±1, we assign the complex of graded R-bimodules F (w) = F (s

ε 1 i 1 ) ⊗ R . . . ⊗ R F (s ε k i k ).
In this context, Rouquier established that if w and w ′ are words representing the same element of B W , then F (w) and F (w ′ ) are homotopy equivalent complexes of R-bimodules. This statement is what we mean by Rouquier's weak categorification of generalized braid groups and what we want, for the type B case, to generalize to virtual braids. But in fact, Rouquier enhanced the above result in [START_REF] Rouquier | Categorification of sl2 and braid groups[END_REF]. He also conjectured his categorification to be faithfull while it will not be true for our generalization to virtual braids; see Remarks 5.2. 5.2. Weak categorification of VB Bn . Our aim is to extend the construction of Rouquier detailed in the former section to the virtual braid group VB Bn of type B n defined in Section 3.

The cochain complexes associated to the generators s ±1 i of VB Bn are the ones defined by (5.1) and (5.2).

To the generator z i we assign the complex of graded R-bimodules concentrated in degree 0

F (z i ) : 0 -→ R s i -→ 0 .
(5.4)

To any word w in the generators s ±1 i and z i of VB Bn we assign the tensor product over R of the complexes associated to the generators involved in the expression of w.

We now state our main result. Relations (3.1)-(3.4) only involve the virtual generators z i . In view of the simple form of the complex F (z i ), the isomorphims of the corresponding complexes directly follow from the first isomorphism in Lemma 4.2.

Similarly, the isomorphism of complexes associated to Relations (3.5) and (3.6) can be constructed as performed in [START_REF] Thiel | Categorification of the virtual braid groups[END_REF] in the case of the categorification of the virtual braid group of type A.

We are left with the mixed relations (3.7)-(3.9). Let us first deal with Relation (3.7). We have to prove that the complexes F (s 0 z 1 z 0 z 1 ) and F (z 1 z 0 z 1 s 0 ) are isomorphic. A simple computation shows that the complex F (z 1 z 0 z 1 s 0 ) is isomorphic to the following:

0 -→ R s 1 s 0 s 1 {2} d -→ R s 1 s 0 s 1 ⊗ R B s 0 -→ 0 , where d(a) = a (α 1 α 0 α 1 (X 0 ) ⊗ 1 + 1 ⊗ X 0 ) = a (X 0 ⊗ 1 + 1 ⊗ X 0 ) for all a ∈ R. The complex F (s 0 z 1 z 0 z 1 ) is isomorphic to 0 -→ R s 1 s 0 s 1 {2} d ′ -→ B s 0 ⊗ R R s 1 s 0 s 1 -→ 0 , where d ′ (a) = a (X 0 ⊗ 1 + 1 ⊗ X 0 )
for all a ∈ R. Lemma 4.2 provides an isomorphism

µ : R s 1 s 0 s 1 ⊗ R B s 0 ∼ = -→ B s 1 s 0 s 1 s 0 s 1 s 0 s 1 ⊗ R R s 1 s 0 s 1 ;
it is given by µ(a⊗b) = a⊗α 1 α 0 α 1 (b), where a, b ∈ R. Now by Relation (2.3) the latter bimodule is equal to B s 0 ⊗ R R s 1 s 0 s 1 . This allows us to build the Now, √ 2X 0 + 2X 1 + 2X 2 belongs to R s 0 ∩ R s 1 s 0 s 1 , so we get the expected equality

ϕ 1 ⊗ ( √ 2X 0 + 2X 1 ) ⊗ 1 = ( √ 2X 0 + 2X 1 ) ⊗ 1 ⊗ 1 .
For X 1 +X 2 , Equality (5.6) follows directly from the definition of ϕ; we have

ϕ 1 ⊗ (X 1 + X 2 ) ⊗ 1 = -X 2 ⊗ 1 ⊗ 1 + 1 ⊗ 1 ⊗ (X 1 + X 2 ) + X 2 ⊗ 1 ⊗ 1 = 1 ⊗ 1 ⊗ (X 1 + X 2 ) .
The last element p ∈ R s 1 s 0 s 1 for which we have to check Equality (5.6) is p = X 1 √ 2X 0 + X 1 . By definition,

