Jean Dolbeault
email: dolbeaul@ceremade.dauphine.fr

Maria J Esteban
email: esteban@ceremade.dauphine.fr

Micha L Kowalczyk
email: kowalczy@dim.uchile.cl

IMPROVED INTERPOLATION INEQUALITIES ON THE SPHERE

Keywords: 2010 Mathematics Subject Classification. 26D10, 46E35, 58E35 Sobolev inequality, interpolation, Gagliardo-Nirenberg inequalities, logarithmic Sobolev inequality, heat equation, hypercontractivity, spectral decomposition

This paper contains a review of available methods for establishing improved interpolation inequalities on the sphere for subcritical exponents. Pushing further these techniques we also establish some new results, clarify the range of applicability of the various existing methods and state several explicit estimates.

where µ is the normalized measure on S d induced by the Euclidean measure on R d+1 and p ∈ [1, 2) ∪ (2, 2 * ) with 2 * = ∞ if d ≤ 2 and 2 * = 2 d d-2 if d ≥ 3. The case p = 2 * is also covered if d ≥ 3 and corresponds to Sobolev's inequality. When p = 1, the inequality is equivalent to the Poincaré inequality. By taking the limit as p → 2, we recover the logarithmic Sobolev inequality

S d u 2 log u 2 u 2 L 2 (S d ) dµ ≤ 2 d ∇u 2 L 2 (S d ) ∀ u ∈ H 1 (S d ) . (2)
The constant d in (1) is optimal: see for instance [START_REF] Dolbeault | Nonlinear Flows and Rigidity Results on Compact Manifolds[END_REF]. When p = 2, it is consistent to define e as the l.h.s. in [START_REF] Arnold | Refined convex Sobolev inequalities[END_REF], that is, e := S d u 2 log u 2 / u 2 L 2 (S d ) dµ. Hence for p ∈ (1, 2 * ), the functionals e and i are respectively the generalized entropy and Fisher information functionals.

In this paper we are interested in improvements of (1) and [START_REF] Arnold | Refined convex Sobolev inequalities[END_REF] in the subcritical range, that is, for 1 < p < 2 * . By improved inequality we mean an inequality of the form

d u 2 L 2 (S d ) Φ e u 2 L 2 (S d ) ≤ i ∀ u ∈ H 1 (S d ) , (3)
for some monotone increasing function Φ such that Φ(0) = 0, Φ (0) = 1 and Φ(s) > s for any s. As a straightforward consequence we get a stability result. Indeed, let us set Ψ(s) := s -Φ -1 (s) .

(4) Hence we get

i -d e = d u 2 L 2 (S d ) Φ -1 i d u 2 L 2 (S d ) -e + d u 2 L 2 (S d ) Ψ i d u 2 L 2 (S d )
, and thus,

i -d e ≥ d u 2 L 2 (S d ) Ψ i d u 2 L 2 (S d )
, where the inequality is a simple consequence of (3). If, additionally, Ψ is nondecreasing, then by reapplying (3) we find that

i -d e ≥ d u 2 L 2 (S d ) (Ψ • Φ) e u 2 L 2 (S d ) ∀ u ∈ H 1 (S d ) . (5)
The function Ψ • Φ is nondecreasing, positive on (0, ∞) and such that (Ψ • Φ)(0) = (Ψ • Φ) (0) = 0. Inequality ( 5) is a stability result since e controls a distance to the optimal functions, which are the constant functions. Our goal is to find the best possible function Φ.

As an important application of the improved version of the inequalities, we can point stability issues. Some straightforward consequences are:

1. the uniqueness of optimal functions when they exists, and a better characterization of the equality cases in the inequalities, 2. stability results in spectral theory with application to problems arising from quantum mechanics, like stability of matter, 3. some additional estimates in variational methods (improved convergence of sequences) with applications for instance when one uses Lyapunov-Schmidt reduction methods, 4. improved convergence rates in evolution problems. The last point is probably the most important in view of applications in physics. In various cases of interest, it allows to prove that before entering an asymptotic regime of, for instance, exponential decay, a system may have an initial regime with an even faster convergence.

Let us briefly review the literature and give some indications of our motivation and main results. Inequality [START_REF] Arnold | Interpolation between logarithmic Sobolev and Poincaré inequalities[END_REF] has been established in [START_REF] Bidaut-Véron | On characterization of solutions of some nonlinear differential equations and applications[END_REF][START_REF] Bidaut-Véron | Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations[END_REF][START_REF]Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality[END_REF]. The limit case p = 2 was known earlier: see for instance [START_REF] Weissler | Logarithmic Sobolev inequalities and hypercontractive estimates on the circle[END_REF] in the case of the circle, and [START_REF] Dolbeault | Sharp interpolation inequalities on the sphere: New methods and consequences[END_REF] for a detailed list of references.

In the case of compact manifolds other than the sphere, the estimates obtained by M.-F. Bidaut-Véron and L. Véron in [START_REF] Bidaut-Véron | Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations[END_REF] (also see [START_REF] Licois | Un théorème d'annulation pour des équations elliptiques non linéaires sur des variétés riemanniennes compactes[END_REF][START_REF] Licois | A class of nonlinear conservative elliptic equations in cylinders[END_REF][START_REF] Bakry | Sobolev inequalities and Myers's diameter theorem for an abstract Markov generator[END_REF][START_REF] Fontenas | Sur les constantes de Sobolev des variétés riemanniennes compactes et les fonctions extrémales des sphères[END_REF][START_REF]Sur les minorations des constantes de Sobolev et de Sobolev logarithmiques pour les opérateurs de Jacobi et de Laguerre[END_REF]) and by J. Demange in [START_REF] Demange | Des équations à Diffusion Rapide aux Inégalités de Sobolev sur les Modèles de la Géométrie[END_REF][START_REF]Improved Gagliardo-Nirenberg-Sobolev inequalities on manifolds with positive curvature[END_REF] can be improved at the leading order term by considering non-local quantities like in [START_REF] Dolbeault | Nonlinear Flows and Rigidity Results on Compact Manifolds[END_REF].

In the case of the sphere, the leading order term is determined by the constant d in [START_REF] Arnold | Interpolation between logarithmic Sobolev and Poincaré inequalities[END_REF]. Looking for improvements therefore makes sense. J. Demange observed in [START_REF]Improved Gagliardo-Nirenberg-Sobolev inequalities on manifolds with positive curvature[END_REF] that Inequality (1) can be improved when p ∈ (2, 2 * ). Moreover, in [START_REF] Demange | Des équations à Diffusion Rapide aux Inégalités de Sobolev sur les Modèles de la Géométrie[END_REF] he noticed the existence of a free parameter. The first purpose of this paper is to clarify the range of the free parameter in the method and optimize on it, in order to get the best possible improvement with respect to that parameter. Recent contributions on nonlinear flows and Lyapunov functionals, or entropies, that can be found in [START_REF] Dolbeault | Sharp interpolation inequalities on the sphere: New methods and consequences[END_REF][START_REF] Dolbeault | Nonlinear Flows and Rigidity Results on Compact Manifolds[END_REF][START_REF] Dolbeault | One-Dimensional Gagliardo-Nirenberg-Sobolev Inequalities: Remarks on Duality and Flows[END_REF], are at the core of our method.

Refined convex Sobolev inequalities have been established in [START_REF] Arnold | Refined convex Sobolev inequalities[END_REF] when p ∈ (1, 2) in the setting of interpolation inequalities involving a probability measure. For a simpler formulation, see [START_REF] Dolbeault | On the Bakry-Emery criterion for linear diffusions and weighted porous media equations[END_REF]. Our second purpose is to adapt the method to interpolation inequalities on the sphere and hence also cover the range p ∈ (1, 2). It is based on refined estimates of entropy decay for a linear heat flow, in the spirit of the Bakry-Emery method.

Our last contribution is inspired by [START_REF] Arnold | Interpolation between logarithmic Sobolev and Poincaré inequalities[END_REF]. Under additional orthogonality conditions, we show that other (and in some cases better) improved inequalities can be established in the range p ∈ (1, 2). The method is based on hypercontractivity estimates for a linear heat flow and a spectral decomposition. It raises an intriguing open question on the possibility of obtaining improvements under orthogonality conditions in the range p > 2. Now let us explain the strategy of the paper. 1) Standard symmetrization techniques allow to decrease the function i while preserving the L 2 and L p norms, and the functional e as well. Thus, there are optimal functions for inequalities (3) and ( 5) which depend only on the azimuthal angle on the sphere and the interpolation inequalities are therefore equivalent to one-dimensional inequalities for the d-ultraspherical operator, where d can now be considered as a real parameter. Details are given in Section 2. We should point out that this first step simplifies the calculations needed to get an improved inequality, but it is not fundamental for our method. After a change of variables we are led to the following expressions

i = 1 -1 |f | 2 ν dν d and e = 1 -1 |f | p dν d 2 p - 1 -1 |f | 2 dν d p -2 ,
where ν d is a probability measure, and ν ≥ 0 is a smooth function (these functions are explicit but their exact form is irrelevant for the moment). Then we define the self-adjoint ultraspherical operator through the identity

1 -1 f 1 f 2 ν dν d = - 1 -1 Lf 1 f 2 ν dν d .
The natural function space for our inequalities is the form domain of L, that is

H := f ∈ L 2 (-1, 1), dν d : 1 -1 |f | 2 ν dν d < ∞ ,
and we shall denote by f q the L q (-1, 1), dν d norm of f .

2) The key to our approach is to combine the ideas of D. Bakry and M. Emery, i.e., take the derivative of i -d e along some flow, with ideas that go back to B. Gidas and J. Spruck in [START_REF] Gidas | Global and local behavior of positive solutions of nonlinear elliptic equations[END_REF] and that were later exploited by M.-F. Bidaut-Véron and L. Véron for getting rigidity results in nonlinear elliptic equations. An unessential but useful trick amounts to write the flow for w with f = w β for some β ∈ R, in the expressions for i and e, as we shall see below.

Let us start with the case β = 1. We consider the manifold

M p = {w ∈ H : w p = 1} .
The main issue is to choose the right flow for our setting. We observe that if κ = p -1, then M p is invariant under the action of the flow

w t = Lw + κ |w | 2 w . (6)
Notice that g = w p evolves according to the equation g t = Lg and we shall therefore refer to the case associated with this equation as the linear case, or the 1homogeneous case. At the level of w, the equation is indeed 1-homogeneous since λ w is a solution to (6) for any λ > 0 if w is a solution to [START_REF] Bakry | Hypercontractivité de semi-groupes de diffusion[END_REF]. Let

2 := 2 d 2 + 1 (d -1) 2 and γ 1 := d -1 d + 2 2 (p -1) (2 # -p) if d > 1 , γ 1 := p -1 3 if d = 1 . (7)
If p ∈ [1, 2) ∪ (2, 2 ]
and w is a solution to (6), then

d dt (i -d e) ≤ -γ 1 1 -1 |w | 4 w 2 dν d ≤ -γ 1 |e | 2 1 -(p -2) e .
Recalling that e = -i, we get a differential inequality

e + d e ≥ γ 1 |e | 2 1 -(p -2) e ,
which after integration implies an inequality of the form d Φ(e(0)) ≤ i(0) .

Details will be given in Sections 3.1 and 5.

3) Now let us consider the case with a general β. The range of p's for which an improved inequality is valid can be extended to any p ∈ (1, 2 * ) by considering the nonlinear flow

w t = w 2-2β Lw + κ |w | 2 w ( 8
)
for some β ≥ 1, with κ = β (p-2)+1. Then M βp is invariant under the flow and our improved functional inequality follows from the computation of d dt (i -d e) written for f = w β , with the additional difficulty that e now differs from -i. Details on (8) will be given in Section 3 and improved inequalities will be established in Section 4. The change of function f = w β , for some parameter β ∈ R, is convenient to compute d dt (i -d e) but also sheds light on the strategy used for proving rigidity according to the method of [START_REF] Bidaut-Véron | Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations[END_REF]. It moreover shows that the computations are equivalent and explains why a local bifurcation result from constant functions can be extended to a global uniqueness property. This is because the flow relates any initial datum to the constants through the monotonically decreasing quantity i -d e.

