Karamoko Diarra
email: karamoko.diarra@univ-rennes1.fr

Frank Loray
email: frank.loray@univ-rennes1.fr

RAMIFIED COVERS AND TAME ISOMONODROMIC SOLUTIONS ON CURVES

Keywords: 1991 Mathematics Subject Classification. 34M55, 34M56, 34M03 Ordinary differential equations, Isomonodromic deformations, Hurwitz spaces

In this paper, we investigate the possibility of constructing isomonodromic deformations by ramified covers. We give new examples and prove a classification result.

Introduction

Let X be a complete curve of genus g over C and D be a reduced divisor on

X: D = [x 1 ] + • • • + [x n
] is equivalent to the data of n distinct points on X. Set N := 3g -3 + n; when N > 0, that we will assume along the paper, then N is the dimension of the deformation space M g,n of the pair (X, D).

Let (E, ∇) be a rank 2 logarithmic connection over X with polar divisor D. In other words, E → X is a rank 2 holomorphic vector bundle and ∇ : E → E ⊗ Ω 1 X (D) a linear meromorphic connection having simple poles at the points of D. By considering the analytic continuation of a local basis of ∇-horizontal sections of E, we inherit a monodromy representation

ρ ∇ : π 1 (X \ D) → GL 2 (C)
(which is well-defined up to conjugacy in GL 2 (C)).

Given a deformation t → (X t , D t ) of the complex structure, there is a unique deformation t → (X t , D t , E t , ∇ t ) up to bundle isomorphism such that the monodromy is constant. For t varrying in the Teichmuller space T g,n , we get the universal isomonodromic deformation (see [START_REF] Heu | Universal isomonodromic deformations of meromorphic rank 2 connections on curves[END_REF]). Considering the moduli space M g,n of quadruples (X, D, E, ∇), isomonodromic deformations define the leaves of a Ndimensional foliation transversal to the natural projection M g,n → M g,n ; (X, D, E, ∇) → (X, D).

The corresponding differential equation is explicitely described in [START_REF] Krichever | Isomonodromy equations on algebraic curves, canonical transformations and Whitham equations. Dedicated to Yuri I. Manin on the occasion of his 65th birthday[END_REF] (via local analytic coordinates on M g,n ) and is known to be polynomial with respect to the algebraic structure of M g,n (it is the non-linear Gauss-Manin connection constructed in [25, section 8]). In the case g = 0, we get the Garnier system (see [START_REF] Okamoto | Isomonodromic deformation and Painlevé equations, and the Garnier system[END_REF]), and for n = 4, the Painlevé VI equation. Solutions (or leaves) are generically transcendental and it is expected that the transcendence increase with N (see [START_REF] Dubrovin | On the reductions and classical solutions of the Schlesinger equations[END_REF]Introduction] for instance). However, there are some tame solutions.

Algebraic solutions of Painlevé VI equation were recently classified in [START_REF] Boalch | Towards a non-linear Schwarz's list[END_REF][START_REF] Lisovyy | Algebraic Solutions of the sixth Painlevé Equation[END_REF]. Some algebraic solutions are constructed in [START_REF] Diarra | Construction et classification de certaines solutions algébriques des systèmes de Garnier[END_REF] for the Garnier case; see the discussion in the introduction of [START_REF] Diarra | Solutions algébriques partielles des équations isomonodromiques sur les courbes de genre 2[END_REF] for higher genus case. Some solutions, called "classical", reduce to solutions of linear differential equations. They are classified in the Painlevé case in [START_REF] Watanabe | Birational canonical transformations and classical solutions of the sixth Painlevé equation[END_REF]. In the Garnier case, such solutions arise by considering deformations of reducible connections (see [START_REF] Okamoto | On particular solutions of the Garnier systems and the hypergeometric functions of several variables[END_REF][START_REF] Mazzocco | The geometry of the classical solutions of the Garnier systems[END_REF]): they can be expressed in terms of Lauricella hypergeometric functions.

There are also "tame solutions" coming from simpler isomonodromy equations (e.g. with lower n or g) . In [START_REF] Mazzocco | The geometry of the classical solutions of the Garnier systems[END_REF], it is proved that, one way of reducing n (when g = 0) is to consider those deformations having scalar local monodromy around some pole. There is another way of reduction, by using ramified covers, and this is what we want to investigate in this note.

Known constructions via ramified covers

Ramified covers of curves have already been used to construct algebraic solutions of the Painlevé VI equation (see [START_REF] Doran | Algebraic and geometric isomonodromic deformations[END_REF][START_REF] Andreev | Transformations RS 2 4 (3) of the ranks ≤ 4 and algebraic solutions of the sixth Painlevé equation[END_REF]) and Garnier systems (see [START_REF] Diarra | Construction et classification de certaines solutions algébriques des systèmes de Garnier[END_REF]). But they have also been used to understand relations between transcendental solutions.

1.1. The most classical case is the quadratic transformation of the Painlevé VI equation (see [START_REF] Kitaev | Quadratic Transformations for the Sixth Painlevé Equation[END_REF][START_REF] Yu | Sixth Painlevé Equation, Universal Elliptic Curve and Mirror of P 2[END_REF][START_REF] Tsuda | Folding transformations of the Painlevé equations[END_REF][START_REF] Mazzocco | Cubic and quartic transformations of the sixth PainlevÃľ equation in terms of Riemann-Hilbert correspondence[END_REF]). We consider a deformation t → (E t , ∇ t ) of a rank 2 connection on P 1 with simple poles at (x 1 , x 2 , x 3 , x 4 ) = (0, 1, t, ∞). At a pole x i , we consider eigenvalues θ 1 i , θ 2 i of the residual matrix and call exponent the difference θ i := θ 1 i -θ 2 i (defined up to a sign). To be concrete, if all θ 1 i + θ 2 i = 0 and the connection is irreducible, then E t is the trivial bundle except for a discrete set of parameters (see [START_REF] Bolibrukh | The Riemann-Hilbert problem[END_REF]) and the connection is just defined by a two-by-two system. If moreover exponents satisfy θ 0 = θ ∞ = 1 2 then after lifting the connection on the two-fold cover P 1

x → P 1 x ; x → x2 we get a connection ( Ẽ0 t , ∇0 t ) having 6 simple poles at x = 0, ±1, ± √ t and ∞ (see figure 1).

Figure 1. Quadratic transformation's cover

Those two poles at ramification points x = 0, ∞ have now integral exponents and therefore scalar local monodromy -I. These singular points are "apparent", i.e. can be erased by a combination of • a rational gauge (i.e. birational bundle) transformation,

• the twist by a rank 1 connection.

This can be done taking into account the deformation, and we get a new deformation t → ( Ẽt , ∇t ) of a rank 2 connection with 4 simple poles x = ±1 and ± √ t on the Riemann sphere P 1

x. This new deformation is clearly isomonodromic if the initial deformation was. Taking into account the exponents, we get a rational two-fold cover

Quad : M 0,4 ( 1 2 , θ 1 , θ t , 1 2 ) 2:
1

-→ M 0,4 (θ 1 , θ 1 , θ t , θ t )
between moduli spaces that conjugates isomonodromic foliations. The map Quad is called quadratic transformation of the Painlevé VI equation.

When exponents satisfy θ

0 = θ 1 = θ ∞ = 1 2
, we can iterate twice the map (after conveniently permuting the poles) and we get the quartic transformation

Quad • Quad : M 0,4 ( 1 2 , 1 2 , θ t , 1 2 ) 4:
1

-→ M 0,4 (θ t , θ t , θ t , θ t ).
Finally, if we consider the Picard parameters

θ 0 = θ 1 = θ t = θ ∞ = 1 2
for Painlevé VI equation, we can iterate arbitrary many times the quadratic tranformation. There is also a cubic transformation in this case (see [START_REF] Mazzocco | Cubic and quartic transformations of the sixth PainlevÃľ equation in terms of Riemann-Hilbert correspondence[END_REF]).

1.3. For Picard parameters (θ 0 , θ 1 , θ t , θ ∞ ) = ( 1 2 , 1 2 , 1 2 , 1
2 ) of Painlevé VI equation, one can modify the construction above as follows. Consider now the elliptic two-fold cover ramifying over the 4 poles of (E t , ∇ t )

φ t : X t = {y 2 = x(x -1)(x -t)} 2:1 → P 1
x ; (x, y) → x and lift-up the connection on the elliptic curve. After birational gauge transformation, we get a holomorphic connection ( Ẽt , ∇t ) that generically split as the direct sum of two holomorphic connections of rank 1. This means that, for these parameters, Painlevé VI solutions actually parametrize isomonodromic deformations of rank 1 connections over a family of elliptic curves. This allow to solve this very special element of Painlevé VI family (originally found by Picard) by means of elliptic functions (see [START_REF] Hitchin | Twistor spaces, Einstein metrics and isomonodromic deformations[END_REF][START_REF] Mazzocco | Picard and Chazy Solutions to the Painlevé VI Equation[END_REF][START_REF] Loray | The Lamé family of connections on the projective line[END_REF]). By the way, we get a birational map

M 0,4 ( 1 2 , 1 2 , 1 2 , 1 2 )
∼ -→ M 1,0 that commutes with isomonodromic flow. This map has been extended to Lamé parameters in [START_REF] Loray | Okamoto symmetry of Painlevé VI equation and isomonodromic deformation of Lamé connections. Algebraic, analytic and geometric aspects of complex differential equations and their deformations[END_REF][START_REF] Loray | Isomonodromic deformation of Lamé connections, Painlevé VI equation and Okamoto symetry[END_REF] as a birational transformation

Lamé : M 0,4 ( 1 2 , 1 2 , 1 2 , θ ∞ ) ∼ -→ M 1,1 (2θ ∞ -1)
also commuting with isomonodromic flow (see figure 2).