ϕ 1 ⊗ X 1 ( √ 2X 0 + X 1 ) ⊗ 1 = -X 2 ⊗ 1 ⊗ √ 2X 0 +1 ⊗ 1 ⊗ √ 2X 0 (X 1 + X 2 ) + X 2 2 ⊗ 1 ⊗ 1 -2X 2 ⊗ 1 ⊗ (X 1 + X 2 ) +1 ⊗ 1 ⊗ (X 1 + X 2 ) 2 = 1 ⊗ 1 ⊗ √ 2X 0 (X 1 + X 2 ) + (X 1 + X 2 ) 2 -X 2 ⊗ 1 ⊗ ( √ 2X 0 + 2X 1 + 2X 2 ) +X 2 2 ⊗ 1 ⊗ 1 .
Now remark that X 2 2 can be written as follows:

X 2 2 = X 1 ( √ 2X 0 + X 1 ) -(X 1 + X 2 ) 2 - √ 2X 0 (X 1 + X 2 ) + ( √ 2X 0 + 2X 1 + 2X 2 )X 2 .
One can check that both

X 1 ( √ 2X 0 + X 1 ) -(X 1 + X 2 ) 2 - √ 2X 0 (X 1 + X 2 )
and √ 2X 0 + 2X 1 + 2X 2 belong to R s 0 ∩ R s 1 s 0 s 1 . So replacing X 2 2 by its latter expression in the expression of ϕ 1 ⊗ X 1 ( √ 2X 0 + X 1 ) ⊗ 1 , we obtain what we expected, namely

ϕ 1 ⊗ X 1 ( √ 2X 0 + X 1 ) ⊗ 1 = 1 ⊗ 1 ⊗ X 1 ( √ 2X 0 + X 1 ) .
The existence of the bimodule isomorphism ϕ leads to an isomorphism between the complexes F (s 0 z 1 s 0 z 1 ) and F (z 1 s 0 z 1 s 0 ). In fact, F (s 0 z 1 s 0 z 1 ) is isomorphic to the complex B s 0 {2} proof of theorem 5.1, this enable us to construct an isomorphism between the complexes F (z 0 s 1 z 0 s 1 ) and F (s 1 z 0 s 1 z 0 ). • As in the type A case (treated in [START_REF] Thiel | Categorification of the virtual braid groups[END_REF]), the complexes F (s i z i ) and F (z i s i ) are isomorphic, which contrasts with the fact that the relation s i z i = z i s i does not hold in VB Bn .

• Although we do not manage to state here that j : VB Bn → VB 2n is injective, it would be interesting to know at least if, for any a ∈ ker(j), the complexes F (a) and F (1) are homotopy equivalent.

Figure 3 .

 3 Figure 3. Virtual Reidemeister moves

Figure 6 .

 6 Figure 6. The symmetric braid diagram s 0

Figure 7 .

 7 Figure 7. The symmetric braid diagram s i (i > 0)

  twice and the mixed Reidemeister move of Figure4twice.

Figure 11 .

 11 Figure 11. A graphical proof of Relation (3.7)

Figure 12 .

 12 Figure 12. A graphical proof of Relation (3.8)

Figure 13 .

 13 Figure13. A graphical proof of Relation (3.9)

Figure 14 .

 14 Figure 14. A forbidden Reidemeister move

  sending a ⊗ b to aw(b), and

  both generators σ i and ζ i . Elements of VB n can be represented by virtual braid diagrams with n strands. Such a diagram is a planar braid diagram with virtual crossings in addition to the usual positive and negative crossings.