In his thesis, J. Demange [START_REF] Demange | Des équations à Diffusion Rapide aux Inégalités de Sobolev sur les Modèles de la Géométrie[END_REF] made a computation which is similar to ours. Compared to his approach, we work in a setting in which the change of function f = w β clarifies the relation of flow methods with rigidity results in nonlinear elliptic PDEs. We give an explicit range for the parameter β and numerically observe that there is no optimal choice of β valid for an arbitrary value of the entropy e when p > 2. We also give a new result when p ∈ (1, 2) and it turns out that only β = 1 can be used in that range. 4) Our last improvement is of a different nature. If p ∈ (1, 2) and assuming that the function u is in the orthogonal of the eigenspaces associated with the lowest positive eigenvalues of the Laplace-Beltrami operator ∆ S d on the sphere, using Nelson's hypercontractivity result, it is possible to obtain another improved inequality.

Here we again use the linear flow (6) corresponding to β = 1. Although it is a rather straightforward adaptation of [START_REF] Arnold | Interpolation between logarithmic Sobolev and Poincaré inequalities[END_REF], such an approach raises an interesting open question. See Section 6.

The nonlinear flow defined by [START_REF] Bakry | Inégalités de Sobolev pour un semi-groupe symétrique[END_REF] and by [START_REF] Bakry | Hypercontractivité de semi-groupes de diffusion[END_REF] when β = 1 is the main conceptual tool for our analysis. It is introduced in Section 3. Explicit stability results rely on generalized Csiszár-Kullback-Pinsker type inequalities, which are detailed in Section 7. Most of our statements, when written for the ultraspherical operator, are valid for d taking real values and our proofs can be adapted without changes. However, for simplicity, we shall assume that d is an integer throughout this paper, unless explicitly specified.

It is very likely that the improved inequalities presented in this paper are not optimal. What could be an optimal Φ (or even if such a question really makes sense) is an open question, but at least we can construct a whole collection of such functions (depending on p and d), which in most cases improve on previously known results. The strategy that we use to prove the improved inequalities builds up on some previous works, but the way we combine these ideas is new and suggests several directions for future investigations.

After all these preliminaries, we are now in position to state our results. Let us start by giving an expression of the function Φ. For each p ∈ (1, 2) ∪ (2, 2 ) and γ 1 defined by [START_REF] Bakry | Diffusions hypercontractives[END_REF], let

ϕ 1 (s) :=      2 γ1+2 (p-2) 1 -(p -2) s - γ 1 2 (p-2) -1 + (p -2) s if p = 2 - γ 1 2 , 1 2-p 1 -(p -2) s log 1 -(p -2) s if p = 2 - γ 1 2 , (9)
where we assume that s is admissible, that is, s > 0 if p ∈ (1, 2) and s ∈ (0,

1 p-2 ) if p > 2. Next, we define γ(β) := - d -1 d + 2 (κ + β -1) 2 + κ (β -1) + d d + 2 (κ + β -1) . ( 10
)
We observe that γ(1) = γ 1 , and that there exists some

β ∈ R such that γ(β) > 0 only if p ≥ 1 if d = 1 or if p ∈ [2, 2 * ] for any d > 1.
Then, we define B(p, d) by

B(p, d) = β ∈ R : γ(β) > 0 , β ≥ 1 , and β ≤ 2 4-p if p < 4 if p > 2 , B(p, d) = {1} if 1 ≤ p ≤ 2 . (11)
A more explicit description of the region B(p, d) will be given in the Appendix. For each β ∈ B(p, d) such that β > 1, let

ϕ β (s) := s 0 exp γ(β) β (β-1) p (1 -(p -2) z) 1-δ(β) -(1 -(p -2) s) 1-δ(β) dz (12)
with

δ(β) := p -(4 -p) β 2 β (p -2) . ( 13
)
The reader is invited to check that lim β→1+ ϕ β (s) = ϕ 1 (s) for any admissible s.

ϕ β (s) . ( 14
)
Theorem 1.1. Assume that one of the following conditions is satisfied:

(i) d = 1 and p ∈ (1, 2) ∪ (2, ∞), (ii) d = 2 and p ∈ (1, 2) ∪ (2, 9 + 4 √ 3), (iii) d ≥ 3 and p ∈ (1, 2) ∪ (2, 2 * ). For any u ∈ H 1 (S d ) be such that u L 2 (S d ) = 1, we have the inequality d Φ u 2 L p (S d ) -1 p -2 ≤ ∇u 2 L 2 (S d ) ,
where Φ is defined by [START_REF]Sur les fonctions extrémales des inégalités de Sobolev des opérateurs de diffusion[END_REF]. Moreover Φ(0) = 0, Φ (0) = 1 and Φ (s) > 0, for a.e. s > 0 if p > 2 or for a.e. s ∈ (0, 1/(2 -p)) if p ∈ (1, 2). If p = 2, then for any u ∈ H 1 (S d ) the following improved logarithmic Sobolev inequality

S d u 2 log |u| 2 u 2 L 2 (S d ) dµ ≤ 4 γ * 1 u 2 L 2 (S d ) log 1 + γ * 1 2d ∇u 2 L 2 (S d ) u 2 L 2 (S d )
holds with γ * 1 = 4 d-1 (d+2) 2 . The reader interested in best constants in logarithmic Sobolev inequalities as in Theorem 1.1 is invited to refer to [START_REF] Brouttelande | The best-constant problem for a family of Gagliardo-Nirenberg inequalities on a compact Riemannian manifold[END_REF][START_REF]On the second best constant in logarithmic Sobolev inequalities on complete Riemannian manifolds[END_REF]. In the case p = 2, a first consequence of the above improved inequality, compared to the standard inequality i ≥ d e, is that i -d e is not only nonnegative, but that it actually measures a distance to the constants in the homogeneous Sobolev norm. With previous notations, one can indeed state that

∇u 2 L 2 (S d ) - d p -2 u 2 L p (S d ) -1 ≥ d Ψ 1 d ∇u 2 L 2 (S d ) for any u ∈ H 1 (S d ) such that u L 2 (S d ) = 1
, where Ψ is defined by (4). We can rephrase this result without normalization as the following corollary.

Corollary 1. Assume that p ∈ (1, 2) ∪ (2, 2 * ).
With the notations of Theorem 1.1, we have

∇u 2 L 2 (S d ) - d p -2 u 2 L p (S d ) -u 2 L 2 (S d ) ≥ d u 2 L 2 (S d ) Ψ ∇u 2 L 2 (S d ) d u 2 L 2 (S d )
,

where Ψ is defined by (4) in terms of Φ, and Φ is given by ( 14).

Now we can use a generalized Csiszár-Kullback-Pinsker inequality to get an estimate of the distance to the constants using a standard Lebesgue norm. We shall distinguish three cases: (i) p ∈ (1, 2), (ii) p ∈ (2, 4) , and (iii) p ≥ 4. Let us define q(p) := 2/p, p/2 and p/(p -2) in cases (i), (ii) and (iii) respectively. Assume also that r(p) := p, 2 and p -2 in cases (i), (ii) and (iii) respectively and let s(p) := max{2, p}.

Proposition 1. Assume that p ∈ (1, 2) ∪ (2, ∞). With the above notations and (q, r, s) = (q(p), r(p), s(p)), there exists a positive constant C, depending only on p, such that for any u ∈ L 1 ∩ L p (Ω, dµ), we have

1 p -2 u 2 L p (S d ) -u 2 L 2 (S d ) ≥ C u 2 (1-r) L s (S d ) u r -ūr 2 L q (S d )
where ū = u L r (S d ) .

The case p = 2 appears as a limit case and corresponds to the standard Csiszár-Kullback-Pinsker inequality. Details on this limit and an explicit estimate of C for all p ≥ 1 will be given in Section 7 (see Corollary 4). Notice that C = 1 when p = 1, and the inequality in Proposition 1 is thus equivalent to a Poincaré inequality. A direct consequence of Proposition 1 and Inequality ( 5) is the following stability result.

Corollary 2. With the notations of Theorem 1.1 and Proposition 1, we have

∇u 2 L 2 (S d ) - d p -2 u 2 L p (S d ) -u 2 L 2 (S d ) ≥ d u 2 L 2 (S d ) (Ψ • Φ) C u 2 (1-r) L s (S d ) u 2 L 2 (S d ) u r -ūr 2 L q (S d ) ∀ u ∈ H 1 (S d ) .
Now let us turn our attention to the statements corresponding to the last class of improvements studied in this paper. The range of p is now restricted to [START_REF] Arnold | Interpolation between logarithmic Sobolev and Poincaré inequalities[END_REF][START_REF] Arnold | Refined convex Sobolev inequalities[END_REF] and we consider the linear flow [START_REF] Bakry | Hypercontractivité de semi-groupes de diffusion[END_REF].

In [START_REF] Beckner | A generalized Poincaré inequality for Gaussian measures[END_REF], W. Beckner gave a method to prove interpolation inequalities between logarithmic Sobolev and Poincaré inequalities in the case of a Gaussian measure. The method extends to the case of the sphere as was proved in [START_REF] Dolbeault | Sharp interpolation inequalities on the sphere: New methods and consequences[END_REF], in the range p ∈ [1, 2), with optimal constants. For further considerations on inequalities that interpolate between Poincaré and logarithmic Sobolev inequalities, we refer to [START_REF] Arnold | Refined convex Sobolev inequalities[END_REF][START_REF] Arnold | Interpolation between logarithmic Sobolev and Poincaré inequalities[END_REF][START_REF] Bakry | Diffusions hypercontractives[END_REF][START_REF] Bakry | Hypercontractivité de semi-groupes de diffusion[END_REF][START_REF] Bolley | Phi-entropy inequalities and Fokker-Planck equations[END_REF][START_REF]Phi-entropy inequalities for diffusion semigroups[END_REF][START_REF] Chafai | Entropies, convexity, and functional inequalities: On Φ-entropies and Φ-Sobolev inequalities[END_REF][START_REF] Lata | Between Sobolev and Poincaré[END_REF] and references therein.

Our purpose is to obtain an improved estimate of the optimal constant C p in

S d |∇u| 2 dµ ≥ C p S d |u| 2 dµ - S d |u| p dµ 2/p ∀ u ∈ H 1 (S d , dµ) . (15)
Without any constraint on u, we have C p = d 2-p , and it is natural to expect that this constant will be improved when we impose additional constraints on the set of admissible u's. It the present case we will assume that u is in the orthogonal complement of the finite dimensional subspace spanned by the spherical harmonics corresponding to the lowest positive eigenvalues of the Laplace-Beltrami operator ∆ S d on the sphere. Under this additional hypothesis we will obtain an improvement of the estimate [START_REF] Bentaleb | A family of integral inequalities on the circle S 1[END_REF], similar to what was done [START_REF] Arnold | Interpolation between logarithmic Sobolev and Poincaré inequalities[END_REF], in a different setting.

Let us introduce some notations and recall some known results. We consider ∆ S d as an operator on L 2 (S d , dµ) with domain H 2 (S d , dµ), whose eigenvalues are

λ k = k (k + d -1) ∀ k ∈ N ,
and denote by E k the corresponding eigenspaces. Recall that

dim(E k ) = (k + d -2)! k! (d -1)! (2 k + d -1) ∀ k ≥ 0 ,
α k := 1 d λ k+1 ,
and functions

Φ k (s) = max α k 1 -(p -1) α k (1 -s), log s log(p -1)
.