1.5. In [START_REF] Heu | Flat rank 2 vector bundles on genus 2 curves[END_REF], a 2-fold ramified cover commuting with isomonodromic flow

M 0,6 ( 1 2 , 1 2 , 1 2 , 1 2 , 1 2 , 1 2
)

2:1 -→ M 2,0
has been constructed by lifting connections on the hyperelliptic cover 3). 1.6. However, for higher genus g > 2 hyperelliptic curve, the similar map

φ r,s,t : X r,s,t = {y 2 = x(x -1)(x -r)(x -s)(x -t)} 2:1 -→ P 1 x ; (x, y) → x (see figure
M 0,2g+2 ( 1 2 , . . . , 1 2
) -→ M g,0 has small image: not only the deformation upstairs is reduced to the hyperelliptic locus (having codimension g -2), but even for a fixed hyperelliptic curve, the image has codimension 2(g -1) in the moduli space of connections.

Results

In this note, we classify all "interesting" maps that can be constructed between moduli spaces like above, using ramified covers of curves. Let us explain.

Let (X, D ∇ , E, ∇) be a logarithmic rank 2 connections and φ : X → X be a ramified cover. Let D φ denotes the set of critical points of φ while D ∇ denotes the set of poles of ∇; they will be not disjoint in many cases. Consider now the universal deformation t → (X t , D t ) of the marked curve (X, D) where D is the union of D φ and D ∇ . There is a unique local deformation t → (X t , D t , E t , ∇ t , φ t ) where t → (X t , D ∇ t , E t , ∇ t ) is isomonodromic, and t → (X t , D φ t , ∇ t ) is topologically trivial (we just deform the critical locus D φ t ). Fibers of the map t → (X t , D ∇ t ) are algebraic deformations, so-called Hurwitz families.

The main remark is that the lift to Xt of the connection:

t → ( Ẽt , ∇t ) := φ * t (E t , ∇ t )
is isomonodromic along the deformation. By applying rational gauge transformation and twisting with a rank 1 isomonodromic deformation, we may assume that ( Ẽt , ∇t ) is an isomonodromic deformation of logarithmic sl 2 -connexion, free of apparent singular points. In fact, this is possible whenever ∇ t has an essential singular point, i.e. with monodromy. Let Dt be the (reduced) polar divisor of ∇t after deleting apparent singular points. Last but not least, assume that

• the connection (E t , ∇ t ), or equivalently ( Ẽt , ∇t ), has Zariski dense monodromy, • the deformation t → ( Xt , Dt , Ẽt , ∇t ) induces a locally universal deformation t → ( Xt , Dt ) of the marked curve.

These are the so-called "interesting" conditions. The second item means that we get a complete isomonodromic deformation after the construction. We thus get a complete parametrisation of a leaf of the isomonodromic foliation. All examples listed in section 1 are examples of such constructions. It is easy to construction many exemples where all conditions but the last one are satisfied. However, the last condition, saying that we get the complete deformation, is so hard to realized that we are able, in section 3, to classify all examples. This is our main result in this note. Appart from above known examples, we have the following three new cases.

2.1. Let s → X s = {y 2 = x(x -1)(x -s)} the Legendre family of elliptic curves and let t → (E t , ∇ t ) an isomonodromic deformation of a rank 2 connection with poles located at x = 0, 1, t, ∞. More rigorously, we should say t → (E t , ∇ t ) where t belongs to the Teichmuller space, given by the universal cover T → P 1

x \ {0, 1, ∞} in this case, and t denotes the projection of t on P 1

x \ {0, 1, ∞}. Now, assume that exponents of ∇ t take the form (θ 0 , θ 1 , θ t , θ ∞ ) = ( 12 , 1 2 , θ, 1 2 ). Therefore, after lifting on the elliptic curve, we get a connection with 3 apparent singular points and two copies of the singular point at x = t. By gauge transformation, we finally get a connection ( Ẽs,t , ∇s,t ) with only two simple poles, but to get a sl 2 -connection we need to shift one of the two exponents (see figure 4). We finally get a rational map

P 1 s × M 0,4 ( 1 2 , 1 2 , θ t , 1 2
) -→ M 1,2 (θ, θ -1).

Figure 4. Ruled deformations via uncomplete elliptic cover

Each isomonodromic deformation thus obtained is parametrized by a combination of a Painlevé VI solution (variable t) and a rational function (variable s). We get a 2-parameter space of such tame isomonodromic deformations; they form a codimension 2 subset in M 1,2 (θ, θ -1), the image of the map above, which is saturated by the isomonodromic foliation. The leaves belonging to this set are ruled surfaces parametrized by a Painlevé transcendent. One should recover the Lamé case of section 1 by restricting the isomonodromic foliation to the locus s = t. We postpone the careful study of this picture to another paper.

Consider now the family of genus 2 curves given by

(s, t) → X s,t = {y 2 = x(x -1)(x -s)(x -t 1 )(x -t 2 )}, s ∈ C, t = (t 1 , t 2 ) ∈ C 2
together with the hyperelliptic cover (see figure 5) φ s,t : X s,t → P 1

x ; (x, y) → x. Let t → (E t , ∇ t ) be an isomonodromic deformation of a rank 2 connection on P 1

x with poles located at five among the six critical values, namely x = 0, 1, t 1 , t 2 , ∞. Assume that all exponents of ∇ t take the form

θ 0 = θ 1 = θ t1 = θ t2 = θ ∞ = 1 2 .

Figure 5. Ruled deformations via uncomplete hyperelliptic cover

After lifting the connection to the curve X s,t , deleting apparent singular points by gauge transformation, we get a sl 2 -connection on X s,t with a single apparent singular point located at x = ∞. This provides a rational map

P 1 s × M 0,5 ( 1 2 , . . . , 1 2
)

-→ M 2,1 (1)
conjugating isomonodromic foliations. Here, the only singular point is apparent and it is not possible to delete it. We can just choose to place it at x = ∞; it is irrelevant for the deformation. Again, isomonodromic deformations obtained by this way are parametrized by rank 2 Garnier solutions ((t 1 , t 2 ) variables) combined with a rational function of s. Again, the corresponding leaves of the isomonodromic foliation are uniruled and form a codimension 2 set.

2.3.

Finally, consider the Legendre family t 1 → X t1 = {y 2 = x(x -1)(x -t 1 )} of elliptic curves and let t = (t 1 , t 2 ) → (E t , ∇ t ) an isomonodromic deformation of a rank 2 connection with poles located at x = 0, 1, t 1 , t 2 , ∞. Assume that exponents of ∇ t take the form

(θ 0 , θ 1 , θ t1 , θ t2 , θ ∞ ) = ( 1 2 , 1 2 , 1 2 , θ, 1
2 ). After lifting and applying gauge transformation, we get a sl 2 -connection on the elliptic curve X t1 with two simple poles over x = t 2 having same exponent θ. This gives us a rational map

Φ θ : M 0,5 ( 1 2 , 1 2 , 1 2 , θ, 1 2 ) -→ M 1,2 (θ, θ)
conjugating isomonodromic foliations (see figure 6). We study this map from the topological (i.e. monodromy) point of view in section 4 and deduce Theorem 1. The map Φ θ is dominant and generically two-to-one.

In other word, almost all rank 2 logarithmic connections with two poles on an elliptic curve is a pull-back of a rank 2 logarithmic connection on P 1 ; in particular, such connections are invariant (up to gauge equivalence) under the hyperelliptic involution permuting the two poles. This construction can be thought as intermediate between the genus two case and the Lamé case of section 1. This is a reminiscence of the hyperelliptic nature of the twice-punctured torus. Figure 6. The two punctured torus 2.4. Classification. We prove in section 3 the following Theorem 2. Let t → (X t , D t , E t , ∇ t ) be an isomonodromic deformation of logarithmic sl 2 -connections. Let φ t : Xt → X t a family of ramified covers. Assume that the pull-back deformation t → ( Xt , Dt , Ẽt , ∇t ) after deleting apparent singular points is locally universal, i.e. the corresponding map t → ( Xt , Dt ) is locally surjective. In particular, the deformation has dimension ≥ 3 • genus( Xt ) -3 + deg( Dt ). Then we are in one of the following cases:

• The monodromy of ∇ t (or equivalently ∇t ) is finite, reducible or dihedral.

• The deformation t → (X t , D t , E t , ∇ t ) is actually trivial, and we get an algebraic isomonodromic deformation by deforming φ t . Up to gauge transformation, we are in the list of Doran [START_REF] Doran | Algebraic and geometric isomonodromic deformations[END_REF] or Diarra [START_REF] Diarra | Construction et classification de certaines solutions algébriques des systèmes de Garnier[END_REF]. In particular,

(X t , D t , E t , ∇ t ) is a rigid hypergeometric system (X t = P 1 , deg(D t ) = 3) and deg(φ t ) ≤ 18. • The deformation t → (X t , D t , E t , ∇ t ) is non trivial, X t = P 1 , deg(φ t ) = 2
or 4, and we are in one of the constructions described in sections 1.1, 1.2, 1.4, 1.5, 2.1, 2.2 and 2.3.

2.5. Complement. In the last section, we complete the picture of section 2.3 when θ = 1 2 by constructing a rational map

Ψ : M 1,2 ( 1 2 , 1 2 ) -→ M 2,0
that conjugates isomonodromic foliations. In order to explain, consider the "bielliptic cover"

Xt1,t2 π2 / / π1 φ " " X t2 φ2 X t1 φ1
/ / P 1

x where φ i : X i → P 1

x is the elliptic two-fold cover branching over x = 0, 1, t i , ∞, for i = 1, 2, and the remaining part of the diagramm is the fiber product of φ 1 and φ 2 . In particular, Xt1,t2 has genus 2 and each π i : Xt1,t2 → X i is a two-fold cover branching over the two points φ -1 i (t j ) (where {i, j} = {1, 2}). By the way, φ : Xt1,t2 → P 1

x is a 4-fold cover ramifying over all five points x = 0, 1, t 1 , t 2 , ∞.