	.7)
	Relations (1.1) and (1.2) are called braid relations and Relations (1.3)-(1.5)
	permutation relations. Relations (1.6) and (1.7) are called mixed relations
	because they involve

  Theorem 5.1. If w and w ′ are words representing the same element of VB Bn , then F (w) and F (w ′ ) are homotopy equivalent complexes of R-bimodules.Proof. It is enough to check that there are homotopy equivalences between the complexes associated to the pair of words appearing in each defining relation of VB Bn .The checking of this for Relations (2.1), (2.2), (2.3) is a consequence of Rouquier's work for generalized braid groups.

Acknowledgements

I thank my thesis advisor C. Kassel for his guidance and his help with writing up this paper.

References

following isomorphism of complexes between F (z 1 z 0 z 1 s 0 ) and F (s 0 z 1 z 0 z 1 ):

The vertical maps (and similarly their inverse) commute with the differentials. Indeed, for a ∈ R s 1 s 0 s 1 ,

Similar arguments allow us to construct an isomorphism of complexes between F (z 0 s 1 z 0 z 1 ) and F (z 1 z 0 s 1 z 0 ), which proves that (3.8) is satisfied on the level of complexes.

In order to handle Relation (3.9), the first step is to find an isomorphism between

Then using the generating set of R w for w ∈ {s 0 , s 1 s 0 s 1 } exhibited in Section 4.2, we can make an isomorphism of R-bimodules

The map ϕ is clearly surjective since any element of R can be written as a sum of products of elements of R s 0 and R s 1 s 0 s 0 . Let us prove that this isomorphism of R-bimodules is well-defined. We have to check that

(5.6)) for p equal to the generating elements of R s 0 (resp. of R s 1 s 0 s 1 ).

For p = X 0 (resp. p = X 2 ) Equality (5.6) (resp. Equality (5.5)) follows directly from the definition of ϕ. For p = X i with i > 2, Equalities (5.5) and (5.6) follow from the fact that

It remains to deal with the elements √ 2X 0 + 2X 1 and X 2 0 of R s 0 , and with the elements

whose differentials are obtained by composing the ones of F (s 0 z 1 s 0 z 1 ) with the isomorphism of Lemma 4.2. More precisely,

for all a, b, c ∈ R. Similarly, the complex F (z 1 s 0 z 1 s 0 ) is isomorphic to

whose differentials are obtained by composing the ones of F (z 1 s 0 z 1 s 0 ) with the isomorphism of Lemma 4.2, namely

for all a, b, c ∈ R. We define an isomorphism between these two complexes using the identity map (up to a sign) for all factors outside of cohomological degree 0 and using ϕ in degree 0. This can be summarized by the following

This morphism of complexes is well-defined: the identities commute with the differentials and for all a, b ∈ R,

Remarks 5.2.

• It is important to work in the homotopy category rather than in the derived category since the complexes F (s i ) and F (s -1 i ) are both quasi-isomorphic to F (z i ) while they are not homotopy equivalent to it.

• We observed in Proposition 3.4 that quite unexpectedly the relation z 0 s 1 z 0 s 1 = s 1 z 0 s 1 z 0 does not hold in VB Bn . Nevertheless the two complexes F (z 0 s 1 z 0 s 1 ) and F (s 1 z 0 s 1 z 0 ) are isomorphic.

The existence of an isomorphism between these complexes follows from the isomorphim of R-bimodules ψ : B s 1 ⊗ R B s 0 s 1 s 0 → B s 0 s 1 s 0 ⊗ R B s 1 .

The isomorphism ψ can be expressed in a similar way as the isomorphism ϕ : B s 0 ⊗ B s 1 s 0 s 1 → B s 1 s 0 s 1 ⊗ B s 0 . Then, just as in the last part of the


Eric Mathieu

Jean-Philippe Farrugia

TECHNICAL REPORT ON THE DEVELOPMENT OF VIDEO GLASSES FOR EYE PROTECTION

Keywords: HMD, laser safety, eye safety, Augmented Reality, image remap, video glasses, Personal Protective Equipment (PPE), video see-through

There is a lack of eye protection for high energy lasers or those having too many wavelengths. Therefore, we have developed a video see-through Head Mounted Display (HMD) to have an indirect vision. The use of video display allows us to add Augmented Reality and help the user in his task. The HMD is composed of off-the-shelf elements. The displays are general market glasses (Sony, Oculus) whereas cameras are industrial highly configurable cameras (IDS -ueye). This paper is a technical description of the device assembly, programming and testing. We highlight the current constraints such as time lag, picture resolution, field of view, weight and autonomy.