With these notations we have:

Theorem 1.2. Assume that p ∈ (1, 2). If u ∈ H 1 (S d ) is such that u L 2 (S d ) = 1 and S d u e dµ = 0 ∀ e ∈ E j , j = 1 , 2 . . . k , (16)
then we have

∇u 2 L 2 (S d ) ≥ d Φ k u 2 L p (S d ) .
We may notice that α → α 1-(p-1) α is an increasing function of α > 1 and thus larger that

α0 1-(p-1) α 0 = 1 2-p since α 0 = 1. Hence for any k ≥ 1, we have d α k 1 -(p -1) α k > d 2 -p ,
thus showing that we have achieved a strict improvement of the constant compared to the one in [START_REF] Bentaleb | A family of integral inequalities on the circle S 1[END_REF] (which is optimal without further assumption). Moreover, lim s→0+ Φ k (s) = +∞, thus showing that when the orthogonality conditions are satisfied, the estimate of Theorem 1.2 is strictly better than the one of Theorem 1.1. See Fig. 4 for an illustration.

2. Symmetrisation and the ultraspherical operator. As we have pointed out in the introduction, the improved inequality on the d-dimensional sphere can be reduced to the inequalities for functions depending only on the azimuthal angle. The interested reader can refer to [START_REF] Dolbeault | Sharp interpolation inequalities on the sphere: New methods and consequences[END_REF]. Alternatively, let us give a sketch of a proof for completeness. More details on the stereographic projection can be found in [START_REF] Dolbeault | Spectral estimates on the sphere, to appear in Analysis & PDE[END_REF]Appendix B.3] We denote by ξ = (ξ 0 , ξ 1 , . . . ξ d ) the coordinates of an arbitrary point in the unit sphere S d ⊂ R d+1 . Consider the stereographic projection Σ :

S d \ {N} → R d , where N denotes the North Pole, that is the point in S d corresponding to ξ d = 1. Here Σ(ξ) = x means x = (1 -ξ d ) -1 (ξ 0 , ξ 1 , . . . ξ d-1 ), and to any u ∈ H 1 (S d ) we associate a function σ[u] := v ∈ H 1 (R d ) such that u(ξ) = (1 -ξ d ) -d-2 2 v(x) ∀ ξ ∈ S d and x = Σ(ξ) . An elementary computation shows that ∇u 2 L 2 (S d ) + d (d-2) 4 u 2 L 2 (S d ) = ∇v 2 L 2 (R d ) , u 2 L 2 (S d ) = 4 (1 + |x| 2 ) -2 v 2 L 2 (R d ) and u 2 L p (S d ) = (2/(1 + |x| 2 )) 1+ d p -d 2 v 2 L p (R d ) .
To v we may apply the standard Schwarz symmetrization and denote the symmetrized function by v * . Let us define

u * := σ -1 (σ[u]) * . Lemma 2.1. Assume that u ∈ H 1 (S d ). Then we have i[u] = S d |∇u| 2 dµ ≥ i[u * ] and for any q ∈ [1, 2 * ], u L q (S d ) = u * L q (S d ) , so that e[u] = e[u * ] .
This symmetry result is a kind of folklore in the literature and we can quote [START_REF] Baernstein | Spherical rearrangements, subharmonic functions, and * -functions in n-space[END_REF][START_REF] Lieb | Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities[END_REF][START_REF]the Poisson semigroup, and analysis on the sphere S n[END_REF] for various related results. Details of the proof are left to the reader. As a straightforward consequence, inf i[u]/Φ(e[u]) is achieved by functions depending only on ξ d , or on the azimuthal angle θ.

Thus, to prove the inequality in Theorem 1.1 it suffices to prove it for

e = 1 p -2 π 0 |v(θ)| p dσ 2 p - π 0 |v(θ)| 2 dσ , i = π 0 |v (θ)| 2 dσ,
and for any function v ∈ H 1 ([0, π], dσ), where

dσ(θ) := (sin θ) d-1 Z d dθ with Z d := √ π Γ( d 2 ) Γ( d+1 2 )
.

The change of variables x = cos θ, v(θ) = f (x) allows to rewrite the expressions for e and i:

e = 1 p -2 1 -1 |f | p dν d 2 p - 1 -1 |f | 2 dν d ≥ , i = 1 -1 |f | 2 ν dν d ,
where dν d is the probability measure defined by

ν d (x) dx = dν d (x) := Z -1 d ν d 2 -1 dx with ν(x) := 1 -x 2 , Z d = √ π Γ( d 2 ) Γ( d+1 2 )
.

We consider the space L 2 ((-1, 1), dν d ) equipped with the scalar product

f 1 , f 2 = 1 -1 f 1 f 2 dν d ,
and recall that the ultraspherical operator is given by

f 1 , Lf 2 = - 1 -1 f 1 f 2 ν dν d . Explicitly we have: Lf := (1 -x 2 ) f -d x f = ν f + d 2 ν f .
With these notations, for any positive smooth function u on (-1, 1), Inequalities ( 1) and ( 2) can be rewritten as

-f, Lf = 1 -1 |f | 2 ν dν d ≥ d f 2 p -f 2 2 p -2 , -f, Lf ≥ d 2 1 -1 |f | 2 log |f | 2 f 2 2 dν d , if p ∈ [1, 2) ∪ (2, 2 *
) and p = 2, respectively. In the framework of the ultraspherical operator, the parameter d can be considered as a positive real parameter, with critical exponent 2

* = 2 d d-2 if d > 2, and 2 * = ∞ if d ≤ 2.
We refer to [START_REF] Mueller | Hypercontractivity for the heat semigroup for ultraspherical polynomials and on the n-sphere[END_REF][START_REF] Bakry | Une suite d'inégalités remarquables pour les opérateurs ultrasphériques[END_REF][START_REF] Bentaleb | A family of integral inequalities on the circle S 1[END_REF][START_REF] Bentaleb | Inégalité de Sobolev pour l'opérateur ultrasphérique[END_REF][START_REF]Sur les fonctions extrémales des inégalités de Sobolev des opérateurs de diffusion[END_REF][START_REF] Fontenas | Sur les constantes de Sobolev des variétés riemanniennes compactes et les fonctions extrémales des sphères[END_REF][START_REF]Sur les minorations des constantes de Sobolev et de Sobolev logarithmiques pour les opérateurs de Jacobi et de Laguerre[END_REF][START_REF] Dolbeault | Sharp interpolation inequalities on the sphere: New methods and consequences[END_REF] for more references. The next lemma gives two elementary but very useful identities: Lemma 2.2. For any positive smooth function u on (-1, 1), we have

1 -1 ( Lw) 2 dν d = 1 -1 |w | 2 ν 2 dν d + d 1 -1 |w | 2 ν dν d , |w | 2 w ν, Lw = d d + 2 1 -1 |w | 4 w 2 ν 2 dν d -2 d -1 d + 2 1 -1 |w | 2 w w ν 2 dν d .

3.

Flows. As we explained in the introduction, the main step in our methodology is to take the derivative of the expression i -d e along the manifold M p of functions whose L p norm is equal to 1, following the evolution given by well-chosen flows. In this section we will describe a special flow which leaves this manifold invariant and we will carry out the computation of the derivative.

3.1. The 1-homogeneous case (β = 1). On (-1, 1), let us consider the flow defined by [START_REF] Bakry | Hypercontractivité de semi-groupes de diffusion[END_REF], that is,

w t = Lw + κ |w | 2 w
, and notice that

d dt 1 -1 w p dν d = (κ -p + 1) 1 -1 w p-2 |w | 2 ν dν d , so that 1 -1 w p dν d is preserved if κ = p -1.
Recall that g = w p obeys to the linear equation g t = Lg. A straightforward computation (using the definition of L and Lemma 2.2) shows that 1 2

d dt 1 -1 |w | 2 ν + d p -2 |w| 2 -w 2 dν d = - 1 -1 |w | 2 ν 2 dν d + 2 d -1 d + 2 κ 1 -1 w |w | 2 w ν 2 dν d - d d + 2 κ 1 -1 |w | 4 w 2 ν 2 dν d , since w = 1 -1 w p dν d 1/p
is independent of t. The r.h.s. is negative if

-γ 1 = d -1 d + 2 κ 2 - d d + 2 κ ≤ 0 , that is, if p ≤ 2 := 2 d 2 +1 (d-1) 2 when d > 1, or p > 1 when d = 1
, and this determines the expression (7) of γ 1 . We have proved the following result.

Proposition 2. For all p ∈ [1, 2 ] if d > 1, p > 1 if d = 1, there exists a constant γ 1 > 0, such that if w(t) is defined by (6), then d dt 1 -1 w p dν d = 0 , - d dt 1 -1 |w | 2 ν + d p -2 w 2 -w 2 dν d ≥ 2 γ 1 1 -1 |w | 4 w 2 ν 2 dν d ,
where γ 1 is given by (7).

3.2. The nonlinear case (β ≥ 1). On (-1, 1), let us consider the flow defined by [START_REF] Bakry | Inégalités de Sobolev pour un semi-groupe symétrique[END_REF], that is,

w t = w 2-2β Lw + κ |w | 2 w
, and notice that

d dt 1 -1 w βp dν d = β p (κ -β (p -2) -1) 1 -1 w β(p-2) |w | 2 ν dν d , so that w = 1 -1 w βp dν d 1/(βp) is preserved if κ = β (p -2) + 1.
Similarly as in the previous case we calculate:

- 1 2 β 2 d dt 1 -1 |(w β ) | 2 ν + d p -2 w 2β -w 2β dν d = 1 -1 |w | 2 ν 2 dν d -2 d -1 d + 2 (κ + β -1) 1 -1 w |w | 2 w ν 2 dν d + κ (β -1) + d d + 2 (κ + β -1) 1 -1 |w | 4 w 2 ν 2 dν d .
The r.h.s. is negative if there exists a β ∈ R such that

-γ = d -1 d + 2 (κ + β -1) 2 -κ (β -1) + d d + 2 (κ + β -1) = d -1 d + 2 β (p -1) 2 - d d + 2 β (p -1) + 1 + β (p -2) (β -1) ≤ 0 ,
i.e., γ given by ( 10) is nonnegative, and in that case we have found that

- 1 2 β 2 d dt 1 -1 |(w β ) | 2 ν + d p -2 w 2β -w 2β dν d ≥ γ 1 -1 |w | 4 w 2 ν 2 dν d . ( 17
)
This defines γ as in [START_REF] Beckner | A generalized Poincaré inequality for Gaussian measures[END_REF]. Since the l.h.s. of the inequality is quadratic in β and evaluates to +1 for β = 0, a necessary and sufficient condition is that the discriminant, which amounts to

4 d (d -2) (d + 2) 2 (p -1) (2 * -p) where 2 * := 2 d d -2 takes nonnegative values, that is 1 ≤ p ≤ 2 * if d ≥ 3 .
In dimension d = 2 and 1, the discriminant is respectively 2 (p -1) and 4 9 (p -1)(p + 2) and takes nonnegative values for any p ≥ 1 (we always assume that p ≥ 1). Altogether, we have proved the following result. Proposition 3. For all p ∈ [1, 2 * ], there exist two constants, β ∈ R and γ > 0, such that if w(t) is defined by (8), then

d dt 1 -1 w βp dν d = 0 , - 1 2β 2 d dt 1 -1 |(w β ) | 2 ν + d p -2 w 2β -w 2β dν d ≥ γ 1 -1 |w | 4 w 2 ν 2 dν d
, where γ is given by [START_REF] Beckner | A generalized Poincaré inequality for Gaussian measures[END_REF].

Note that the flow described in Section 3.1 is a special case of the flow defined by [START_REF] Bakry | Inégalités de Sobolev pour un semi-groupe symétrique[END_REF] corresponding to β = 1. We treat the cases β = 1 and β > 1 separately as they cover different ranges of p, namely (1, 2 # ) and (2, 2 * ) respectively. 4. Improved inequalities in the range p ∈ (2, 2 * ). In this section, we establish the expression of ϕ β that is used for stating Theorem 1.1. The computation corresponds to the one of J. Demange in [START_REF]Improved Gagliardo-Nirenberg-Sobolev inequalities on manifolds with positive curvature[END_REF] when β = 4 6-p . In [START_REF] Demange | Des équations à Diffusion Rapide aux Inégalités de Sobolev sur les Modèles de la Géométrie[END_REF], J. Demange noticed that there is a free parameter, which is equivalent to the parameter β in our setting. Our purpose is to clarify the range of admissibility of β and then optimize on it.