Figure 7. Bi-elliptic cover

The map Φ θ of section 2.3 comes from the elliptic covering π 1 , while the map Ψ above, from φ 1 in the bi-elliptic diagramm. In Theorem 11, we characterize the image of Ψ and

Ψ • Φ 1 2 : M 0,5 ( 1 2 , . . . , 1 2
) -→ M 2,0 in terms of the monodromy representation. Mind that, contrary to the previous constructions, we do not get complete isomonodromic deformations (of holomorphic sl 2 -connections on genus 2 curves) but isomonodromic deformations over the codimension 1 bi-elliptic locus in the moduli space M 2,0 . This last construction was inspired by [START_REF] Machu | Monodromy of a class of Logarithmic Connections on an Elliptic Curve[END_REF], where isomonodromic deformations of dihedral logarithmic sl 2 -connections are constructed in M 1,2 ( 1 2 , 1 2 ) as direct image of rank 1 holomorphic connections on the bi-elliptic cover X t1,t2 .

Classification of covers

Here, we follow ideas of [START_REF] Diarra | Construction et classification de certaines solutions algébriques des systèmes de Garnier[END_REF][START_REF] Diarra | Solutions algébriques partielles des équations isomonodromiques sur les courbes de genre 2[END_REF], replacing connections by their underlying orbifold structure à la Poincaré.

Let φ : X → X be a ramified cover where X is a genus g hyperbolic orbifold with n singularities of order 2 ≤ ν 1 ≤ • • • ≤ ν n ≤ ∞ (i.e. having angle α i = 2π νi ). Pulling-back by φ, we get a branched orbifold structure on X: orbifold points have angle α = 2πk ν where k is the branching order of φ (i.e. φ ∼ z k ) and • ν = ν i over i th orbifold point of X, • ν = 1 over a regular point. Denote by g the genus of X, and by b the number of branching points on X.

The volume of X with respect to the orbifold metric is given by

aire(X) = 2π(2g -2) + i=1 n(2π -α i );
we get the analogous formula for X with respect to the pull-back metric (even if α i need not be < 2π) and aire( X) = d • aire(X) where d = deg(φ). This yields (after division by 2π)

(1) d • 2g -2 + n i=1 1 - 1 ν i ≤ 2g -2 + ñ j=1 1 - k j ν i(j) -b
If branching points are simple (with branching order 2) then we get an equality. We want to classify cases for which, by deforming simultaneously X and φ, we get the local universal deformation of X. The dimension of the deformation space of X is given by 3g -3 + n ≥ 0 (positivity ollows from hyperbolicity). For X, since we are more involved in the differential equation than in the orbifold structure, we do not take into account the branching points in the deformation, and dimension is given by 3g -3 + ñ. The dimension of deformation of the ramified cover φ is given by the number of "free" critical values (outside orbifold points) and thus bounded by b. We thus want [START_REF] Boalch | Towards a non-linear Schwarz's list[END_REF] 3g

-3 + n + b ≥ 3g -3 + ñ
On the other hand, it is reasonable to ask

(3) 0 < 3g -3 + n ≤ 3g -3 + ñ
first because inequality 3g -3 + n = 0 corresponds (in the hyperbolic case) to hypergeometric (g, n) = (0, 3) that has been treated in [START_REF] Diarra | Construction et classification de certaines solutions algébriques des systèmes de Garnier[END_REF][START_REF] Diarra | Solutions algébriques partielles des équations isomonodromiques sur les courbes de genre 2[END_REF]; right inequality just tells us that we are looking for reductions of isomonodromic equations. Throughout the paper, we will also ask d ≥ 2 not to deal with trivial covers. Let us first roughly reduce (1) combined with [START_REF] Boalch | Towards a non-linear Schwarz's list[END_REF]. In view of this, let us denote ν = ν n the maximum orbifold order (that might be infinite). Then

n i=1 1 - 1 ν i ≥ n -1 2 + 1 - 1 ν .
By the same way, we have

ñ j=1 1 - k j ν i(j) ≤ ñ 1 - 1 ν .
We thus get (4)

(2d -3)g + d -2 2 n + g + ñ ν ≤ d 3 2 + 1 ν -2.
In fact, we have implicitely assumed n = 0. In the case n = 0, we automatically get ñ = 0 and inequality becomes (2d -3)g + g ≤ 2d -2;

however, we must have 2 ≤ g ≤ g (hyperbolicity and growth of genus by ramified covers) that gives us (2d -2)g ≤ 2d -2, contradiction.

3.1. First bounds. From the classical Riemann-Hurwitz formula, we necessarily get g ≥ g. After (4), we thus get

(2d -2)g ≤ d 3 2 + 1 ν -2 ≤ 2d -2.
Therefore, we promptly deduce g ≤ 1. But when g = 1, the rough inequality (4) must be an equality, yielding g = g = 1 and thus (still following Riemann-Hurwitz formula) n = ñ = 0 and b = 0. This case is however non hyperbolic. We can therefore assume g = 0 from now on. In particular, n ≥ 4 from (3), and in case n = 4, hyperbolicity implies ν ≥ 3.

We can also assume that either ν ≤ d, or ν = ∞. Indeed, as soon as ν > d, all points of the fiber φ -1 (p n ) are orbifold; we can therefore modify the orbifold structure of X, replacing ν by ∞, without modifying the numbers n and ñ of orbifold points, and thus without changing dimensions involved in our problem.

Assume ν = ∞. Then (4) gives

d -2 2 n + g ≤ 3d 2 -2
and thus

d ≤ 2 n -2 - g n -3 ≤ 2 n -2 n -3 .
Since d ≥ 2, we promptly deduce g ≤ 1, and more precisely, we are in one of the following cases Assume ν = 2; in this case, n ≥ 5 because of hyperbolicity. Then (4) gives

• d = 2,
d n 2 -2 ≤ n -2 -g - ñ 2 ≤ n -2 -g - ñ 3 ≤ 2n 3 - 2
where right inequality follows from (3) 3g + ñ ≥ n. This gives us

d ≤ 4 3 n -3 n -4 < 3
(because n ≥ 5) and therefore d = 2. Taking into account (4), we get

g + ñ 2 ≤ 2.
This gives us the following possibilities

• g = 2 and ñ = 0,

• g = 1 and ñ ≤ 2,

• g = 0 and ñ ≤ 4. Assume finally 3 ≤ ν ≤ d. Then (4) yields

d n 2 -2 + 1 2 - 1 ν ≤ n -2 -g - ñ ν ≤ n -2 - n ν - ν -3 ν g
where right inequality again follows from (3) 3g + ñ ≥ n. We deduce

d ≤ 2 (n -2)ν -n (n -3)ν -2 .
For each n > 4, right-hand-side is an increasing function of ν with asymptotic 2 n-2 n-3 ≤ 3 2 when ν → ∞. Since ν < ∞ here, we get d < 3 and thus d = 2; by the way, ν ≤ d ≤ 2 and this case is empty. For n = 4, right-hand-side is 4 whatever is the value of ν. Taking into account (4) for n = 4 and d = 3, 4, we get

• g = 1, ñ = 1 (and ν = 3),

• g = 0 and ñ = 4.

Degree d = 2.

Here, φ branches over 2g + 2 points; recall that g ≤ 2. At any orbifold point p i , except when ν i = 2 and φ branches over p i , we can assume ν i = ∞. In other words, we have say

• n 1 points with ν i = 2 over which φ branches,

• n 2 = n -n 1 points with ν i = ∞ (over which φ needs not branching). In the case ν = 2, i.e. n = n 1 and n 2 = 0, we have already seen that g ≤ 2, and thus n ≤ 2g + 2 ≤ 6. By hyperbolicity, we must have n ≥ 5 and we get only two possibilities: X is an orbifold with 5 or 6 conical points ν i = 2 and φ : X → X is a genus g = 2 branching over all conical points. We get examples of sections 1.5 and 2.2 respectively.

Let us now assume n 2 = 0 and thus ν = ∞. Coming back to (1) more carefuly, together with (2), we get

n 1 + 2n 2 + g ≤ 2 + n but since n = n 1 + n 2 , we finally get n 2 + g ≤ 2.
Using hyperbolicity assumption (and n ≥ 3), we find the following solutions.

• g = 1, n 2 = 1 and 3 ≤ n 1 ≤ 4,

• g = 0, n 2 = 2 and n 1 = 2. In the first case, we decompose

• n 1 = 4, φ branches precisely over these 4 points and ñ = 2,

• n 1 = 3, φ branches over these 3 points and one free, and ñ = 2,

• n 1 = 3, φ branches over 4 orbifold points and ñ = 1. We respectively get examples of sections 2.3, 2.1 and 1.4. In the second case, φ branches over the two orbifold points of order 2 and ñ = 4 and we get example of section 1.1.

Degree d = 3. We can assume orbifold points of 3 types:

• ν i = 2 and φ branches at the order 2 over this point; therefore, the preimage consists in one regular point (critical for φ) and a copy of the orbifold point. • ν i = 3 and φ branches at order 3 over this point; therefore, the preimage consists in one regular point (critical for φ). • ν i = ∞ and φ is arbitrary over this point; the preimage consists in 1, 2 or 3 copies of this point. Denote by n 2 , n 3 and n ∞ the number of these points respectively, n 2 +n 3 +n ∞ = n. A combination of (1) together with (2) yields (with above notations)

g + n + n ∞ = g + n 2 + n 3 + 2n ∞ ≤ 4
This gives us n = 4 and g = n ∞ = 0. But in this case, the only orbifold points up-stairs have order 2 and there are at most 4 such points. This contradict hyperbolicity assumption.