INTRODUCTION

When doing an eye-hazardous activity such as soldering or using high energy lasers, workers currently have the possibility to wear filters glasses. Two types of filters exist: Intensity filters reduce the whole light spectrum to an acceptable level, and wavelength selective filters let part of the visible spectrum reach the eyes (and stop a narrow bandwidth laser for example). The filters cannot protect from (i) too high energy lights (case of high energy lasers) and (ii) too many visible wavelengths at high energy (multiple lasers or white lasers). Consequently, they result in color blindness.

To avoid these inconvenient, we developed a new closed protective helmet with indirect viewing through stereo cameras. This has multiple advantages: (i) full protection as the light cannot reach the eyes, (ii) possibility to add additional information via augmented reality in the display to help the user, (iii) full color vision.

To preserve 3D perception, we used a stereo system with 2 cameras and 2 displays in front of each eye. This project has been initiated in 2012 in the CETHIL lab which is using high energy lasers. Since 2013, the LIRIS lab is also involved to bring computer vision and computer graphics expertise into the project. Until 2015, the project's goal was to demonstrate the relevance of the concept and its feasibility.

This technical report will be structured as followed. After this brief introduction, the hardware and software structure of the system will be presented through sections 3 and 4. Section 5 will present user tests in working conditions, with various configurations. Section 6 will conclude with future developments and perspectives.

HARDWARE CHOICE

In this section, we will present the hardware selection process. Knowing the constrains, we selected the best suited hardware available on the public market (military market was not an option).

CONSTRAINS

We identified the following constrains for the hardware selection:

• High refreshing rate for visual comfort (at least 30 fps for each eye). By experience, 50 fps would be even more comfortable. Advanced virtual reality systems such as the Oculus HMD recommend 75fps

[1]. • Minimum lag between picture recording and display. Less than 20ms is recommended by the oculus HMD [2], although, it becomes sensible to the user (and give sickness) above 48ms [START_REF] Bailey | Latency requirements for head-worn display S/EVS applications[END_REF].

• Highly sensitive cameras for dark environments.

• Good angle resolution. To meet human vision, we should display with a resolution of 1 arcminute (approximately 0.02°) [START_REF] Yanoff | Opthalmogy 3rd Edition[END_REF].

• Large field of view, especially when moving, in order not to hit objects around the user. Ideal field of view would be 175° (as the human eyes [START_REF] Wandell | Foundations of Vision[END_REF]).

• Light weight. At least for the part that will be worn on the head.

HARDWARE CHOICE

For ease of development and maintenance, the software part of the project runs on a laptop with USB connected cameras and different off-the-shelf video displays. The code itself could run on a smartphone or tablet, but these devices actually lack a high speed USB connection for the cameras.

For the first prototype, we used a VUZIX Wrap 920 AR mask already equipped with 2 cameras (640*480px) and 2 displays (640*480px, horizontal FOV of 30°). This prototype has demonstrated the potential of the technology but had the following strong limitations for our usage:

-Cameras with rolling shutter (webcams): not compatible with pulsed lasers -Low resolution -Narrow Field of View

CAMERAS

Our next choice was driven by the need of high speed port (USB3) for high resolution cameras with high frame rate (>50fps). We chose the IDS UI-3251LE

COMPUTATION DEVICE (PC)

For real time computation, most of the image processing is performed on the graphic processor (GPU). Part of the code is using the CUDA language, which is dedicated to NVIDIA processors. Thus, the choice of the PC was mainly driven by: having 2 distinct USB3 ports, a NVIDIA GPU, being compact and light for carrying. We chose a Gigabyte P34G-3 PC equipped with an Intel i7 4700HQ quad-core and an Nvidia gtx 765m GPU. This laptop is well equipped in a small size (14 inches and less than 2kg). The development has been done in a Microsoft Windows environment.