From here on, we assume that β > 1. In Appendix A we show that the set B(p, d) of admissible β's defined by ( 11) is non-empty if and only if one of the following conditions (Fig. 1) holds:

(i) d ≥ 3 , p ∈ (2, 2 * ) , (ii) d = 2 , p ∈ (2, 9 + 4 √ 3) , (iii) d = 1 , p > 2 ,
and thus from now on we will take for granted that one of these conditions holds. In order to get the improved inequality, we make use of [START_REF] Bianchi | A note on the Sobolev inequality[END_REF] to get a lower bound for d dt (-i + d e). We note that the factor γ is strictly positive for β ∈ B(p, d). However, notice that there exist β / ∈ B(p, d) such that γ(β) > 0. Additional restrictions on the set of the admissible β's indeed appear in what follows.

Let us show that we can get a lower bound for

1 -1 |w | 4
w 2 ν 2 dν d by two different methods, using three points interpolation Hölder's inequalities.

1. With 1 2 + β-1 2β + 1 2β = 1, Hölder's inequality shows that

1 -1 |w | 2 ν dν d = 1 -1 |w | 2 w ν • 1 • w dν d ≤ 1 -1 |w | 4 w 2 ν 2 dν d 1 2 1 -1 1 dν d β-1 2β 1 -1 w 2β dν d 1 2β
, from which we deduce that

1 -1 |w | 4 w 2 ν 2 dν d 1 2 ≥ 1 -1 |w | 2 ν dν d 1 -1 w 2β dν d 1 2β
, because dν d is a probability measure.

With

1 2 + β-1 β(p-2) + 2-(4-p)β 2β(p-2) = 1 and under the condition that 1 < β ≤ 2 4 -p if p < 4 , (18)
(but no condition if p ≥ 4, except β > 1), Hölder's inequality shows that

1 β 2 1 -1 |(w β ) | 2 ν dν d = 1 -1 w 2(β-1) |w | 2 ν dν d = 1 -1 |w | 2 w ν • w p(β-1) p-2 • w 2βδ dν d ≤ 1 -1 |w | 4 w 2 ν 2 dν d 1 2 1 -1 w βp dν d β-1 β(p-2) 1 -1 w 2β dν d δ with δ = δ(β) := p-(4-p) β 2 β (p-2)
as in [START_REF] Bentaleb | Inégalité de Sobolev pour l'opérateur ultrasphérique[END_REF], from which we deduce that

1 -1 |w | 4 w 2 ν 2 dν d 1 2 ≥ 1 β 2 1 -1 |(w β ) | 2 ν dν d 1 -1 w 2β dν d δ ,
because we have chosen 1 -1 w βp dν d = 1. By combining the two estimates we have proved the following. Lemma 4.1. Assume that (18) holds. For all w ∈ H 1 (-1, 1), dν d , such that

1 -1 w βp dν d = 1, 1 -1 |w | 4 w 2 ν 2 dν d ≥ 1 β 2 1 -1 |(w β ) | 2 ν dν d 1 -1 |w | 2 ν dν d 1 -1 w 2β dν d δ .
Next, we can use the above estimates to show that there is a differential inequality relating e and i. From the computations in Section 3.2, and Proposition 3, we find that

- 1 2 β 2 d dt 1 -1 |(w β ) | 2 ν + d p -2 w 2β -w 2β dν d ≥ γ 1 -1 |w | 4 w 2 ν 2 dν d with w = 1 -1 w p dν d 1/p = 1, hence -i + d e ≥ 2 γ 1 -1 |(w β ) | 2 ν dν d 1 -1 |w | 2 ν dν d 1 -1 w 2β dν d δ = - γ 2 β 2 i e (1 -(p -2) e) δ .
Let us suppose that ψ β is a function that satisfies the ODE

ψ β (e) ψ β (e) = - γ 2 β 2 (1 -(p -2) e) -δ ,
and such that ψ β > 0. It is elementary to show that such function exists when β > 1. The l.h.s. of the inequality

i -d e - γ 2 β 2 i e (1 -(p -2) e)
δ ≤ 0 can then be considered as a total derivative, namely

d dt i ψ β (e) -d ψ β (e) ≤ 0 , thus proving after an integration in t ∈ [0, ∞) that d ψ β (e 0 ) ψ β (e 0 ) ≤ i 0 .
Next, we let ϕ β (e) := ψ β (e) ψ β (e) .

It is then elementary to check that ϕ β satisfies the ODE

ϕ β = 1 -ϕ β ψ β (e) ψ β (e) = 1 + ϕ β γ 2 β 2 (1 -(p -2) e) -δ
and ϕ β (0) = 0. Solving this linear ODE, we find that ϕ β is given by [START_REF]Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality[END_REF]. Notice that ϕ β is defined for any e ∈ [0, 1 p-2 ) and ϕ β (e) > 0, for e > 0. From the equation satisfied by ϕ β we get that ϕ β (e) > 1 and ϕ β (e) > 0, e > 0, hence ϕ β (e) > e for any admissible β. As a consequence we also get that the functions ϕ -1 Numerically we observe that ϕ(e) = ϕ β(e) (e) for an optimal β = β(e), which explicitly depends on e. For future reference we observe that ϕ (e) > 1 and ϕ (e) > 0, for a.e. e > 0, hence ϕ(e) > e and the functions ϕ -1 and i → i -ϕ -1 (i) are increasing.

β and i → i -ϕ -1 β (i)
From the preceding considerations it follows that we have the inequality:

d ϕ(e) ≤ i .
Now we recall that this inequality is obtained under the assumption

u 2 L p (S d ) = 1, namely e = 1 p-2 1 -u 2 L 2 (S d ) .
Then, if we do not normalize the L p norm we get in general:

d u 2 L p (S d ) ϕ e u 2 L p (S d ) ≤ i .
Now, remembering the definition of the improved inequality, we need to find the relation between the l.h.s. of this estimate, which is a function of e/ u 2 L p (S d ) , and the function d u 2 L 2 (S d ) Φ e/ u 2 L 2 (S d ) . This is quite easy since we have

u 2 L p (S d ) u 2 L 2 (S d ) = 1 + (p -2) s with s = e u 2 L 2 (S d )
.

By straightforward manipulations we get from this

u 2 L p (S d ) ϕ e u 2 L p (S d ) = 1 + (p -2) s ϕ s 1 + (p -2) s =: Φ(s)
where Φ is the function under consideration in Theorem 1.1. Based on the properties of ϕ, it is easy to check that Φ(0) = 0, Φ (0) = 1 and Φ (s) > 0. In Fig. 2 we show the measure of improvement between the improved inequality and the standard inequality. This ends the proof of Theorem As a conclusion for this section, let us comment on the literature and emphasize the new results. In [START_REF]Improved Gagliardo-Nirenberg-Sobolev inequalities on manifolds with positive curvature[END_REF] J. Demange gives a proof by considering a flow which corresponds to [START_REF] Bakry | Inégalités de Sobolev pour un semi-groupe symétrique[END_REF], in the special case β = 4 6-p . Here we generalize it to a larger but explicit range of β's, as was done in [32, pages 122-130] (see in particular Proposition 3.12.5). Moreover we explicitly show how to optimize the interpolation inequalities with respect to β and we specify the range of admissible β's. See Appendix A for more details.

5.

Improved inequalities in the range p ∈ (1, 2 # ): Proof of Theorem 1.1 when p ∈ (1, 2) and when p = 2. In this section we adapt the Bakry-Emery approach (which amounts to write a differential inequality for i = -e ) and improve it by taking into account the remainder term as in [START_REF] Arnold | Refined convex Sobolev inequalities[END_REF]. Here we assume that β = 1. Our result is primarily an improvement of the existing results for p ∈ (1, 2), but we work in the larger range p ∈ (1, 2 ) (see Section 3.1). We will see that the limitation on the exponent appears naturally since we do not allow any freedom for β. This special exponent 2 was already noticed in [START_REF] Bakry | Inégalités de Sobolev pour un semi-groupe symétrique[END_REF][START_REF] Dolbeault | Sharp interpolation inequalities on the sphere: New methods and consequences[END_REF]. In the range p > 2, the limitation p ≤ 2 # is equivalent to require that β = 1 ∈ B(p, d). When p ∈ (2, 2 # ], the computations of this section are a limiting case of those in Section 4. Since estimates have to be adapted and since the range in p is anyway different, we handle this case separately.

Assuming that u p = 1 and u is a solution of ( 6) and following the calculations of Section 3.

Proof. Let us define h(t)

:= 1 -(p -2) e(t). When p ∈ (2, 2 # ) we have h = -(p -2) e > 0 , h = -(p -2) e < 0 ,
while when p ∈ (1, 2) these inequalities are changed to their opposities. Also h(t) ∈ (0, 1), h(+∞) = 1, h (+∞) = 0. Our differential inequality takes form:

- 1 p -2 (h + d h ) - γ 1 2 (p -2) 2 |h | 2 h ≥ 0 ,
which upon multiplying by p-2 h and rearranging leads to

d dt log h + γ 1 2 (p -2) d dt log h ≤ -d if p > 2 , ≥ -d if p < 2 ,
and after one integration from 0 to t yields

h (t) h(t) γ 1 2 (p-2) ≤ e -dt h (0) h(0) γ 1 2 (p-2) if p > 2 , ≥ e -dt h (0) h(0) γ 1 2 (p-2) if p > 2 .
Integrating now from 0 to ∞ we find

2 (p -2) γ 1 + 2 (p -2) 1 -h(0) γ 1 2 (p-2) +1 ≤ 1 d h (0) h(0) γ 1 2 (p-2) ,
from which it follows that 2d (p -2)

γ 1 + 2 (p -2) h(0) -γ1/2 (p-2) -h(0) ≤ h (0) .
This is the desired inequality since h (0) = (p -2) i 0 . This ends the proof of Proposition 4, as a particular case, and completes the proof of Theorem 1.1 in the case p ∈ (1, 2).

An interesting consequence is that the limit case p = 2 gives an improvement of the logarithmic Sobolev inequality, which is stated in Theorem 1.1 for p = 2. We check that as p → 2, γ 1 converges to γ * 1 = (4 d -1)/(d + 2) 2 . The conclusion holds by observing that e converges to

1 2 1 -1 w 2 log |w| 2 / w 2 2 dν d , while lim p→2 2d γ 1 + 2 (p -2) (1 -(p -2) e 0 ) - γ 1 2 (p-2) -1 + (p -2) e 0 = 2d γ * 1 e γ * 1 e0/2 -1 .
6. Improved inequalities based on spectral estimates. In this section, we first adapt the results and the proofs of [START_REF] Arnold | Interpolation between logarithmic Sobolev and Poincaré inequalities[END_REF] to the setting of the ultraspherical operator. The method is the same, but gives a point of view which complements the other improved inequalities of this paper in the range p ∈ (1, 2). For completeness, we shall give short proofs and refer to [START_REF] Arnold | Interpolation between logarithmic Sobolev and Poincaré inequalities[END_REF] for more details and references in the framework of probability measures. We shall next review some results in the critical case p = 2 * when d > 2 or its counterpart when d ≤ 2 (Section 6.2), in order to raise an open question (Section 6.3).

6.1. Improved inequalities for reduced classes of functions in the range p ∈ (1, 2). We will now establish Theorem 1.2 by proving a series of intermediate results. We recall that E j denotes the eigenspace associated with the eigenvalue λ j of the Laplace-Beltrami operator ∆ S d on the sphere (see the introduction for more details).