3.4. Degree d = 4. We can assume orbifold orders of 4 types:

• ν i = 2 and φ branches at least once at order 2 over this point; then the preimage consiste consists in one regular point (critical for φ) and either a second one, or two copies of the orbifold point. • ν i = 3 and φ branches atb order 3 over this point; then the preimage consiste consists in one regular point (critical for φ) and a copy of the orbifold point. • ν i = 4 and φ branches at order 4 over this point; then the preimage consiste consists in one regular point (critical for φ). • ν i = ∞ and φ is arbitrary over this point; therefore, the preimage consists in 1, 2, 3 or 4 copies of this point.

Denote by n 2 , n 3 , n 4 et n ∞ the number of these points respectively,

n 2 + n 3 + n 4 + n ∞ = n.
A combination of (1) together with (2) yields (with above notations)

g + n 2 + 2n 3 + 2n 4 + 3n ∞ + ñ2 2 ≤ 6
(here, ñ2 is the number of orbifold points of X over the n 2 points of order 2). By hyperbolicity, we get n ≥ 4 and, when n = 4, at least one of the orbifold points is not of minimal order 2, yielding n

+ n 2 + n 3 + n 4 ≥ 5.
Assume first n ∞ = 0; then, inequalities allow the only possibility n = 4 with (n 2 , n 3 , n 4 , n ∞ ) = (3, 0, 0, 1), g = 0 and ñ2 = 0. We get the quartic transformation for Painlevé VI (see section 1.2).

Let us now assume n ∞ = 0. Recall that we want 3g -3+ ñ = 3g -3+ ñ2 +n 3 ≥ 1 if n = 4 and ≥ 2 if n ≥ 5. From these inequalities, the only possibility is n = 4 with (n 2 , n 3 , n 4 , n ∞ ) = (3, 1, 0, 0), g = 1 and ñ2 = 0. The covering φ branches only over the 4 orbifold points, is totally ramified at the order 2 over the 3 points of order 2 and has a single order 3 branching point over the point of orbifold order 3. Its monodromy, taking values into the symmetric group Σ 4 , is generated by 3 double-transpositions (ij)(kl), {i, j, k, l} = {1, 2, 3, 4}, whose composition has order 3. However, in Σ 4 , double-transpositions form a group (together with the identity) and cannot generate an order 3 element: such a cover does not exist.

4.

From the five-punctured sphere to the twice-punctured torus Fix distinct points 0, 1, t, λ, ∞ ∈ P 1 , and consider the elliptic cover

φ : X λ := {y 2 = x(x -1)(x -λ)} → P 1 x ; (x, y) → x;
denote by {t 1 , t 2 } := φ -1 (t) the preimage of the fifth point (mind that we change notations). The orbifold fundamental group of P 1 \ {0, 1, t, λ, ∞} is defined by

Γ := γ 0 , γ 1 , γ t , γ λ , γ ∞ | γ 0 γ 1 γ t γ λ γ ∞ = γ 2 0 = γ 2 1 = γ 2 λ = γ 2 ∞ = 1 .
On the other hand, the fundamental group of the twice punctured torus X λ \{t 1 , t 2 } is given by

Γ := α, β, δ 1 , δ 2 | αβ = δ 1 βαδ 2 .
The elliptic cover induces a natural monomorphism φ * : Γ → Γ identifying Γ with an index two subgroup of Γ: the subgroup generated by γ t and words of even length in letters γ 0 , γ 1 , γ λ , γ ∞ . In fact, a careful study of the topological cover yields Lemma 3. The morphism φ * is defined by

       φ * (α) = γ1 • γt • γλ φ * (β) = γλ • γ∞ φ * (δ 1 ) = γt φ * (δ 2 ) = γ∞ • γ t • γ -1 ∞
One easily check the compatibility between relations defining Γ and Γ.

Proof. If p ∈ P 1 \ {0, 1, t, λ, ∞} denotes the base point used to compute the fundamental group on the sphere, denote by p and p the two lifts on the elliptic curve. For i = 0, 1, λ, ∞, the loop γ i lifts as paths (half loops)

• γi from p to p ,

• γ i from p to p. On the other hand, the loop γ t lifts as loops

• γt based at p, • γ t based at p . Then, carefully drawing the picture, we get

       α = γ1 • γ t • γ λ β = γλ • γ ∞ δ 1 = γt δ 2 = γ∞ • γ t • γ -1 ∞
We check that these loops indeed satisfy αβ = δ 1 βαδ 2 by using relations

γi • γ i = 1 for i = 0, 1, λ, ∞
and those which lift as

γ 0 • γ 1 • γ t • γ λ • γ ∞ = 1 namely γ0 • γ 1 • γt • γλ • γ ∞ = 1 and γ 0 • γ1 • γ t • γ λ • γ∞ = 1.
We get the result by projection on P 1

x . Lemma 4. The unique elliptic involution of X t1 that permutes t 1 and t 2 acts as follows on the fundamental group:

α ↔ α -1 β ↔ β -1 γ 1 ↔ γ 2
We note that the relation αβ = δ 1 βαδ 2 is indeed invariant by the involution.

Proof. We have to take care that the base point p is not fixed. In fact, the involution permutes p and p and acts on γ i lifts as follows γi ↔ γ i for i = 0, 1, t, λ, ∞.

In particular, if we denote

α = γ 1 • γt • γλ β = γ λ • γ∞
then involution acts on these loops as α ↔ α and β ↔ β .

We bring back these new loops to the base point p by conjugating (for instance) with γ∞ , which gives us

α ↔ γ∞ • α • γ-1 ∞ β ↔ γ∞ • β -1 • γ-1 ∞ γt ↔ γ∞ • γ t • γ-1 ∞
We thus get δ 1 ↔ δ 2 and, by a direct computation, using relations between γi and γ i , we check that α ↔ α -1 and β ↔ β -1 .

In order to prove Theorem 1, it is enough to prove that the map Φ θ is dominant, generically two-to-one. By the Riemann-Hilbert correspondance, it is equivalent to work with the corresponding spaces of monodromy representations. Let us denote by R θ the space of monodromy representations for M 0,5 ( 1 2 , 1 2 , 1 2 , θ, 1 2 ):

R θ :=    (M 0 , M 1 , M t , M λ , M ∞ ) ∈ SL 2 (C) 5 ; M 0 M 1 M t M λ M ∞ = I trace(M i ) = 0 for i = 0, 1, λ, ∞ trace(M t ) = 2 cos(πθ)    / ∼
where the equivalence relation ∼ is the diagonal adjoint action by SL 2 (C) on quintuples. Recall that, in SL 2 (C), we have

trace(M ) = 0 ⇔ M 2 = -I
and the corresponding PSL 2 (C)-representations are actually representations

Γ → PSL 2 (C).
On the other hand, consider the space Rθ of monodromy representations of M 1,2 (θ, θ)

Rθ := (A, B, D 1 , D 2 ) ∈ SL 2 (C) 4 ; AB = D 1 BAD 2 trace(D 1 ) = trace(D 2 ) = 2 cos(πθ) / ∼
The natural map φ 1 * : R θ → Rθ induced by φ 1 is described by

Corollary 5. We have φ 1 * (M 0 , M 1 , M t , M λ , M ∞ ) = (A, B, D 1 , D 2 ) with        A = M 1 M t M λ , B = M λ M ∞ , D 1 = M t , D 2 = M ∞ M t M -1
∞ . Proof. From Lemma 3, we know that AB = ±D 1 BAD 2 ; we just have to check that we have the right sign, and thus a representation

π 1 (X λ \ {t 1 , t 2 }) → SL 2 (C)
and we must have trace(D 1 ) = trace(D 2 ) = 2 cos(πθ) (= trace(M t )).

We now want to prove that the map φ 1 * : R θ → Rθ just defined is generically one-to-one. This follows from the following

Theorem 6. Let A, B, D 1 , D 2 ∈ SL 2 (C) such that AB = D 1 BAD 2 and D 1 , D 2 = ±I.
Assume moreover that the subgroup < A, B > generated by A and B is irreducible, i.e. without common eigendirection. Then there is a matrix M ∈ SL 2 (C), unique up to a sign, such that

M AM -1 = A -1 , M BM -1 = B -1 and M D 1 M -1 = D 2 . Moreover, M 2 = -I and (A, B, D 1 , D 2 ) = φ 1 * (M 0 , M 1 , M t , M λ , M ∞ ) for            M 0 = -AM M 1 = ABD -1 2 M M t = D 1 M λ = -BM M ∞ = M
First recall well-known results concerning SL 2 (C). Proof. If A et B have a common eigenvector, then we can assume < A, B > is triangular and the commutator will be a unipotent matrix, thus having trace 2.

Conversely, assume that A and B have no common eigenvector. Therefore, an eigenvector v for AB will not be eigenvector for A or for B. If ABv = γv, then in the base (v, -γBv), matrices take the form

A = a -1 1 0 and B = 0 1 γ -γ b
where a = trace(A) and b = trace(B). We check that

[A, B] = a 2 + b 2 + γ 2 -abc γ -2 (a -bγ) a -bγ -1 γ -2 and thus trace([A, B]) = a 2 + b 2 + c 2 -abc -2, c = γ + γ -1 = trace(AB).
Finally, these matrices A and B have a common eigenvector if, and only if, Proof. This is a consequence of formulae from the preceeding proof. Proof. It suffices to notice that trace(A) = trace(A -1 ) and trace(AB) = trace(BA) for all matrices A, B ∈ SL 2 (C). We deduce, under our assumptions, that trace(A) = trace(A -1 ), trace(B) = trace(B -1 ) and trace(AB) = trace(A -1 B -1 ).

a 2 + b 2 + c 2 -abc -2 = 2.
Therefore, there exists an M satisfying the first part of the statement. But M 2 has to commute to A and B. Thus M 2 must fix all eigendirections of all elements of the group < A, B >. There are at least three distinct such directions and M 2 is projectively the identity: M 2 = ±I. But M = ±I is impossible since M AM -1 = A -1 = A (A = ±I otherwise < A, B > would be reductible). Thus M 2 = I and M 2 = -I. If matrices A and B are given in the normal form like in the proof above, then M is given by ( 5)