HARDWARE ASSEMBLY

The 2 cameras have been mounted on a light steel plate which permits to adjust their yaw and pitch angles. The cameras were mounted with a lateral distance of 65mm, which is the commonly agreed average interpupillary distance. At a first step, and for easy integration, we put the cameras in front of the helmet, but it is suggested that the use of a mirror is preferable [START_REF] Park | Hand-Eye Coordination Using a Video-see-through Augmented Reality System[END_REF] and can improve the user motion speed. With the mirror, the cameras could be places at the same distance to the objects as the eyes.

The cameras were slightly tilted to converge at a distance of 4 meters. This gives more comfort for the user, but generates keystone distortion [START_REF] Woods | Image distortions in stereoscopic video systems[END_REF]. Nevertheless, we couldn't feel this keystone distortion.

We For ease of movement, the Sony headset was connected to a portable battery. Using a laptop, we may get rid of of power cables. The laptop and the portable battery may then be taken and used in a backpack.

SOFTWARE DEVELOPMENTS

The software to control the camera and the display to the user has been developed in C++ in a Microsoft Windows environment, using OpenCV library for image processing. C++ is the core language of IDS cameras SDK, Occulus SDK and OpenCV. It is a sufficiently low level language to have a good control of the memory management to optimize the realtime processing.

IMAGE PROCESSING

The pictures captured by the camera are not directly sent to the display. Our goal is to facilitate the manipulation of laser hardware in a complex environment, so we need to enhance visual perception of the environment. We chose to simulate a wider field of view by keeping undistorted the central vision and compressing the side vision. Therefore, the pictures recorded from the cameras needed the following treatment:

• Crop (to adjust the FOV of the cameras to the display • Center area with compressed sides (for certain tests, to add compressed information on the side)

• Distortion correction (not applied currently), to correct optics aberrations All these operations can be computed in a single pass by a remap function (available in the Open-CV library). We used the GPU version of the remap function (using Cuda code) in order to speed up the process.

As shown on figure 15, the compression of the sides of the picture is exponential. The more on the side, the more the picture is compressed.

PROFILES AND SETTINGS

A User Interface (UI) has been developed to control the camera basic parameters such as exposure time and gain. Through this simple UI, the user can also control the software and switch between display profiles or set new profiles. This is especially useful during the development process to select the best profiles (zoom or not, compressed sides or not, etc.)

The user can navigate in the UI using a presentation remote controller with 2 buttons. The UI is generated directly on the GPU, using OpenGL.

The different display profiles are described in an xml file which is read at the program initialization. The user can easily create a new xml profile if necessary.

HIGH DYNAMIC RANGE (HDR)

To insure optimal visibility in certain applications, such as soldering, we tried to implement HDR visualisation.

The HDR process builds an high dynamic range image using 2 low dynamic range images taken with different exposure times. The goal is to get a clear vision of low light areas as well as very bright areas. This could be very valuable for soldering work for instance. The user could have a clear view of the soldering arc as well as the environment.

Our IDS cameras can work in a 2 sequential exposure mode with 2 different exposure times. The main difficulty was to make this HDR in real time.

Our tests have been performed using OpenCV/Cuda library. Unfortunately, the best frame rate we could get was 10 fps, but there is room for optimization. This frame rate is far too slow to allow practical usage. Some teams have developed HDR computation on FPGA board for real-time processing [START_REF] Mann | REALTIME HDR (HIGH DYNAMIC RANGE) VIDEO FOR EYETAP WEARABLE COMPUTERS[END_REF], this could be a simple and robust solution to this problem.