Proposition 5. Let k ≥ 1 be an integer. Assume that u ∈ L 2 (S d , dµ) is such that (16) holds. Then the improved inequality

S d |∇u| 2 dµ ≥ d α k 1 -(p -1) α k S d |u| 2 dµ - S d |u| p dµ 2/p (19) holds for any p ∈ [1, 2).
Proof. To establish the inequality, we proceed in two steps. 1 st step: Nelson's hypercontractivity result. The method is exactly the same as in [START_REF] Dolbeault | Sharp interpolation inequalities on the sphere: New methods and consequences[END_REF]Proposition 5]. Although the result can be established by direct methods, we follow here the strategy of Gross in [START_REF] Gross | Logarithmic Sobolev inequalities[END_REF], which proves the equivalence of the optimal hypercontractivity result and the optimal logarithmic Sobolev inequality.

Consider the heat equation of S d , namely

∂f ∂t = ∆ S d f , with initial datum f (t = 0, •) = u ∈ L p (S d
), for some p ∈ (1, 2), and let

F (t) := f (t, •) L λ(t) (S d ) .
The key computation goes as follows.

F F = d dt log F (t) = d dt 1 λ(t) log S d |f (t, •)| λ(t) dµ = λ λ 2 F λ S d v 2 log v 2 S d v 2 dµ dµ -4 λ -1 λ S d |∇v| 2 dµ with v := |f | λ(t)/2 . Assuming that 4 λ-1 λ = 2 d so that F ≤ 0 by the logarithmic Sobolev inequality (2), that is λ λ -1 = 2 d , we find that log λ(t) -1 p -1 = 2 d t if we require that λ(0) = p < 2. Let t * > 0 be such that λ(t * ) = 2, i.e. t * = - log(p -1) 2 d .
As a consequence of the above computation, we have

f (t * , •) L 2 (S d ) ≤ u L p (S d ) because 1 p -1 = e 2 d t * . (20)
2 nd step: Spectral decomposition. Let u = ū + j>k f j be a decomposition of the initial datum on the eigenspaces of -∆ S d so that -∆ S d f j = λ j f j . Let a j = f j 2 L 2 (S d ) , for any j > 0, and a 0 = ū2 . As a straightforward consequence of this decomposition, we know that

u 2 L 2 (S d ) = a 0 + j>k a j , ∇u 2 L 2 (S d ) = j>k λ j a j , f (t * , •) 2 L 2 (S d ) = a 0 + j>k a j e -2 λj t * .
Using [START_REF] Bolley | Phi-entropy inequalities and Fokker-Planck equations[END_REF], it follows that

S d |u| 2 dµ - S d |u| p dµ 2/p 2 -p ≤ S d |u| 2 dµ - S d |f (t * , •)| 2 dµ 2 -p = 1 2 -p j>k λ j a j
1 -e -2 λj t * λ j .

Since λ → 1-e -2 λ t * λ is decreasing, we can bound 1-e -2 λ j t * λj by 1-e -2 λ k+1 t * λ k+1

for any k ≥ 1. This proves that

S d |u| 2 dµ - S d |u| p dµ 2/p 2 -p ≤ 1 -e -2 λ k+1 t * (2 -p) λ k+1 j>k λ j a j = 1 -e -2 λ k+1 t * (2 -p) λ k+1 ∇u 2 L 2 (S d ) .
The conclusion easily follows.

Using the same spectral method as in the proof of Proposition 5, as in [START_REF] Arnold | Interpolation between logarithmic Sobolev and Poincaré inequalities[END_REF], we will next a establish a more general interpolation inequality. Proposition 6. Let k ≥ 1 be an integer. Assume that u ∈ L 2 (S d , dµ) is such that

S d u e dµ = 0 ∀ e ∈ E j , j = 1 , 2 . . . k .
Then, with the same notations as in Proposition 5, the improved inequality

∇u 2 L 2 (S d ) ≥ d α k 1 -(p -1) α k γ/2 u 2 L 2 (S d ) -u γ L p (S d ) u 2-γ L 2 (S d ) (21)
holds for any p ∈ [1, 2) and any γ ∈ (0, 2).

Proof. The computations are analogous to the ones of Proposition 5. If f is a solution to the heat equation with initial datum u, then

u 2 L 2 (S d ) -f (t * , •) γ L 2 (S d ) u 2-γ L 2 (S d ) = a 0 j>k a j -   a 0 + j>k a j e -2λj t   γ 2   a 0 + j>k a j   2-γ 2
can be estimated using Hölder's inequality by

a 0 + j>k a j e -γλj t * = a 0 + j>k a j e -2λj t * γ 2 • a 2-γ 2 j ≤   a 0 + j>k a j e -2λj t *   γ 2   a 0 + j>k a j   2-γ 2 .
Nelson's estimate [START_REF] Bolley | Phi-entropy inequalities and Fokker-Planck equations[END_REF] shows that

u 2 L 2 (S d ) -u γ L p (S d ) u 2-γ L 2 (S d ) ≤ j>k a j 1 -e -γλj t * ≤
1 -e -γλ k+1 t * λ k+1 j>k λ j a j using the decay of λ → 1 -e -γλt * /λ and conclude as before with j>k λ j a j = S d |∇u| 2 dµ.

An optimization on γ can also be done, as in [START_REF] Arnold | Interpolation between logarithmic Sobolev and Poincaré inequalities[END_REF].

Corollary 3. Let k ≥ 1 be an integer and p ∈ (1, 2). Assume that u ∈ L 2 (S d , dµ) is such that

S d u e dµ = 0 ∀ e ∈ E j , j = 1 , 2 . . . k .
Then the following estimate holds

S d |∇u| 2 dµ ≥ 2 d log(p -1) u 2 L 2 (S d ) log u L p (S d ) u L 2 (S d ) (22)
if

u L p (S d ) ≤ (p -1) α k /2 u L 2 (S d ) .
Proof. We optimize the r.h.s. of Inequality (21) w.r.t. γ ∈ (0, 2), that is, we maximize

γ → d α k u 2 L 2 (S d ) u 2 L 2 (S d ) -u γ L p (S d ) u 2-γ L 2 (S d ) [1 -(p -1) α k γ/2 ] u 2 L 2 (S d ) = d α k u 2 L 2 (S d ) h(γ) , with h(γ) := 1 -a γ 1 -b γ , a = f L p (S d ) f L 2 (S d ) ≤ 1 and b = (p -1) α k /2 ≤ 1 .
We write h(γ) = g(b γ ) with g(y) := (1 -y log a log b )/(1 -y). For a < b < 1 the function g(y) is monotone increasing because y → y log a/ log b is convex. Hence, h(γ) is monotone decreasing. Analogously, h is monotone increasing for b < a < 1. Hence, the maximum of the function h(γ) on [0, 2] is either h(2) (if a > b) or lim γ→0 h(γ) (in the case a < b). This yields the conclusion.

Summarizing, Inequalities ( 19), ( 21) and ( 22) can be written respectively with i = 1, 2, and 3 as

S d |∇u| 2 dµ ≥ d u 2 L 2 (S d ) χ (i) k u L p (S d ) u L 2 (S d ) ,
where Here x ∈ (0, 1] and the proof of Corollary 3 amounts to

χ (1) k (x) := α k 1 -(p -1) α k (1 -x 2 ) , χ (2)
k (x) := α k 1 -(p -1) γα k /2 (1 -x γ
χ (2) k (x) ≤ max χ (1) k (x), χ (3)
k (x) . This completes the proof of Theorem 1.2. See Fig. 4.

6.2.

A review of some results in the critical case p = 2. The question of improvements under orthogonality constraints is a topic which has been studied in various contexts. Although our method does not apply to p ≥ 2, for completeness let us mention a few results that are concerned with the critical case p = 2, or with Onofri related inequalities in dimension less or equal than 2.

Let us start with the case d = 1. In [START_REF] Osgood | Compact isospectral sets of plane domains[END_REF][START_REF]Extremals of determinants of Laplacians[END_REF] (also see [START_REF]Compact isospectral sets of surfaces[END_REF]), B. Osgood, R. Here dω is the measure induced by Lebesgue's measure on S 2 ⊂ R 3 (without normalization). This inequality has been established in [START_REF] Moser | A sharp form of an inequality by N. Trudinger[END_REF] (without optimal constant) and in [START_REF] Onofri | On the positivity of the effective action in a theory of random surfaces[END_REF] with sharp constant α = 1 when u is an arbitrary function in H 1 (S 2 ). In [START_REF] Chang | Prescribing Gaussian curvature on S 2[END_REF], S.-Y. A. Chang and P.C. Yang asked the question whether the inequality is true with α = 1/2 if

S 2
e u x j dω = 0 ∀ j = 1 , 2 , 3 .

A partial answer has been given in [START_REF] Ghoussoub | On the best constant in the Moser-Onofri-Aubin inequality[END_REF] by N. Ghoussoub and C.-S. Lin, who showed that in such a case α < 2/3. In dimension d ≥ 3, there are no explicit results, as far as we know, but G. Bianchi and H. Egnell show in [START_REF] Bianchi | A note on the Sobolev inequality[END_REF] that improvements (without optimal constants) can also be achieved if we require the appropriate orthogonality conditions. A more precise statement, although still not fully explicit in terms of spectral estimates, has been established in [START_REF] Cianchi | The sharp Sobolev inequality in quantitative form[END_REF].

6.

3. An open problem. The results of Sections 6.1 and 6.2 raise a straightforward question: how can spectral estimates together with well-chosen orthogonality constraints provide improved inequalities in the subcritical range p ∈ (2, 2 * ) when d > 2, or in the range p > 2 when d ≤ 2 ? 7. Csiszár-Kullback-Pinsker inequalities. The standard form of the Csiszár-Kullback-Pinsker inequality is

Ω f log f f 0 dµ ≥ 1 4 M f -f 0 2 L 1 (Ω,dµ) ,
for any nonnegative integrable functions f and f 0 such that S d f dµ = S d f 0 dµ = M . See [START_REF] Csiszár | Information-type measures of difference of probability distributions and indirect observations[END_REF][START_REF] Kullback | On the convergence of discrimination information[END_REF][START_REF] Pinsker | Information and Information Stability of Random Variables and Processes, Translated and[END_REF] for details. Here we need a generalized form of this inequality for p = 2. As will be made clear in the proof, the result is based on Taylor expansions and does depend neither on the domain of integration Ω nor on the positive measure dµ. We shall therefore assume that Ω is a measurable subset in a submanifold of the Euclidean space, and that dµ is a probability measure on Ω, without further notice. In practice (Ω, dµ) is either the d-dimensional sphere S d endowed with the probability measure induced by Lebesgue's measure, or (-1, 1) and dµ = dν d corresponds to the setting of the ultraspherical operator.

The following result is somewhat standard and the interested reader is invited to read for instance [START_REF] Cáceres | Nonlinear stability in L p for a confined system of charged particles[END_REF][START_REF] Unterreiter | On generalized Csiszár-Kullback inequalities[END_REF]. It allows us to prove various kinds of Csiszár-Kullback-Pinsker type inequalities. For completeness, we also provide a proof. Proposition 7. Assume that q ∈ [1, 2] and consider ψ(s) = s q if q > 1 or ψ(s) = s log s if q = 1. Let f and g be two nonnegative functions in L 1 ∩ L q (Ω, dµ). Then

e ψ [ f |g ] := Ω ψ(f ) -ψ(g) -ψ (g)(f -g) dµ ≥ q (q -1) 2 2/q min f q-2 L q (Ω,dµ) , g q-2 L q (Ω,dµ) f -g 2 L q (Ω,dµ) . (23)
Remark 1. The case q = 1 is the well-known Csiszár-Kullback-Pinsker inequality, cf. [START_REF] Csiszár | Information-type measures of difference of probability distributions and indirect observations[END_REF][START_REF] Kullback | On the convergence of discrimination information[END_REF][START_REF] Pinsker | Information and Information Stability of Random Variables and Processes, Translated and[END_REF][START_REF] Arnold | On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations[END_REF]. The case q = 2 is a consequence of the expansion of the square.