M = ± γ 2 -1 2γ a-bγ 2γ aγ-b 2 -γ 2 -1 2γ
Proof of Theorem 6. We want now to prove that the unique (up to a sign) matrix M satisfying M AM -1 and M BM -1 also satisfy

M D 1 M -1 = D 2 and thus M D 2 M -1 = D 1 (M 2 = -I). From relation AB = D 1 BAD 2 , this is equivalent to AB = D 1 BAM D 1 M -1 ⇔ (BAM D 1 ) 2 = -I ⇔ trace(BAM D 1 ) = 0. Rewrite the relation AB = D 1 BAD 2 into the form [A, B] = D 1 BAD 2 A -1 B -1 = D 1 D 2 with D 2 = (BA)D 2 (BA) -1 .
Note that This is the monodromy space of those connections on the elliptic curve X λ having logarithmic poles with exponent 1 2 at t 1 and t 2 . Let us now consider the 2-fold ramified cover π : Xt,λ → X λ ramifying over t 1 and t 2 and let us study the associated map which is given by (see also [START_REF] Machu | Monodromy of a class of Logarithmic Connections on an Elliptic Curve[END_REF])

(BAM ) 2 = BAM BAM = BAB -1 A -1 M 2 = -BAB -1 A -1 = -[A, B]
π * : M 1,2 ( 1 2 , 1
Lemma 10. We have π * (A, B, C 1 , C 2 ) = (A 1 , B 1 , A 1 , B 2 ) with        A 1 = A, B 1 = B, A 2 = C -1 1 AC 1 , B 2 = C -1
1 BC 1 . Conversely, we can characterize the image of π * as follows To apply the Lemma, we just need to prove that the two traces c i := A i B i coincide for i = 1, 2. But the relation [A 1 , B 1 ][A 2 , B 2 ] = I implies that the two commutators are inverse to each other, and thus share the same trace. By the commutator trace formula in the proof of Lemma 7, we get

(c 1 -c 2 )(c 1 + c 2 -ab) = 0.
The image of π * has codimension 2 in R . We also see that generic fibers of π * consist in 2 points.

Figure 2 .

 2 Figure 2. Lamé's cover

Figure 3 .

 3 Figure 3. Genus 2 cover

g ≤ 1

 1 and n arbitrary, • d = 3, g = 0 and n = 4 or 5, • d = 4, g = 0 and n = 4.In particular, we get d ≤ 4 in this case.

Lemma 7 .

 7 Two matrices A, B ∈ SL 2 (C) generate a reducible group if, and only if, trace[A, B] = 2 where [A, B] = ABA -1 B -1 is the commutator.

Lemma 8 .

 8 Let A, B, A , B ∈ SL 2 (C) and assume trace[A, B] = 2. There exists M ∈ SL 2 (C) such that M AM -1 = A and M BM -1 = B if, and only if, trace(A) = trace(A ), trace(B) = trace(B ) and trace(AB) = trace(A B ).

Corollary 9 .

 9 If trace[A, B] = 2, then there exists M ∈ SL 2 (C), unique up to a sign, such that M AM -1 = A -1 and M BM -1 = B -1 . Moreover, M 2 = -I.

- 1 andC 1 =

 11 therefore (BAM ) 2 D 1 = -(D 2 ) -1 and trace((BAM ) 2 D 1 ) + trace(D 1 ) = 0. Now, recall that in SL 2 (C) we have universal relationstrace(M 1 M 2 ) + trace(M 1 M -1 2 ) = trace(M 1 ) • trace(M 2 ). Applying this to M 1 = BAM and M 2 = BAM D 1 , we get 0 = trace((BAM ) 2 D 1 ) + trace(D 1 ) = trace(BAM D 1 ) • trace(BAM ). But, trace(BAM ) = 0 otherwise (BAM ) 2 = -[A, B] -1 = -I, i.e. [A, B] = I,that would contradict irreducibility. Thus trace(BAM D 1 ) = 0, what we wanted to prove. Finally, we easily check that matrices M i given by the statement are indeed inversing preceeding formulae of Lemma 5 by using relation AB = D 1 BAD 2 and properties of M .5. Bielliptic coversLet us now assume θ = 0 and rewriteR1/2 := (A, B, C 1 , C 2 ) ∈ SL 2 (C) 4 ; [A, B] = C 1 C 2 trace(C 1 ) = trace(C 2 ) = 0 / ∼where we have modified generators of the fundamental group for convenience: D 1 and C 2 = (BA) -1 D 2 (BA).

2 )

 2 → M 2,0 on the monodromy side of the Riemann-Hilbert correspondance. Denote byR := (A 1 , B 1 , A 1 , B 2 ) ∈ SL 2 (C) 4 ; [A 1 , B 1 ][A 2 , B 2 ] = I / ∼the space of monodromy representations associated to M 2,0 . Then we get a map π * : R1/2 -→ R

Theorem 11 . 2 =

 112 Let A 1 , B 1 , A 2 , B 2 ∈ SL 2 (C) such that [A 1 , B 1 ][A 2 , B 2 ] = I.Assume that there exists a matrix M ∈ SL 2 (C) such thatM A 1 M -1 = A 2 , M B 1 M -1 = B 2 and M 2 = -I. Then (A 1 , B 1 , A 2 , B 2 ) = π * (A, B, C 1 , C 2 ) M -1 [A 1 , B 1 ], If moreover trace[A 1 , B 1 ] = 2 then (A 1 , B 1 , A 2 , B 2 ) is in the image of π • φ, i.e. comes from a representation of the 5-punctured sphere. Remark 12. From Lemma 8, we see that existence of M is almost equivalent to trace(A 1 ) = trace(A 2 ) =: a and trace(B 1 ) = trace(B 2 ) =: b.

Remark 13 .

 13 If we fix A 1 and B 1 generic, we obtain: (1) the set {M ∈ SL 2 (C) ; M 2 = -I and M conjugates [A 1 , B 1 ] to its inverse} has dimension 1, (2) the set {A 1 , B 1 , M -1 .A 1 .M, M -1 .B 1 .M } has also dimension 1 up to conjugacy. Thus we can freely choose (A 1 , B 1 ) in the image of π * .


Olivier Kihl
email: olivier.kihl@ensea.fr

David Picard
email: david.picard@ensea.fr

Philippe-Henri Gosselin
email: philippe.gosselin@inria.fr

Local polynomial space-time descriptors for actions classification

Keywords: Action classification, Visual descriptors, Motion

In this paper we propose to tackle human actions indexing by introducing a new local motion descriptor based on a model of the optical flow. We propose to apply a coding step to vector field before the modeling. We use two modeling, a spatial model and a temporal model. The spatial model is computed by projection of optical flow onto bivariate orthogonal polynomials. Then, the time evolution of spatial coefficients is modeled with a one dimension polynomial basis. To perform the action classification, we extend recent still image signatures using local descriptors to our proposal and combine them with linear SVM classifiers. The experiments are carried out on the well known UCF11 dataset and on the more challenging Hollywood2 action classification dataset and show promising results.

Introduction

Human action recognition has become an important research area in computer vision since it concerns several key applications like video indexing, video surveillance or human computer interaction. The typical setup for this task involves the extraction of highly discriminant features localized in both space and time. A wide variety of such descriptors have been introduced recently [START_REF] Wang | Action recognition by dense trajectories[END_REF][START_REF] Wang | Evaluation of local spatio-temporal features for action recognition[END_REF][START_REF] Laptev | Learning realistic human actions from movies[END_REF][START_REF] Schuldt | Recognizing human actions: A local svm approach[END_REF], and have become essential tools of the action classification framework. These descriptors are then aggregated into a single vector using the extension to video of the well known "Bag of Words" image signature approaches [START_REF] Sivic | Video google: A text retrieval approach to object matching in videos[END_REF]. To further improve the results, most action classification systems use the combination of several complementary descriptors.

This paper is a revised and extended version of earlier work presented in [START_REF] Kihl | Local polynomial space-time descriptors for actions classification[END_REF]. The proposed descriptor is localized spatially and temporally in a space-time tube, in order to capture characteristic atoms of motion. The Serie of Polynomial Approximation of Flow (SoPAF) space-time motion descriptor is based on polynomial decomposition of the optical flow [START_REF] Kihl | Local polynomial space-time descriptors for actions classification[END_REF].

We propose to extend this descriptor by coding the vector field with the half-wave rectification proposed by Efros et al. [START_REF] Efros | Recognizing action at a distance[END_REF]. Moreover, we study two different functions basis (polynomial and sine) for modeling the temporal evolution of spatial polynomial coefficients.

The paper is organized as follows. In section 2 we present the most popular space-time feature descriptors in the literature. Then, in section 3 we present the SoPAF descriptor and our extension. Finally, in section 4 we carry out experiments on two well known action classification datasets.

Related work

The recognition of human action and activity is an important area in several fields such as computer vision, machine learning and signal processing. A popular way of comparing videos is to extract a set of descriptors from video, to find a transformation that maps the set of descriptors into a single vector, and then to measure the similarity between the obtained vectors.

We first present several works related to descriptors extraction, and then we present the most popular signature approaches.

Video descriptors

In the early work on action recognition, silhouette based descriptors, also called motion appearance models, were used. These descriptors are computed from the evolution of a silhouette obtained by background subtraction methods or by taking the difference of frames (DOF). From a sequence of binary images, Bobick and Davis [START_REF] Davis | The representation and recognition of action using temporal templates[END_REF] propose descriptors called Motion Energy Image (MEI) representative of the energy of movement and Motion History Image (MHI) providing information about the chronology of motion. These two descriptors are modeled by seven Hu moments. Kellokumpu et al. use histograms of Local Binary Patterns (LBP) to model the MHI and MEI images [START_REF] Kellokumpu | Texture Based Description of Movements for Activity Analysis[END_REF]. In [START_REF] Kellokumpu | Human activity recognition using a dynamic texture based method[END_REF], they propose an extension of the LBP directly applied on the image pixels with successful results. Wang and Suter [START_REF] Wang | Learning and matching of dynamic shape manifolds for human action recognition[END_REF] use two other descriptors, namely the Average Motion Energy (AME) and the Mean Motion Shape (MMS). The AME is a descriptor close to the MHI representing the average image of silhouettes. The MMS is defined from boundary points of the silhouette in complex coordinates with the origin placed at the centroid of the 2D shape. As time is an important information in video, Gorelick et al. study the silhouettes as space-time volumes [START_REF] Blank | Actions as space-time shapes[END_REF][START_REF] Gorelick | Actions as space-time shapes[END_REF]. Space-time volumes are modeled with Poisson equations. From these, they extract seven spatio-temporal characteristic components.