AR

One major advantage of our video-see-through glasses is the possibility to add Augmented Reality. The first applications we designed were for laser use. We planned to add simple figures to the scene to show invisible beams and highlight hazardous areas. We also designed the AR capability to display text messages. This text could be used to display and make real-time adjustments, like laser power for instance. For each frame, the augmented reality process requires 2 steps: 1. Find the cameras position in space (camera registration) 2. Compute the position of the shapes in each view, corresponding to each eye and draw the virtual shapes in the picture For the first step, we used markers recognition. We could use markers because we want to use our glasses in a known environment (a lab) with elements always at the same position.

For the markers detection and spatial position computation, we used the ARToolKitPlus library [START_REF] Wagner | ARToolKitPlus for Pose Tracking on Mobile Devices[END_REF]. The computation is performed on the CPU and takes 7ms.

Step 2 is totally performed on the GPU with OpenGL (freeGlut for execution loops, GLU for perspective definition). It takes 1.5 ms to compute with 5 shapes to draw (see figure above).

PROCESSING TIMES

The measurement of the total lag (between reality and display) has been done with the following procedure:

• We displayed a timer on a screen • The cameras were pointing on the timer • The camera display (after whole treatment) was displayed in the same screen, on the side of the timer • We take a screenshot

The difference between the time displayed by the timer and the camera display is the total lag. The maximum error is directly linked to the refreshing rate of the screen: 60hz = 17ms. The measurement of each step of the program was done by using the std::clock function.

Here is a detail of the whole processing time between the image acquisition to its display.

STROBOSCOPIC ISSUE WITH PULSED LASERS

Because high energy lasers are generally pulsed lasers, we face stroboscopic problems. With rolling shutter cameras, we had rolling overexposed areas (see above). With global shutter cameras, the flashes can be synchronized to the recorded images, or not. Generally, we see an alternate flashing.

We thought of 2 ways to handle this problem:

1. Replace an overexposed image by the previews image 2. Synchronize, or desynchronize the cameras using an external trigger Solution 1 can be easily tested, but the laser presence on the picture was not so obvious to determine as it is sometimes only a tiny part of the picture which is over-exposed. The best solution was to check the average exposure of one color canal only, because the laser has a stronger impact in one color only (green in our case). Because this development has been done prior to having an operational second prototype, we have not tested this solution in a real case yet.

Solution 2 is possible with our IDS cameras, which may be externally triggered. To get the laser frequency, a photodiode could be added on the glasses. We have not developed this system yet.

TESTS IN CONDITION

USER'S FEELING: FIELD OF VIEW, IMAGE RATE

These tests have been difficult to handle because of the lag between reality and display. Without reducing this lag, it is difficult for the user to judge if the feeling is worst or not in different configuration. Nevertheless, with the first prototype, we have conducted real tests with 15 different people. They were divided into 3 groups: A. large field of view (120°) without stereo vision, displayed in the 30° FOV display (thus having the sensation to have everything far from them). B. The second group had a scale 1 display in the whole display area. Thus having no information of what is situated on the sides of the field of view. C. The third group had a scale 1 display in the center with a shrinked display of the wide field of view on the sides.

Each candidate should follow the same procedure:

Step 1: get used to the display during 10 minutes, then:

Step 2: Mobility phase • Walk in a corridor • Enter a room (by opening a closed door)

• Walk in the room between tables, catch an object • Walk back to sit

Step 3: static phase • sit and put washers on a screw All these steps had to be performed in a minimum of time. All collisions to furniture were recorded.

The results are the following: Mobility phase For mobility, it appears that a lateral view is necessary (faster move and less collisions). Group B has worst results than the 2 others. Surprisingly, the group A, which has a non stereo vision and a large field of view in a shallow display can handle pretty well this test.

In the static phase, we have an opposite result: group B is faster to put washers on the screw. The higher display resolution in that case is crucial to make precise movements. It is also clear that the group A has more difficulties for this exercise. The resolution is probably too low for the needed precision.

In conclusion, we confirm that it is necessary to have a wide FOV for mobility and a high resolution (large FOV unnecessary?) for static and precise work.

Figure 1 :

 1 Figure 1: Example of a laser protection mask using filter

Figure 3 :

 3 Figure 3: Augmented Reality application example on a dye laser. Visualization of invisible beam, delimitation of dangerous area, display of laser power during adjustments...