Proof. Assume first that f > 0. By a Taylor expansion at order two, we get

e ψ [ f |g ] = 1 2 Ω ψ (ξ) |f -g| 2 dµ ≥ A 2 Ω ξ q-2 |f -g| 2 dµ , (24)
where ξ lies between f and g. By Hölder's inequality, for any h > 0 and for any measurable set A ⊂ Ω, we get

A |f -g| q h -α h α dµ ≤ A |f -g| 2 h q-2 dµ q/2 A h αs dµ 1/s with α = q (2 -q)/2, s = 2/(2 -q). Thus, A |f -g| 2 h q-2 dµ q/2 ≥ A |f -g| q dµ A h q dµ (q-2)/2
.

We apply this formula to two different sets.

1) On

A = A 1 = {x ∈ Ω : f (x) > g(x)}, use ξ q-2 > f q-2 and take h = f : A1 |f -g| 2 ξ q-2 dµ q/2 ≥ A1 |f -g| q dµ f -(2-q) q/2
L q (Ω,dµ) .

2) On A = A 2 = {x ∈ Ω : f (x) ≤ g(x)}, use ξ q-2 ≥ g q-2 and take h = g:

A2 |f -g| 2 ξ q-2 dµ q/2 ≥ A2 |f -g| q dµ g -(2-q) q/2
L q (Ω,dµ) .

To prove [START_REF]On the second best constant in logarithmic Sobolev inequalities on complete Riemannian manifolds[END_REF] in the case f > 0, we just add the two previous inequalities in [START_REF] Cáceres | Nonlinear stability in L p for a confined system of charged particles[END_REF] and use the inequality (a + b) r ≤ 2 r-1 (a r + b r ) for any a, b ≥ 0 and r ≥ 1. To handle the case f ≥ 0, we proceed by a density argument and conclude by using Lebesgue's convergence theorem.

Next we give the proof of some generalized Csiszár-Kullback-Pinsker inequalities for various values of p, all easily derived from the above proposition. The case of p ∈ [1, 2) can be handled with q = 2/p. For p ∈ (2, 4], one can use the inequality of Proposition 7 written for q = p/2 and control u 2 -ū2 L q (S d ) . For p ≥ 4, the control is achieved in terms of u p-2 -ūp-2

L p/(p-2) (S d ) . For each range, the average ū has to be defined specifically. We do not claim originality for the following result, as it has probably been discovered in other settings. Let us just mention a few additional references: in the case p > 2, see: [START_REF] Carrillo | Asymptotic L 1 -decay of solutions of the porous medium equation to self-similarity[END_REF][START_REF] Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions[END_REF] and [START_REF] Dolbeault | Large time behaviour of solutions to nonhomogeneous diffusion equations[END_REF]Proposition 2] for related results. A recent contribution in a similar spirit can be found in [START_REF] Carlen | Stability estimates for the lowest eigenvalue of a Schrödinger operator[END_REF].

Corollary 4. For any u ∈ L 1 ∩ L p (Ω, dµ), we have Proof. When p ∈ [1, 2), we apply Proposition 7 with q = 2/p, ψ(s) := s 2/p , to f = u p and g = u p L p (Ω,dµ) . The result follows using u L p (Ω,dµ) ≤ u L 2 (Ω,dµ) . If p = 2, Proposition 7 directly applies, with ψ(s) := s log s. If p ∈ (2, 4), we apply Proposition 7 with q = p/2, ψ(s) := s p/2 , to f = u 2 and g = u 2 L 1 (Ω,dµ) = ū2 , so that p -not defined 9 -4 √ 3

u 2 L 2 (Ω,dµ) -u 2 L p (Ω,dµ) ≥ κ(p) u 2 (1-p) L 2 (Ω,dµ) u p -ūp 2 L 2/p (Ω,dµ) if p ∈ [1, 2) , Ω |u| 2 log |u| 2 u 2 L 2 (Ω,dµ) dµ ≥ 2 |κ (2)| u 2 L 2 (Ω,dµ) u 2 -ū2 2 L 1 (Ω,dµ) if p = 2 , u 2 L p (Ω,dµ) -u 2 L 2 (Ω,dµ) ≥ κ(p) u -2 L p (Ω,dµ) u 2 -ū2 2 L p/2 (Ω,dµ) if p ∈ (2, 4) , u 2 L p (Ω,dµ) -u 2 L 2 (Ω,dµ) ≥ κ(p) u 2 (3-p) L p (Ω,dµ) u p-2 -ūp-2 2 L p p-2 (Ω,dµ) if p ≥ 4 ,
u p L p (Ω,dµ) -u p L 2 (Ω,dµ) ≥ 2 -2-

1 . 2 L 2 (

 122 Introduction. On the d-dimensional sphere, for any real valued function u in H 1 (S d ), let us consider the inequality d e ≤ i where e := u 2 L p (S d ) -u S d ) p -2 and i := ∇u 2 L 2 (S d ) ,

  Finally we define Φ(s) := 1 + (p -2) s ϕ s 1 + (p -2) s where ϕ(s) := sup β∈B(p,d)

according to [ 16 ,

 16 Corollary C-I-3, page 162]. Notice that E 0 is generated by the function 1. Finally, for k = 1, 2, . . . let us define constants

Figure 1 .

 1 Figure 1. Generically γ(β) = 0 has two solutions, β±(p, d). Plot of p → β±(p, d) with d = 1 (left) and d = 2 (right) on the first row, d = 3 (left) and d = 4 (right) on the second row, and d = 5 (left) and d = 10 (right) on the third row. The dotted line corresponds to p → 4 6-p and we have β-(p) ≤ 4 6-p ≤ β+(p) if and only if 18 d 17 d-2 ≤ p ≤ 2 * . The dashed lines correspond to β = 1 and β = 2 4-p , and the interior of B(p, d) corresponds to the grey area: see definition (11) of B(p, d). Also see Fig. 5 for more details in the case d = 2.

  are increasing. Let us define (cf. introduction) ϕ(e) := sup ϕ β (e) : β ∈ B(p, d) .

2 |e | 2 1 - 1 - 1 Figure 3 . 1 ξγ 1 p- 2 -1 ξ 2 - 1 represents

 2111311221 Figure 3. The plot of ξ → 2 (2-p) 2 (2-p)-γ 1

Figure 4 .

 4 Figure 4. Plot of ξ → χ (2) k (ξ)/χ (1) k (ξ) for various values of γ and of ξ → χ

k

  (ξ) (bold curve) when d = 2, p = 3/2, k = 1 and α1 = 3. The plot of Fig. 3-left appears as a dashed line. Notice that an additional assumption required in Theorem 1.2, namely the orthogonality to the eigenfunctions associated with λ1 = d.

k

  Phillips and P. Sarnak established the inequality, known as the first Lebedev-Millin inequality [40, Section 5.| φ(k)| 2if, additionally, 2π 0 e φ e iθ dθ = 0. This inequality has been improved to log e imθ dθ = 0 for any m = 1, 2. . . n by H. Widom in[START_REF] Widom | On an inequality of Osgood, Phillips and Sarnak[END_REF].The case d = 2 is not understood as well. Consider the inequality log

4 p

 4 whereκ(p) = 2 1-p 2-p p 2 , ū = u L p (Ω,dµ) if p ∈ [1, 2) , |κ (2)| = 1 8 , ū = u L 2 (Ω,dµ) if p = 2 , κ(p) = 2 -1-4 p (p -2) , ū = u L 2 (Ω,dµ) if p ∈ (2, 4) , κ(p) = 2 (p -2) -2 , ū = u L p-2 (Ω,dµ) if p ≥ 4 .

4 √ 3 ) ∪ ( 9 + 4 √ 3 , 4 √ 3 , 9 + 4 √ 3 ) 5 .Figure 6 .

 439434394356 Figure 6. Horizontal axis: d, vertical axis: p. Bold plain curve: d → p+(d), bold dotted curve: d → p-(d). Other curves shown on the picture correspond to d = 1, d = 2 and p = 2 d d-2 . We notice that a(p, d) < 0 if and only if either d = 1 and p ∈ (2, ∞), or d = 2 and p ∈ (9 -4 √ 3, 9 + 4 √ 3), or d = 3 and p ∈ ( 9 4 , 6). We may also notice that lim d→1 p-(d) = 2 and lim d→1 p+(d) = +∞ (if we consider p± as a function of d ∈ R).

  4 p p (p -2) u p-4 L p (Ω,dµ) u 2 -ū2 2 L p/2 (Ω,dµ) . Since t → 1-t-2 p (1-t p/2 ) is convex, nonnegative, with t = u 2 L 2 (Ω,dµ) / u 2 L p (Ω,dµ) , we can write that

	u 2 L p (Ω,dµ) -u 2 L 2 (Ω,dµ)
	= u 2 L p (Ω,dµ) (1 -t) ≥ u 2 L p (Ω,dµ)

Table 1 .

 1 Values of a and b for d ∈ N, d ≥ 1.

	9 4	3	not defined

c 2013 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.

Acknowledgments. J.D. and M.J.E. have been partially supported by the ANR grant NoNAP. Part of this work was completed during M.K.'s visit at Ceremade, Université Paris-Dauphine. J.D. participates in the AmSud QUESP project and thanks for support. J.D. has also been partially supported by ANR grants STAB and Kibord, and the ECOS project C11E07. M.K. has been partially supported by the FONDECYT grant 1130126, the ECOS project C11E07 and Fondo Basal CMM. M.L. has been partially supported by the NSF grant DMS-1301555. J.D. thanks the organizers of the Conference on Nonlinear Elliptic and Parabolic Partial Differential Equations held in Milano in June 2013, which has provided the opportunity for completing this paper.

We conclude by using the convexity of t → 1-t-2 p (1-t p/2 ) with t = ū2 / u 2 L p (Ω,dµ)

and, by Hölder's inequality, the fact that ū = u L p-2 (Ω,dµ) ≥ u L 2 (Ω,dµ) .

Proposition 1 is a straightforward consequence of Corollary 4. Details are left to the reader. Remark 2. In the proof of Corollary 4, p = 4 is a threshold case. In [START_REF] Dolbeault | Large time behaviour of solutions to nonhomogeneous diffusion equations[END_REF]Lemma 2], it is p = 3 that plays a special role. Many other estimates can be derived for p > 2 and the value p = 4 a priori plays no special role, as it is shown by the following computation. Let

Two differentiations show that

On the one hand we have f (1) = f (1) = 0 and, on the other hand,

for any w ≥ 1 if we assume that p ≥ 3 and p 0 ≥ 1 .

Thus f is convex and therefore nonnegative on (1, +∞). Now, if we define ūp-p0 = Ω u p-p0 dµ, by integrating the inequality f (u/ū) ≥ 0 with respect to the measure ūp dµ, we arrive at

Hence, if p -p 0 ≥ 2, then the inequality ū = u L p-p 0 (Ω,dµ) ≥ u L 2 (Ω,dµ) and the convexity of t → 1 -t -2 p (1 -t p/2 ) applied with t = ū2 / u 2 L p (Ω,dµ) allows us to conclude that

with ū = u L p-p 0 (Ω,dµ) .

Appendix A. A discussion on the range of admissible p and β. Consider [START_REF] Beckner | A generalized Poincaré inequality for Gaussian measures[END_REF] we have that -γ(β) = a β 2 -2 b β + 1 . Provided a = 0 (this will be discussed below), the two roots of the equation γ(β) = 0 are

See Fig. 6 for a plot and Table 1 for a summary of the values of a and b depending on the value of d, when d is an integer.

Hence we know that a = 0 if and only if In these cases, the sign of b = d+3-p d+2 matters: This concludes the proof by discussing the cases depending whether a = 0 (and the range of β is determined by b) or a has a strict sign and β ± defines the admissible range for β in order that µ is nonpositive.


Thanh-Ha Do

Salvatore Tabbone
email: tabbone@loria.fr

Oriol Ramos
email: oriolrt@cvc.uab.cat

Document Noise Removal using Sparse Representations over Learned Dictionary

Keywords: J.m [Computer Applications]: Miscellaneous Sparse Representation, Learned Dictionary, K-SVD, Normalized Cross Correlation, Noise Suppression

In this paper, we propose an algorithm for denoising document images using sparse representations. Following a training set, this algorithm is able to learn the main document characteristics and also, the kind of noise included into the documents. In this perspective, we propose to model the noise energy based on the normalized cross-correlation between pairs of noisy and non-noisy documents. Experimental results on several datasets demonstrate the robustness of our method compared with the state-of-the-art.