The main drawback of all these methods is the computation of silhouettes. Indeed, this computation is not very robust, making these methods only relevant in controlled environments such as the Weizmann dataset [START_REF] Blank | Actions as space-time shapes[END_REF] or the KTH dataset [START_REF] Schuldt | Recognizing human actions: A local svm approach[END_REF]. Moreover, they tend to fail on more realistic data-sets such as UCF11 [START_REF] Liu | Recognizing realistic actions from videos in the wild[END_REF] or Holly-wood2 [START_REF] Laptev | Learning realistic human actions from movies[END_REF].

Assuming that action recognition is closely linked to the notion of movement, many authors have proposed descriptors based on the modeling of optical flow. The optical flow encodes the displacement of pixels from two consecutive frames. The result can be represented by vector fields with two components U and V. Here, U denotes the horizontal component of motion and V the vertical component. Early works with respect to this approach were proposed by Polana and Nelson [START_REF] Polana | Low level recognition of human motion[END_REF]. The vector field is first decomposed according to a spatial grid. Then, in each cell of the grid, the magnitude of motion is accumulated. This method can only process periodic actions such as running or walking.

Efros et al. propose a descriptor computed on a figure-centric spatio-temporal volume for each person in a video [START_REF] Efros | Recognizing action at a distance[END_REF]. The vector field representing the motion between two consecutive frames of the volume is computed with the Lucas and Kanade optical flow algorithm [START_REF] Lucas | An iterative image registration technique with an application to stereo vision[END_REF]. The two components U and V of the vector field are decomposed with a half-wave rectification technique. The resulting four components are blurred using a Gaussian filter and normalized. They are directly used as a descriptor. The obtained descriptors are compared using the normalized correlation measure. This descriptor is used and/or extended by several authors in [START_REF] Fathi | Action by learning midlevel motion features[END_REF][START_REF] Danafar | Action recognition for surveillance applications using optic flow and svm[END_REF].

Tran et al. have proposed the motion context descriptor [START_REF] Tran | Human activity recognition with metric learning[END_REF]. It is also a figure-centric descriptor based on the silhouette extraction. They use the vector field and the binary silhouette as three components. The components of the field are blurred with a median filter. Then, the three components are subdivided with a grid of 2 × 2 cells. Each cell is decomposed in 18 radial bins, each covering 20 degrees. Inside the radial bins, the sum of each component is computed. This provides, for each component, 4 histograms composed with 18 bins. The concatenation of these histograms provides a 216-dimensional vector which is the movement pattern of a given field. From this pattern, the Motion Context is created. It is composed of the 216-dimensional vector of the current frame plus the first 10 vectors of the PCA models of the 5 previous frames, the first 50 vectors of the PCA models of 5 current frames and finally the first 10 vectors PCA models of 5 next frames.

Ali and Shah first compute many kinematic features on the field, and then compute kinematic modes with a spatio-temporal principal component analysis to create a figure-centric descriptor [START_REF] Ali | Human action recognition in videos using kinematic features and multiple instance learning[END_REF].

Figure-centric descriptors are dependent on the person detector associated with them. Moreover, they don't take into account the context in the video that can add relevant information to action recognition. Consequently, these methods tend to fail on more realistic data-sets such as UCF11 [START_REF] Liu | Recognizing realistic actions from videos in the wild[END_REF] or Hollywood2 [START_REF] Laptev | Learning realistic human actions from movies[END_REF] datasets.

An other approach is proposed in [START_REF] Shechtman | Space-time behavior based correlation[END_REF][START_REF] Shechtman | Space-time behavior based correlation or how to tell if two underlying motion fields are similar without computing them[END_REF] that allows to compute the similarity between motions of videos segments without computing motion fields. This method do not have to use video background subtraction. However, this method requires a set of training video centered on the action to recognize.

Finally, the descriptors that have emerged in recent years are the extention to video of still image descriptors [START_REF] Wang | Action recognition by dense trajectories[END_REF]. The most commonly used are SIFT [START_REF] Lowe | Distinctive image features from scale-invariant keypoints[END_REF], SURF [START_REF] Bay | Surf: Speeded up robust features[END_REF] and Histogram of oriented gradient (HOG) [START_REF] Dalal | Human detection using oriented histograms of flow and appearance[END_REF]. SIFT and SURF are both interest points detector and local image descriptor. In this paper, we only consider the descriptors. SIFT and HOG descriptors rely on a histogram of orientation of gradient. Locally, the orientation of the gradient is quantized in o orientations (typically 8). For a given spatial window, a HOG (or a SIFT) descriptor is computed by decomposing the window with a grid of N × N cells. Each cell contains the histogram of orientations of the gradient. The descriptor is obtained by the concatenation of the N × N histograms. HOF is the same as HOG but is applied to optical flow instead of gradient. The MBH models the spatial derivatives of each component of the optical flow vector field with a HOG.

Recently, Wang et al. propose to model these usual descriptors along dense trajectories [START_REF] Wang | Action recognition by dense trajectories[END_REF]. The time evolution of trajectories, HOG, HOF and MBH is modelled using a space time grid following pixels trajectories. The use of dense trajectories for descriptor extraction increases the performances of popular descriptors (HOG, HOF and MBH).

Signatures

Once a set of descriptors is obtained from the video, a popular way of comparing images (or videos) is to map the set of descriptors into a single vector and then to measure the similarity between the obtained vectors (for example in [START_REF] Sánchez | Modeling the spatial layout of images beyond spatial pyramids[END_REF], [START_REF] Wang | Evaluation of local spatio-temporal features for action recognition[END_REF] and [START_REF] Wang | Action recognition by dense trajectories[END_REF]). The most common method for such embeddings is inspired by the text retrieval community and is called the "Bag of Words" (BoW) approach [START_REF] Sivic | Video google: A text retrieval approach to object matching in videos[END_REF]. It consists in computing a dictionary of descriptor prototypes (usually by clustering a large number of descriptors) and then computing the histogram of occurrences of these prototypes (called "Visual Words") within the set.

In still images classification, these approaches have been formalized in [START_REF] Wang | Locality-constrained linear coding for image classification[END_REF] by a decomposition of the mapping into two steps. The first step, namely the "coding step", consists in mapping each descriptor into a codeword using the aforementioned dictionary. The second step is to aggregate the codewords into a single vector and is called the "pooling step". Structural constraints such as sparsity [START_REF] Yang | Linear spatial pyramid matching using sparse coding for image classification[END_REF] or locality [START_REF] Wang | Locality-constrained linear coding for image classification[END_REF] can be added to the coding process to ensure most of the information is retained during the pooling step. Common pooling processes include averaging the codewords or retaining the entry-wise maximum among the codewords (max pooling). Extensions of the BoW model have been recently proposed to include more precise statistical information. In [START_REF] Avila | Bossa: Extended bow formalism for image classification[END_REF], the authors propose to model the distribution of distances of descriptors to the clusters centers. In the coding/pooling framework, each descriptor is coded by 1 in the bin corresponding to its distance to the cluster's center to which it belongs, and 0 otherwise. The pooling is simply the averaging over all codewords.

In [START_REF] Jégou | Aggregating local descriptors into a compact image representation[END_REF], the authors proposed a coding process where the deviation between the mean of the descriptors of the set and the center of the cluster to which they belong to is computed. The whole mapping process can be seen as the deviation between a universal model (i.e. the dictionary) and a local realization (i.e. the set of descriptors). Using this model deviation approach, higher order statistics have been proposed, like "super-vectors" in [START_REF] Zhou | Image classification using super-vector coding of local image descriptors[END_REF], "Fisher Vectors" in [START_REF] Jégou | Aggregating local image descriptors into compact codes[END_REF] or "VLAT" in [START_REF] Picard | Efficient image signatures and similarities using tensor products of local descriptors[END_REF][START_REF] Picard | Improving image similarity with vectors of locally aggregated tensors[END_REF]. Fisher Vectors are known to achieve state of the art performances in image classification challenges [START_REF] Chatfield | The devil is in the details: an evaluation of recent feature encoding methods[END_REF].

To compare the performances of descriptors, in this paper, we consider a compressed version of VLAT which is known to achieve near state of the art performances in still images classification with very large sets of descriptors [START_REF] Negrel | Using spatial pyramids with compacted vlat for image categorization[END_REF]. In our case, the dense sampling both in spatial and temporal directions leads to highly populated sets, which is consistent with the statistics computed in VLAT signatures. Given a clustering of the descriptors space with C clusters computed on some training set, the first and second order moments µ c and τ c are computed for each cluster c:

µ c = 1 |c| i r ν rci (1)
τ c = 1 |c| i r (ν rci -µ c )(ν rci -µ c ) T (2)
with |c| being the number of descriptors ν rci of video i in cluster c, for all videos in the training set. The eigen decomposition of the covariance matrix τ c for each cluster c is then performed:

τ c = V c D c V ⊤ c (3)
Using this decomposition, descriptors are projected on the subspace generated by the eigenvectors V c . The compressed VLAT signature τ i,c of video i is computed for each cluster c with the following equation:

τ i,c = r (V c (ν rci -µ c ))(V c (ν rci -µ c )) ⊤ -D c (4)
τ i,c are then flattened into vectors v i,c . The complete VLAT signature x i of video i is obtained by concatenation of v i,c for all clusters c:

v i = (v i,1 . . . v i,C ) (5)
It is advisable to perform a normalization step for best performance. (thanks to the matrices τ i,c being symmetric, only half of the coefficients are kept).