  Figure 2: System Layout

Figure 4 :

 4 Figure 4: the Vuzix Wrap 920 AR glasses Figure 5: the Vuzix camera and displays mounted on a protective mask

Figure 6 :

 6 Figure 6: laser saturation example with rolling shutter cameras

Figure 7 :Figure 8 :

 78 Figure 7: IDS UI3251 camera

Figure 9 :

 9 Figure 9: Oculus DK2 virtual reality mask

Figure 10 :

 10 Figure 10: Cameras mounted on the Sony HMD Figure 11: cameras mounted with protective plastic

  used 3 set of lenses: 1. Large FOV: focal 2.95mm. Horizontal field angle of 101°. Corresponds to the oculus display 2. Medium FOV : focal 4mm. Horizontal field angle of 84°. Used with the Sony HMZ display, with a crop in the picture to fit the 45° FOV of the display. 3. Small FOV: focal 12mm. Horizontal field angle of 33°.

Figure 12 :

 12 Figure 12: FOV with the 2.95mm lens Figure 13: FOV with the 4mm lens Figure 14: FOV with the 12mm lens

Figure 15 :

 15 Figure 15: Pixel displacement between original and display. With lateral compression

Figure 18 :

 18 Figure 18: example of different shapes that can be displayed with our AR tool

Figure 19 :

 19 Figure 19: total lag measurement with screenshot of a timer

  Note: we tested the minimum lag time for different types of cameras (onboard webcam, usb webcam, other IDS camera). It is interesting to see that the lag is always around 60ms (+-17ms)! This looks like an incompressible time for our PC configuration.

	Operation		Time [ms]
	Total time of the software (from image reception to	Around 35
	availability for display). In which:
	Markers detection		7
	Image copy to GPU		Up to 10
	Remap treatment		2
	AR computation and drawing		1.5
	Incompressible time, not linked to our software.	Around 60
	(Camera image reception, through USB3, and display
	to the HMD)
	TOTAL time		100 ms
	The Incompressible times of the cameras were measured using the IDS cockpit software provided by IDS. We
	change the mode of picture acquisition from Device Independent = through CPU to GPU modes (Direct3D and
	OpenGL).
	Acquisition mode	Total lag
		(between reality and display)
	Device Independent (CPU)	60ms
	Direct3D	60ms
	OpenGL	110 ms !!

Table 1 mobility tests : total time for the run

 1

	Minimum time Maximum time Average time
	Group A 35 s	1min 16 s	49,8 s
	Group B 50 s	1 min 20 s	56,4 s
	Group C 42 s	52 s	47,2 s

Table 2 : Number of collisions

 2

	# of collisions	0	1	2	3	4
	Group A	5	0	0	0	0
	Group B	2	1	1	0	1
	Group C	3	2	0	0	0
	Static phase
	Tableau 3

Static phase : number of washers put on the screw

		Minimum	Maximum	Sum of washers
		number	number	put
	Groupe A	1 + 1/3	2 + 2/3	10
	Groupe B	2+ 2/3	4 + 1/3	17 +2/3
	Groupe C	2+2/3	3+2/3	16+2/3
	Conclusions :

AUGMENTED REALITY TESTING

The first usage of Augmented reality is to add information into the display in order to help the user. Our first application is a safety guidance for laser use. We added the following figures:

1. semi-transparent surface over a dye laser, to highlight an hazardous area 2. blue laser beam to show the UV laser beam path. Camera position tracking is done by sticking reference markers into the scene. We put markers over the laser and on the walls. We encountered the following issues:

• Blinking of the drawn figures due to a defective analysis of the markers • Bad matching between the figures and the real object because the camera distortion is not taken into account. On the sides, the drawn figures are not coincident to the real image.

The first tests nevertheless demonstrated that AR is possible and does not slow down the image processing and display in a sensible way (AR treatment takes 8.5ms). It could be a very convenient tool for the user, but it still requires further improvements to be comfortable:

• Improve the markers detection, especially when contrast is weak (dark areas)

• Take into account camera distortion.