INTRODUCTION

The performance of many pattern recognition techniques applied on images depends on an accurate control of the noise included into the image. In the case of document image analysis applications, the kind of noise is different compared to the noise found in natural scenes images that are generated by devices like digital cameras or similar.

Therefore, the problem of document denoising has been tackled since the very beginning. There is a vast literature on methods proposing solutions such as median filter [START_REF] Davies | Machine Vision: Theory, Algorithms and Practicalities[END_REF], morphological filters [START_REF] Marrgos | Morphological filters, part 2: Their relations to median, order-statistic, and stack filters[END_REF], or curvelets transform [START_REF] Starck | The curvelet transform for image denoising[END_REF]. Median filter replaces each pixel in the noisy image by the median of pixels in a neighborhood of that pixel, while morphological filter carries out dilation and erosion operations as denoisy operators. In addition, morphological filtering can discriminate between positive and negative noise spikes, whereas median filter cannot [START_REF] Marrgos | Morphological filters, part 2: Their relations to median, order-statistic, and stack filters[END_REF]. Both filters, median and morphological, are appealing because they are easy to implement and perform well with the presence of impulse noise. However, neither of them are efficient for other types of noise like white noise or noise arising from printing, photocopying and scanning processes. This kind of noise not only generally causes undesirable document appearance, but also has a bad influence on document processing performance.

Recent researches obtain good performance in denoising gray-scale images with white noise using Multi Resolution Analysis (MRA) methods [START_REF] Starck | The curvelet transform for image denoising[END_REF]. Sparse transforms and MRA methods are applied to a wide range of image processing problems as image compression, image restoration and image denoising [START_REF] Starck | The curvelet transform for image denoising[END_REF]. These methods have proven to perform well in terms of Mean Square Error (MSE) measure as well as Peak Signal-to-Noise Ratio (PSNR) measure for essentially white noise (additive and following a Gauss distribution). Moreover, sparse transforms, like curvelets, contourlets, wedgelets, bandelets, or steerable wavelets, have also been successfully applied in document images for removing noisy edges, showing that sparse representation can effectively be used for denoising purposes. Sparse transforms represent images as linear combinations of (atom) functions of a given particular family of functions (dictionary). However, the overall performance of these methods depends on two factors: first of all, a ap r i o r iknowledge about images which drives the choice of dictionary functions and secondly, the kind of noise found in document images. Recently, curvelets transform has been applied in document denoising with a relative high degree of success [START_REF] Hoang | Edge noise removel in bilevel graphical document images using sparse representation[END_REF]. In that approach, the authors take advantage of directional properties of curvelets transform to denoise degraded graphic documents. The results obtained in that work show an improvement in removing noise for document images comparing with other state-of-the-art methods. However, curvelet is one of the pre-defined dictionaries, and therefore can only work well with some kinds of noise and cannot be adapted to arbitrary noise models.

In this paper, we address the task of document denoising by proposing a method which overcomes the difficulties found in denoising methods based on MRA. On the one hand, we apply the K-SVD algorithm to learn a proper set of atom functions adapted to both document characteristics and document noise. On the other hand we propose an energy noise model which allows us to easier set the threshold required for noise removal even if the noise model is unknown.

The remainder of this paper is organized as follows. We introduce the theoretical framework of sparse representations in section 2 and the learning algorithm in section 3. Then, we recall the document degradation models used in experiments (section 4) and we propose the energy noise model in section 5. Next, we discuss the experimental results in section 6 and finally, we give our conclusions in section 7.

SPARSE REPRESENTATION

The main idea of the proposed method is to find a dictionary adapted to the properties of the data which will allow us to obtain a denoised version of the original degraded images using sparse representations.

A sparse representation is a linear combination of few atoms (basis functions) of a given dictionary. Mathematically, given a dictionary A and a signal h, we consider the under-determined linear system of equations h = Ax,w i t h

A = {a1,a2,...,am}∈R n×m , h ∈ R n , x ∈ R m , m ≫ n.
If A is a full-rank matrix, there will be infinitely many different sets of values for the xi's that satisfy all equations simultaneously. The set of x can be described using mathematical language. However, from an application point of view, one of the main tasks in dealing with the above system of equations is to find the proper x that can explain h well comparing with others. To gain this well-defined solution, a function f (x) is added to assess the desirability of a would-be solution x, with smaller values being preferred:

(P f ):min x f (x)s u bj e c tt oAx = h (1)
If f (x)i st h el0 pseudo-norm x 0 (number nonzero elements in vector x), then the problem (P f ) becomes finding the sparse representation x of h satisfying: (P0):min

x x 0 subject to Ax = h (2)
In general, solving equation ( 2) is often difficult (NP-hard problem) and one of the choices is to look for an approximate solution using greedy algorithms, such as Matching Pursuit (MP) [START_REF] Mallat | Matching pursuits with time-frequency dictionaries[END_REF], Orthogonal MP (OMP) [START_REF] Pati | Orthogonal matching pursuit: Recursive function approximation with applications to wavelet decomposition[END_REF], Weak MP [START_REF] Temlyakov | Weak greedy algorithms[END_REF] and Thresholding algorithm [START_REF] Elad | Sparse and redundant representation: From theory to applications in signal and images processing[END_REF]. A greedy algorithms is an algorithms that follows the problem solving heuristic of making the locally optimal single term updates with the hope of finding a global optimum. In our case, the set of active columns started from empty is maintained and expanded by one additional column of A at each iteration. The chosen column is a column that maximally reduces the residual l2 error in approximating h from the currently active columns. The residual l2 error is evaluated after constructing an approximate including the new column; if it is bounded below a specified threshold, the algorithm terminates.

The other choice is to relax the l0-norm by replacing it with the lp-norms for some p ∈ (0, 1] or by smooth functions such as

i log(1 + αx 2 i ), i x 2 i /(α + x 2 i ), or i (1 - exp(-αx 2 i ))
. The interesting algorithm of this family is the FOcal Under-determined System Solver (FOCUSS) [START_REF] Gonzalez | Sparse signal reconstruction from limited data using focuss: a re-weighted minimum norm algorithm[END_REF]. In this algorithm, the lp-norm (for some fixed p ∈ (0, 1])isrepresented as a weighted l2-norm by using Iterative Reweighed Least Squares (IRLS) method [START_REF] Daubechies | Iteratively reweighted least squares minimization for sparse recovery[END_REF].

Another popular strategy is to replace the l0-norm by the l1-norm proposed by Donoho et al [START_REF] Donoho | Optimally sparse representation in general (nonorthogonal) dictionaries via l 1 minimization[END_REF] (P1):min

x W -1 x 1 subject to Ax = h (3)
The matrix W is a diagonal positive-definite matrix. A natural choice for each entry in W is w(i, i)=1 / ai 2 1 . Let x = W -1 x, then equation ( 3) is re-formulated as (P1):min

x x 1 subject to h = AW x = Ãx (4)
in which à is the normalized version of A. Equation ( 4) is the classic basis pursuit format, and the solution x can be found by de-normalizing x.T h u s ,( P1) is usually used with a normalized matrix. The solution for (P1)p r o b l e mc a nb e found by some existing numerical algorithms, such as Basis Pursuit by Linear Programming [START_REF] Chen | Atomic decomposition by basis pursuit[END_REF] or IRLS (for p =1).

If there exists some appropriate conditions on A and x, like

x 0 ≤ 1 2 (1 + 1 max i =j |a T i a j | a i 2 a j 2 )
then Basic Pursuit as well as OMP give the unique solution of ( 4) and it is also the unique solution of (P0).

However, when signals are perturbed by noise it becomes more useful to relax the exact constraint: h = Ax by using instead the quadratic penalty function Q(x)= Ax -h2 2 ≤ ǫ,w i t hǫ ≥ 0 being the error tolerance. Therefore, an errortolerant version of (P0) is defined by:

(P ǫ 0 ):min x x 0 subject to Ax -h 2 ≤ ǫ (5)
In (P ǫ 0 )t h el2-norm is used for evaluation, and the error Ax -h can be replaced by other options, as l1,l2,o rl∞.

Observe that problem (P ǫ 0 ) as defined above is useful for the given task. Assuming that signal h has noise e with finite energy e 2 2 ≤ ǫ 2 , h = Ax + e, solving (P ǫ 0 ) can help us to find the solution x. Then, we can recover the unknown denoised signal ĥ as ĥ = Ax. Similarly, when relaxing l0norm to an l1-norm, we get (P ǫ 1 ) known in the literature as basis pursuit denoising (BPDN) [START_REF] Donoho | Optimally sparse representation in general (nonorthogonal) dictionaries via l 1 minimization[END_REF] (P ǫ 1 ):min

x x 1 subject to Ax -h 2 ≤ ǫ (6)

LEARNED METHODOLOGY FOR DIC-TIONARY AND K-SVD ALGORITHM

The optimal solution of the problem (P ǫ 1 ) directly depends on the used dictionary A. Our working hypothesis is that if we are able to learn A from a training dataset, then A should be well adapted to the document characteristics and noise. So, in this section, we review learning algorithms, in general, used for constructing a dictionary A and in particular, the learning algorithms we have used: the K-SVD algorithm.

In a general learning methodology, a family l signals {hj} l j=1 is considered as the training database. Our goal is to find a dictionary A in which each signal hj ∈ R n has an optimally sparse approximation hj ⋍ Axj satisfying hj -hj 2 ≤ ǫ, or finding:

min A,x j l j=1 xj 1 subject to hj -Axj 2 ≤ ǫ, for all j =1, .., l (7)
This dictionary can be obtained by the learning process that iteratively adjusts A via two main stages: sparse coding stage and update dictionary stage. In the sparse coding stage, all sparse representations X = {xj} l j=1 ∈ R m×l of H = {hj} l j=1 ∈ R n×l are found by solving equation ( 6), on the condition that A is fixed. In the update dictionary stage, an updating rule is used to optimize the sparse representations of the training signals. In general, the way to update the dictionary is different from one learning algorithm to another. There are two well-known dictionary-learning algorithms, named Method of Optimal Directions (MOD) by Engan et al [START_REF] Engan | Frame based signal compression using method of optimal directions (mod)[END_REF], and K-SVD by Aharon et al [START_REF] Aharon | An algorithm for designing overcomplete dictionaries for sparse representation[END_REF]. These two algorithms behave similarly with a small advantage to the K-SVD [START_REF] Elad | Sparse and redundant representation: From theory to applications in signal and images processing[END_REF]. So, we chose the K-SVD algorithm in our paper to construct the learned dictionary.

In K-SVD algorithm [START_REF] Aharon | An algorithm for designing overcomplete dictionaries for sparse representation[END_REF], the updating rule is to make a modification on dictionary's columns. At this step, we handle to update sequentially each columns aj 0 of A such that the residual error ( 8) is minimized, where X and {a1, ...aj 0 -1 ,aj 0 +1, ..., am} are fixed,

H -AX 2 F = H - m j=1 ajx T j 2 F = (H - j =j 0 ajx T j ) -aj 0 x T j 0 2 F = Ej 0 -aj 0 x T j 0 2 F (8)
In equation ( 8), x T j 0 ∈ R l is the k-th row in X and the notation . F stands for the Frobenius norm. Because X and all the columns of A are fixed excepted the column aj 0 ,s o Ej 0 = Hj =j 0 ajx T j is fixed. It means that the minimum error H -AX 2 F depends only on the optimal aj 0 and x T j 0 . This is the problem of approximating a matrix Ej 0 with another matrix which has a rank 1 based on minimizing the Frobenius norm. The optimal solutions ãj 0 , xT j 0 can be given by the Singular Value Decomposition (SVD) of Ej 0 of rank r1,n a m e l y ãj 0 xT j 0 = Q1 ŨQ2 where Q1 = {q 1 1 , ..., q 1 n }, U =d i a g ( σ1,σ2, ..., σr 1 ), Q2 = {q 2 1 , ..., q 2 l } is the SVD of Ej 0 : Ej 0 = Q1UQ2;a n d Ũ is the same matrix as U except that it contains only one singular values σ1 (the other singular values are replaced by zero). This means ãj 0 = q 1 1 and xT j 0 = σ1q 2 1 . However, the new vector xT j 0 is very likely to be filled, implying that we increase the number of non-zeros in the representation of X,o rt h e condition about the sparsity of X can be broken.