∀j, v ′ i [j] = sign(v i [j])|v i [j]| α , (6)
x i = v ′ i v ′ i (7)

Series of Polynomial Approximation of Flow (SoPAF)

We propose to extend the SoPAF descriptor [START_REF] Kihl | Local polynomial space-time descriptors for actions classification[END_REF]. SoPAF descriptor models the vector field of motion between two frames using projection on an orthogonal basis of polynomials. This polynomial model is used in [START_REF] Kihl | Human activities discrimination with motion approximation in polynomial bases[END_REF] to recognize movements in a video. The modeling is applied to the entire field and each frame is processed separately. In another context, this polynomial model is locally used to detect singularities such as vortex or saddle point in fluid motion [START_REF] Kihl | Multivariate orthogonal polynomials to extract singular points[END_REF]. Since motion can succesfully be modeled by polynomials, we propose to use such models on a local neighborhood in order to obtain densely extracted local motion descriptors. We use two successive polynomial models. At first, the spatial vector field is modeled with two dimensional polynomial basis. Then, time evolution of spatial coefficients are modeled with a one dimensional basis. We propsose to extend the descriptor using the half-wave rectification technique proposed by Efros et al. [START_REF] Efros | Recognizing action at a distance[END_REF]. Moreover, we propose to evaluate sine functions basis in addition to of polynomial functions.

Spatial modeling using a polynomial basis

Let us consider the descriptor M(i, j, t) located in frame at coordinates (i, j) and in video stream at time t. Descriptors are computed using space and time neighborhood around location (i, j, t), denoted as window W (i, j, t). An example of W (i, j, t) is shown in Fig. 1a. We propose to model the vector field of motion inside the window W (i, j, t) by a finite expansion of orthogonal polynomials. Let us define the family of polynomial functions with two real variables as follows:

P K,L (x 1 , x 2 ) = K k=0 L l=0 a k,l x k 1 x l 2 (8)
where k ∈ {0..K}, l ∈ {0..L}, K ∈ N + and L ∈ N + are respectively the maximum degree of the variables (x 1 , x 2 ) and {a k,l } are the polynomial coefficients. The global degree of the polynomial is

D = K + L. Let B = {P k,l } k∈{0.
.K},l∈{0..L} be an orthogonal basis of polynomials. A basis of degree D is composed by n polynomials with n = (D + 1)(D + 2)/2 as follows:

B = {P 0,0 , P 0,1 , • • • , P 0,L , P 1,0 , • • • • • • , P 1,L-1 , • • • , P K-1,0 , P K-1,1 , P K,0 } (9)
We can create an orthogonal basis using the following three terms recurrence:

           P -1,l (x) = 0 P k,-1 (x) = 0 P 0,0 (x) = 1 P k+1,l (x) = (x 1 -λ k+1,l )P k,l (x) -µ k+1,1 P k-1,l (x) P k,l+1 (x) = (x 2 -λ k,l+1 )P k,l (x) -µ k,l+1 P k,l-1 (x) (10)
where x = (x 1 , x 2 ) and the coefficients λ k,l and µ k,l are given by

λ k+1,l = x1P k,l (x)|P k,l (x) P k,l (x) 2 λ k,l+1 = x2P k,l (x)|P k,l (x) P k,l (x) 2 µ k+1,l = P k,l (x)|P k,l (x) P k-1,l (x) 2 µ k,l+1 = P k,l (x)|P k,l (x) P k,l-1 (x) 2 (11)
and • | • is the usual inner product for polynomial functions:

P 1 | P 2 = Ω P 1 (x)P 2 (x)w(x)dx (12)
with w the weighting function that determines the polynomial family and Ω the spatial domain covered by the window W (i, j, t). We use Legendre polynomials (w(x) = 1, ∀x).

Using this basis, the approximation of the horizontal motion component U is:

Ũ = D k=0 D-k l=0 ũk,l P k,l (x) P k,l (x) (13)
The polynomial coefficients ũk,l are given by the projection of component U onto normalized B elements:

ũk,l = U | P k,l (x) P k,l (x) (14)
Similarly, vertical motion polynomial coefficients ṽk,l are given by computing the projection of vertical component V onto B elements. Using the polynomial basis B of degrre D, the vector field asociated to window W (i, j, t) is modelled by (D + 1) × (D + 2) coefficients.

Time modeling using a polynomial basis

Since an action is performed along more than two frames, we propose to model motion information in longer spacetime volumes.

Let us consider the descriptor located in frame at coordinates (i, j) and in video stream at time t 0 . We consider the same spatial domain as previously defined (see Fig. 1a). Moreover, we now consider the space-time tube defined by all the window W (i, j, t 0 ) to W (i, j, t 0 + τ ), with τ being the length of our descriptors temporal domain (see Fig. 1b). For each frame at time t between t 0 and t 0 + τ , we propose to model the vector field of motion inside the windows W (i, j, t) of the tube by the coefficients ũk,l and ṽk,l , as defined in the previous section.

Then all coefficients ũk,l (i, j, t) (respectively ṽk,l (i, j, t)) for t = t 0 to t = t 0 + τ are grouped in a vector defined as u k,l (i, j, t 0 ) = [ũ k,l (i, j, t 0 ), . . . , ũk,l (i, j, t 0 + τ )] (

We model the time evolution of the coefficients ũk,l (i, j, t) (resp. ṽk,l (i, j, t)) by projecting u k,l (i, j, t 0 ) (resp. v k,l ) onto a one dimension orthogonal function basis. In [START_REF] Kihl | Local polynomial space-time descriptors for actions classification[END_REF], we use Legendre polynomial basis of degree d defined by

               P -1 (t) = 0 P 0 (t) = 1 T n (t) = (t -tP n-1 (t)|P n-1 (t) )P n-1 (t) -P n-2 (t) P n (t) = T n (t) |T n | (16)
In this work, we also use Sine basis for time evolution modeling. Using such basis (polynomial or sine) with degree d, the approximation of u k,l (i, j, t) is:

ũk,l (i, j, t) = d n=0
ũk,l,n (i, j, t) P n (t)

P n (t) (17)
The model has d + 1 coefficients ũk,l (i, j, t) given by ũk,l,n (i, j, t) = u k,l (i, j, t) | P n (t)

P n (t) (18)
The time evolution of a given coefficient ũk,l (i, j) (respectively ṽk,l (i, j)) is given by the vector m l,k (i, j, t 0 ) (respectively n l,k (i, j, t 0 )) as defined in equation ( 19) m l,k (i, j, t 0 ) = [ũ k,l,0 (i, j, t 0 ), ũk,l,1 (i, j, t 0 ), . . . , ũk,l,d (i, j, t 0 )]

The feature descriptor ν(i, j, t 0 ) for the whole spacetime volume beginning at time t 0 and centered at position (i,j) is given by

ν(i, j, t 0 ) = [m 0,0 , m 0,1 , • • • , m 0,L , m 1,0 , • • • , m 1,L-1 , • • • • • • , m K-1,0 , m K-1,1 , m K,0 , n 0,0 , n 0,1 , • • • • • • , n 0,L , n 1,0 , • • • , n 1,L-1 , • • • n K-1,0 , n K-1,1 , n K,0 ] (20)
Here, m k,l (i, j, t 0 ) and n k,l (i, j, t 0 ) are written as m k,l and n k,l for clarity reasons. The size of the descriptor

ν ( i, j, t 0 ) is (D + 1) × (D + 2) × d.
We name Series of Polynomial approximation of Flow the descriptor as it is defined in [START_REF] Kihl | Local polynomial space-time descriptors for actions classification[END_REF]. If Sine basis is used to model the motion vector field evolution, we name the descriptor SoPAF+Sine. Note, for the spatial modeling of the vector field, only polynomial basis are used.

Series of local Polynomial Approximation of Rectified Flow

We propose an extension of the SoPAF descriptor by using the half-wave rectification coding proposed by Efros et al. in [START_REF] Efros | Recognizing action at a distance[END_REF] and used in several works. The half-wave rectification coding produces a four dimension code from the horizontal component U and the horizontal component V of the vector field. The code is defined as:

U + (x) = U (x) if U (x) > 0 0 else (21) U -(x) = U (x) if U (x) < 0 0 else (22) V + (x) = V(x) if V(x) > 0 0 else (23) V -(x) = V(x) if V(x) < 0 0 else ( 24
)
This coding is applied to motion vector field before the modeling steps of SoPAF descriptor. This preprocessing doubles the dimensions of the obtained descriptor. We

Trajectories

As proposed in [START_REF] Wang | Action recognition by dense trajectories[END_REF], we use trajectories to follow the spatial position of the window along time axis.

In our case the window W (i 1 , j 1 , t 0 +1) at time t 0 +1 is selected as the best matching block with respect to the window W (i 0 , j 0 , t 0 ) from time t 0 . This matching is performed using a three step search block matching method from [START_REF] Koga | Motion-compensated interframe coding for video conferencing[END_REF]. The temporal evolution of spatial coefficients is thus modeled on tubes instead of volumes.

Experiments

We carry out experiments on two well known human action recognition datasets. The first one is the UCF11 dataset [START_REF] Liu | Recognizing realistic actions from videos in the wild[END_REF], and the second one is the Hollywood2 Human Actions dataset [START_REF] Laptev | Learning realistic human actions from movies[END_REF].

In this section, we first introduce the two datasets. Second, we evaluate parameters of our descriptor on the UCF11 dataset. Third, we compare our descriptor to literature results on UCF11 and Hollywood2 datasets. For the parametrization, we use the best results obtained on UCF11 evaluation.