This marker hasn't enough contrast and is not detected

The plane doesn't follow the real shape of the laser

CONCLUSION AND PERSPECTIVES

PROMISING TECHNOLOGY

The indirect view through cameras is probably the only solution to have a full eye protection at all energies and for the whole light spectra. The technology furthermore allows Augmented Reality. These virtual elements can be very useful in a professional environment, for example by visualising hazardous areas or hazardous beam which could be invisible.

CONSTRAINTS

The current technology has 4 strong constraints: 1. Processing time (lag) between the reality and the display 2. Cameras and display resolutions 3. Field of view 4. Weight and autonomy

The main constraints are known and could be solved in a near future by the natural technology improvements (HMD and cameras are very rapidly improving) or by the development of a dedicated system with an electronic and mechanical design totally built for this usage.

LAG ISSUE

The time between reality and display is called "lag time". This should be minimized as much as possible. We confirmed that 60ms is already sensible. The lag has a strong influence on the sickness feeling of the user [START_REF] Bailey | Latency requirements for head-worn display S/EVS applications[END_REF].

We determined an incompressible lag of 60ms due to USB port, data transfers and OS management. Adding our computation, the lag was increased to 100ms, which is too high for a comfortable usage of the glasses. This is one improvement to be taken into account in the future.

ANGLE RESOLUTION ISSUE

The Sony, as well as the Oculus HMD, have a lower spatial resolution than human eye. It is difficult to read small characters (such as keyboard for instance). To be more comfortable, the display should have a higher resolution.

Nevertheless, the possibility to switch between a large field of view with low angle resolution (Oculus, 9.6px/degree) to a narrower view with high resolution (Sony, 28.4px/degree) will improve the capabilities of the glasses and make them usable for displacement as well as working.

FIELD OF VIEW

The maximum field of view available on the market is currently the oculus mask (100° FOV) which is still much lower than human vision (175°). Although the human eye doesn't need high resolution on the side since it is only used for navigation.

To date (July 2015), we have not yet tested our prototype with the oculus DK2. The whole image processing is operational, we now need to implement the Oculus SDK.

WEIGHT AND AUTONOMY

Our prototype is based on a laptop PC. The total weight of the system (laptop + HMD controller + HMD glasses + external battery for HMD) is around 10kg. This is definitely a system that couldn't be used in real conditions. The cables between the HMD and the rest of the system are also uncomfortable. The final and operational version of the protection glasses should be all weared on the head. It could be a complete helmet with a total weight around 1kg.

The autonomy should also be able to cover one day of work (8 hours).

NEXT STEPS VALIDATE THE CONFIGURATIONS (FIELD OF VIEW, RESOLUTION, ETC.)

The tests with different people and different configurations should be pursued with the new prototype. For instance, it is still unsure if the addition of lateral compressed information is useful or not when working (static work) with the glasses. Optionally, a small lateral portion of the picture could be compressed to help the user to perceive what lies in the dead angle.

BUILD A MECHANICAL DEVICE TO SWITCH OPTICAL CONFIGURATION

When the optimal configuration will be determined, we will design a mechanical system to switch between the large FOV with low angular resolution to shallow FOV with high angle resolution. We will also add the possibility to use the glasses (or helmet) with laser filters or without any protection. Thus, the user will be able to switch between different shield levels, depending on the hazard and the work to perform.

TRANSFER THE TECHNOLOGY TO A LIGHT WEIGHT DEVICE

To come to a versatile system, we must implement the cameras and image processing on a light system to be included on the helmet. A smartphone-based device would fit most of our constraints. Some smartphones have powerful GPUs such as the Nvidia Tegra. The issue to solve is the communication port between the cameras and the smartphone. The standard MIPI port cannot handle 2 high resolution and high frame rate cameras. Another way to explore would be to design a dedicated electronic board with FPGA technology for instance. It will reduce the lag to a minimum, but raise the question of implementing a complex process like augmented reality on such a system.