We use a Horn and Schunk optical flow algorithm [START_REF] Horn | Determining optical flow[END_REF] for motion extraction with 25 iterations and the regularization parameter λ is set to 0.1. We extract the mo-tion fields at 5 scales for UCF11 and 7 for Hollywood2, the scale factor is set to 0.8.

For experiments, we use VLAT indexing method to obtain signatures from descriptors. We train a linear SVM for classification.

Datasets

UCF11 dataset

The UCF11 [START_REF] Liu | Recognizing realistic actions from videos in the wild[END_REF] dataset is an action recognition data set with 11 action categories, consisting of realistic videos taken from youtube (Fig. 3). The data set is very challenging due to large variations in camera motion, object appearance and pose, object scale, viewpoint, cluttered background and illumination conditions. The videos are grouped into 25 groups, where each group consists of more than 4 action clips. The video clips in the same group may share some common features, such as the same person, similar background or similar viewpoint. The experimental setup is a leave one group out cross validation.

Hollywood dataset

The Hollywood2 [START_REF] Laptev | Learning realistic human actions from movies[END_REF] dataset consists of a collection of video clips and extracts from 69 films in 12 classes of human actions (Fig. 4). It accounts for approximately 20 hours of video and contains about 150 video samples per actions. It contains a variety of spatial scales, zoom camera, deleted scenes and compression artifact which allows a more realistic assessment of human actions classification methods. We use the official train and test splits for the evaluation. In this section, we evaluate our descriptor. The spatial size of space-time volumes are set to 32 × 32 pixels and the length is set to 15. These parameters are defined according to results of the evaluation of parameters of HOG, HOF and MBH in [START_REF] Wang | Action recognition by dense trajectories[END_REF]. The spatial step for dense extraction is set to 10 pixels and the time step is set to 5 frames. In Fig. 5, we show the results of our evaluation. In Fig. 5(a), we show the results for the SoPAF with spatial degree varied from 2 to 4, and time degree varied from 0 to 5. The best results are obtained for spatial degree 4 and time degree 1. In Fig. 5(b), we show the results for SoPAF+Sine with spatial degree varied from 2 to 4, and time degree varied from 0 to 5. The best results is obtained for spatial degree 3 and time degree 1. In Fig. 5(c), we show the results for the SoPARF with spatial degree varied from 2 to 4, and time degree var-ied from 0 to 5. The best results is obtained for spatial degree 4 and time degree 2. This result is clearly better than results of SoPAF and SoPAF+Sine. In Fig. 5(d), we show the results for the SoPARF+Sine with spatial degree varied from 2 to 4, and time degree varied from 0 to 5. The best results is obtained for spatial degree 4 and time degree 1. This result is slightly lower than SoPARF but clearly better than those of SoPAF and SoPAF+Sine. We compare now our descriptors with HOF, since it models the same information as ours. In order to compare our descriptor with HOF, we evaluate HOF for space grid from 2 × 2 to 4 × 4 cells and time grid from 1 to 4 cells. We show the results of this evaluation in Fig. 6. Note we obtain at best 80.4%, which is better than Wang et al. in [START_REF] Wang | Action recognition by dense trajectories[END_REF], albeit with a different configuration of the HOF descriptor. Our best setup is obtained for a grid of 3 × 3 cells and a time grid of 2 cells. With SoPARF and SoPARF+Sine descriptors, we obtain significantly better results (83.9% and 83.6% resp.) than HOF (80.4%).

Comparison of descriptors computational time

We compare the computation of our four best setups to the computation time of the best HOF descriptor in the previous evaluation. The computation of descriptors is performed with an Intel(R) Xeon(R) E5-2620 0 @ 2.00GHz processor. We compute all the descriptors of the video called "v biking 01 01" of the UCF11 dataset.

In order to be fair in comparison, we use the same space-time dimensions of descriptors. We use a dimen- The results of computational time by frame (in seconds) are reported in Table 1. As one can see, descriptors that don't use rectification coding are comparable to HOF in computational time. When we use the rectification coding, the computational time clearly increase. However, the best descriptors in our evaluation is the SoPARF and its computational time is still acceptable for real datasets.

Experimental results

In this section we compare descriptors to the literature on the two datasets. For each dataset, we show the results with our SoPARF and SoPARF+Sine descriptors alone and with a HOG and MBH descriptors On Table 3, we show the results obtained on Hol-lywood2 dataset. With our SoPARF descriptor, we obtain better results than the related HOG, HOF and MBH descriptors of [START_REF] Wang | Action recognition by dense trajectories[END_REF] and than our implementation of HOG, HOF and MBH descriptors. Especially, we improve by 4% the HOF of Wang et al. and by 6% our impementation of HOF. The SoPARF improves the SoPAF by 3%, although this comes at the price of slightly increasing the computational time and dimension of the resulting descriptor. When combining SoPARF with HOG and HOF, we obtain a mAP of 58.6% with linear classifier, slightly better than the results obtain by combining 4 descriptors in [START_REF] Wang | Action recognition by dense trajectories[END_REF].

Conclusion

In this paper, we introduced a novel family of local motion descriptors using polynomial approximations of the optical flow and time evolution modeling. For a given spatial window, after projecting the components of the optical flow on an orthogonal bivariate polynomial basis, we model the temporal evolution of spatial coefficients with one dimension polynomial basis. In order to model homogenous motion patterns, our space-time volumes follows trajectories of associated image patches. The use of the half-wave rectification coding improve the results of SoPAF descriptor. Moreover, we show the the possibility of using other basis for modeling the time evolution of spatial coefficients.

We carry out experiments on the well known UCF11 and Hollywood2 datasets, using recent signatures method from image classification techniques. We obtain improved results over popular descriptors such as HOG, HOF and MBH which highlight the soundness of the approach.

4 Olivier

 4 Kihl et al.

WFig. 1 :

 1 Fig. 1: Localisation in space and space-time domains. (a) Localisation in space domain; (b) Localization example in space-time domain with τ = 3.

Fig. 2 :

 2 Fig. 2: Half-wave rectification

Fig. 5 :

 5 Fig. 5: Evaluation of space and time degree for our descriptor on UCF11 dataset; The horizontal axis represents the degree of the temporal functions basis and the vertical axis represents the average accuracy

Fig. 6 :Fig. 3 :

 63 Fig. 6: Evaluation of HOF descriptor on UCF11 dataset ; The horizontal axis represent the number of cells along the time axis and the vertical axis represent the average accuracy

Fig. 4 :

 4 Fig. 4: Example of videos from Hollywood2 dataset

Table 1 :

 1 Computational time of the best descriptors from the evaluation presented in section 4.2

	descriptor	parameters	parameters	times/frame
	HOF	3 × 3 cells	2 cells	0.68 s
	SoPAF	poly degree 4 poly degree 1	0.85 s
	SoPAFs	poly degree 3	sine degree 1	0.62 s
	SoPARF	poly degree 4 poly degree 2	1.12 s
	SoPARFs poly degree 4	sine degree 1	1.26 s
	combination. Let us note that our approach uses linear
	classifiers, and thus leads to better efficiency both for
	training classifiers and classifying video shots, on the
	contrary to methods [38] and [11].
	On

Table 2 ,

 2 we show the results obtained on UCF11 dataset, and compare them to recent results from the literature. We obtain good results only using the proposed SoPARF or SoPARF+Sine descriptors. The SoPARF improves the results of Wang et al. HOF descriptor by 11% and our implementation of HOF by 3%. The SoPARF provides the same results than the MBH of Wang et al. and improve by 4% the SoPAF. When using SoPARF, HOG and MBH combination or SoPARF+Sine, HOG and MBH combination we obtain 86.0% of average accuracy, which is above state of the art performances while using a linear classifier and combining less descriptors.

Table 2 :

 2 Classification average accuracy on the UCF11 dataset ; ND means the number of descriptors used ; NL stands for non-linear classifiers

	Method	ND NL Results
	Ikizler [14]	6		75.2%
	Wang [38](trajectory)	1	X	67.2%
	Wang [38](HOG)	1	X	74.5%
	Wang [38](HOF)	1	X	72.8%
	Wang [38](MBH)	1	X	83.9%
	Wang [38](all)	4	X	84.2%
	HOG	1		81.1%
	HOF	1		80.4%
	MBH	1		83.1%
	SoPAF	1		79.6%
	SoPAF+Sine	1		79.7%
	SoPARF	1		83.9%
	SoPARF+Sine	1		83.6%
	HOG+HOF+MBH	3		84.7%
	SoPARF+HOG+MBH	3		86.0%
	(SoPARF+Sine)+HOG+MBH	3		86.0%

Table 3 :

 3 Mean Average Precision on the Hollywood2 dataset ; ND : number of descriptors ; NL : non-linear classifiers ; ⋆ In[START_REF] Ullah | Improving bag-offeatures action recognition with non-local cues[END_REF] HOG/HOF descriptors are accumulated on over 100 spatio-temporal regions each one leading to a different BoW signature

	Method	ND	NL Results
	Gilbert [11]	≃ 3	X	50.9%
	Ullah [37] HOG+HOF	2	X	51.8%
	Ullah [37]	2(≥ 100 ⋆ )	X	55.3%
	Wang [38] traj	1	X	47.7%
	Wang [38] HOG	1	X	41.5%
	Wang [38] HOF	1	X	50.8%
	Wang [38] MBH	1	X	54.2%
	Wang [38] all	4	X	58.3%
	HOG	1		49.6%
	HOF	1		48.4%
	MBH	1		53.1%
	SoPAF	1		51.3%
	SoPARF	1		54.8%
	SoPARF+Sine	1		53.7%
	HOG+HOF+MBH	3		56.4%
	SoPARF+HOG+MBH	3		58.6%
	SoPARF(+Sine)+HOG+MBH	3		58.5%

Further improvement would be to use this framework to model gradient field of images or optical flow as in HOG and MBH and extending the coding step with other approaches.