This problem can be overcome as follows. Define the group of indexes where x T j 0 is nonzero:

ωj 0 = {i|1 ≤ i ≤ l, x T j 0 (i) =0}.
and a matrix Ωj 0 ∈ R l×|ω j 0 | is defined Ωj 0 (ωj 0 (i),i)=1and zeros elsewhere. Let

1. x R j 0 = x T j 0 Ωj 0 , x R j 0 ∈ R |ω j 0 | 2. E R j 0 = Ej 0 Ωj 0 , E R j 0 ∈ R n×|ω j 0 |
and following equation ( 8), the minimization is equivalent to

Ej 0 Ωj 0 -aj 0 x T j 0 Ωj 0 2 F = E R j 0 -aj 0 x R j 0 2 F (9)
Note that the solution of (9) xR j 0 has the same support as the original xT j 0 , and the optimal values xR j 0 , ãj 0 can be obtained by finding SVD of a subset of the columns of the error matrix E R j 0 of rank r2: E R j 0 = SDV T . The solution for ãj 0 is defined as the first column of S,a n dt h ec o e ffi c i e n t vector xR j 0 as the first column of V multiplied by d1,w i t h D =d i a g ( d1,d2, ..., dr 2 ). More details about the K-SVD algorithm can be found in algorithm 1.

Algorithm 1 Learning algorithm K-SVD

INPUT: A (0) ∈ R n×m ; {hi} l i=1 ; k =0; 1. Initialize: Normalization the columns of matrix A (0) ; 2. Main Iteration k = k +1; while ( H -A (k) X (k) 2
F is not small enough) do -S o l v e( P ǫ 1 ) to find all sparse representation {xi} l i=1 of {hi} l i=1 for (j0 =1tom) do -Define:

ωj 0 = {i|1 ≤ i ≤ l, x T j 0 (i) =0} -C a l c u l a t eEj 0 via equation Ej 0 = H -j =j 0 ajx T j -L e tE R
j 0 is the sub-matrix of Ej 0 on the columns corresponding to ωj 0 -C a l c u l a t eS V Do fE R j 0 : E R j 0 = SDV T . -Updating: aj 0 = s1 and x R j 0 = d1v1 with S = {s1, ..., sn}, V = {v1, ..., v |ω j 0 | }, D = diag(d1,d2, ..., dr 2 ) end for end while OUTPUT: The result A (k)

DOCUMENT DEGRADATION MODELS

Inspired by several authors like in [START_REF] Kanungo | Global and local document degradation models[END_REF][START_REF] Hoang | Edge noise removel in bilevel graphical document images using sparse representation[END_REF], we have used document noise models for evaluating the proposed method. In 1993, Kanungo et al [START_REF] Kanungo | Global and local document degradation models[END_REF] introduced a statistical model for document degradation showing quite realistic results, which gave the following three reasons justifying research on noise models. First of all, noise modeling allows to study recognition algorithms in general, as a function of the perturbation of the input data. Secondly, it permits the evaluation of any algorithm depending on the degradation level. Thirdly, a knowledge of the degradation model can enable us to design algorithms for image restoration. More recently, [START_REF] Barney | Modeling image degradations for improving ocr[END_REF] has proposed the Noise Spread model, inspired on the physics of image acquisition process. According to this model the acquired image is obtained as a result of convolving the source image with the sensor function defined by the Point Spread Function with white noise.

Since the Kanungo noise model [START_REF] Kanungo | Global and local document degradation models[END_REF][START_REF] Kanungo | A statistical, nonparametric methodology for document degradation model validation[END_REF] is a statistical model, widely used to verify the robustness of document image analysis methods to noise, we use it as noise model in this paper. However, the extension to another model is easy as our method is enabled to denoise documents for which the model of noise is unknown. One of the advantages of degradation models is they permit to generate degraded images controlled by the parameter models. The qualitative results of these images range from quite realistic noisy images to unrealistic, or even highly degraded images in function of the parameters set up. For instance, in rameters α0, α, β0, β, η,a n dk as a function of the pixel distance to the shape boundaries to degrade a binary image. α and α0 control the probability of flipping a foreground pixel to a background pixel, in other words, these two parameters provide the probability of changing a black pixel to a white one. Similarly, parameters β and β0 control the probability of changing a background (white) pixel with a foreground (black) pixel. In addition, these two probabilities exponentially decay on the distance of each pixel to the nearest boundary pixel. In contrast, the parameter η is a constant value added to all pixels regardless their relative position to shape boundaries. Finally, the last parameter k is the size of the disk used in the morphological closing operation. The whole process of image degradation can be summarized in the following three steps:

1. Use standard distance transform algorithms to calculate the distance d of each pixel from the nearest boundary pixel.

2. Each foreground pixel and background pixel is flipped with probability p(0|1,d,α0,α)=α0e -αd 2 + η,a n d p(1|0,d,β0,β)=β0e -βd 2 + η 3. Use a disk structuring element of diameter k to perform a morphological closing operation.

ENERGY NOISE MODEL

For denoising images using basis pursuit denoising, we need to decide which kind of dictionary A is used along with the best value ǫ in equation [START_REF] Davies | Machine Vision: Theory, Algorithms and Practicalities[END_REF]. From Section 3 we know how to learn a dictionary A adapted to noisy data. In this Section, we explain how to choose the best value ǫ when we apply equation ( 6) on the image patches with size w.

Denoising in MRA methods assumes that images have been corrupted by an additive white noise. For such noisy images, the energy of noise η is proportional to both the noise variance and size image. For Noise Spread model, authors in [START_REF] Hoang | Edge noise removel in bilevel graphical document images using sparse representation[END_REF] empirically supposed that the optimal energy of noise depends on the noise spread relation. However, neither of these two assumptions can be applied to document images where the noise follows a Kanungo noise model instead of white noise. The reason is that in Kanungo model pixels near the shape boundaries have higher probability to be affected by noise than pixels far from the shape boundaries. On the contrary, white noise model assumes statistical independence between the noise and the image. In fact, the probability that a pixel is perturbed by noise depends only on the variance of the model and does not depend on the position of a pixel in the image. In addition, the parameters used in Noised Spread model are different from the Kanungo's ones, so we cannot use the noise spread relation proposed in [START_REF] Hoang | Edge noise removel in bilevel graphical document images using sparse representation[END_REF] to decide the value for ǫ.

Therefore, we propose an energy noise model inspired by [START_REF] Lewis | Fast normalized cross-correlation[END_REF]. Thus, we evaluate the noise level using the peak values of the normalized cross-correlation between noisy and cleaned documents. Let D being a training dataset including 2t documents (D c i ,D n i ),i =1, ..., t where D c i is a cleaned document and D n i is its noisy version. We define ri as the peaks of normalized cross-correlation between D n i and D c i , then the tolerance error value is defined by:

ǫ = cwr ( 10
)
where c is a constant value set experimentally, w is the size of patches, r is the mean value of the peaks ri.

We can summarize the procedure for document denoising using sparse representation of a learned dictionary A as follows:

1. Create a training database using a sliding window of size w × w to scan the corrupted image y ∈ R M ×N with scanning step set to 1 pixel in both directions. All (M -w +1)(N -w + 1) obtained patches {hj} l j=1 , hj ∈ R w×w are considered as the training database.

2. Create a learned dictionary using the K-SVD algorithm to create the learned dictionary A from {hj} l j=1 . 4. Binarise ŷ to get the final result ỹ.

Combine the learned dictionary

EXPERIMENTAL RESULTS

We firstly evaluated our algorithm on the GREC 2005 dataset. This dataset has 150 different symbols which have been degraded using the Kanungo's method to simulate the noise introduced by the scanning process. Six sets of parameters are used to obtain six different noise levels as shown in Figure [START_REF] Aharon | An algorithm for designing overcomplete dictionaries for sparse representation[END_REF].

At each level of noise, a dataset containing 2 × 50 noisy and cleaned symbols are used to calculate the value of r and ǫ (Section 5). We empirically found that the best results in denoising bilevel images are achieved when c in equation [START_REF] Gonzalez | Sparse signal reconstruction from limited data using focuss: a re-weighted minimum norm algorithm[END_REF] The learning dictionary was produced using K-SVD algorithm with 50 iterations, and the training dataset includes all 8×8 patches. Those patches were taken from a corrupted image h. The ratio of the dictionary is 1/4 (m =4× n).

To examine the effectiveness of the proposed method, we compare it with three existing methods for denoising binary images: median filtering, morphological operators (opening and closing), and curvelet transform. The median filtering is performed with a window size of 3 × 3. The morphological operators use a 3 × 3 structuring element. Curvelets transform has been verified upon the best value of η = c × √ MN × σ 2 , σ ∈ [0.02, 0.1] for each noise level, where M, N is the size of the image. The criterion to choose these best values σ is the average MSE (Mean Squared Error ).

Moreover, using the traditional MSE to estimate the quality change between the original document and the reconstructed one, all algorithms are evaluated by Jaccard 's similarity measure [START_REF] Choi | A survey of binary similarity and distance measures[END_REF]. This measure is computed based on the three values a, b, c as below:

S = a a + b + c (11)
where

a = |{(i, j)|y 0 (i, j)=1, ỹ(i, j)=1, 1 ≤ i ≤ M, 1 ≤ j ≤ N }| b = |{(i, j)|y 0 (i, j)=0, ỹ(i, j)=1, 1 ≤ i ≤ M, 1 ≤ j ≤ N }| c = |{(i, j)|y 0 (i, j)=1, ỹ(i, j)=0, 1 ≤ i ≤ M, 1 ≤ j ≤ N }|
a means 'right matches', and b, c mean 'mismatches'; y 0 , ỹ ∈ R M ×N are cleaned and denoised images, respectively. The maximal value of the Jaccard measure is one when two images are identical.

Figure [START_REF] Chen | Atomic decomposition by basis pursuit[END_REF] shows an exemple of denoised images and table (2) and (3) show the average results obtained by the four methods on 6 levels of noise that are respectively evaluated by MSE and Jaccard 's measures. A paired Wilcoxon signed test with a significance level of 5% is used also to check whether the difference between the results obtained by our method and the ones obtained by the other methods is significant. In these tables, an entry mark by (-) indicates that the corresponding method performs worst than our method.

Similarly, an entry marked by (+) indicates that the corresponding method outperforms the proposed method, and an entry marked by (=) indicates that results obtained by the both methods are not significantly different. Table (2) and ( 3) also show that at level 5 and level 6, none of the four methods are good enough but other methods are worse than ours. We further examine the set of noisy images at level 4 and we found that the set of noisy patches of the corrupted image cannot provide a good training data since most of patches are trivial (zeros value), making not enough discrimination between A (0) and A (k) (see algorithm 1). This can explain why the curvelets transform method is better than the proposed method at this level of noise. Although at level 1 the Wilcoxon signed test indicates that results obtained by the curvelets and our method are not significantly different, when we zoom the denoised images we found that the intersection of edges are not well restored with curvelet as shown in Figure [START_REF] Choi | A survey of binary similarity and distance measures[END_REF]. Since curvelets are smooth functions, they are not well adapted for singularity points.

We also verified our method on real scanned documents created by printing the original documents and scanning printed documents at the different resolutions. By using 15 original documents, we got 12 scanned datasets with different resolutions. Figure [START_REF] Davies | Machine Vision: Theory, Algorithms and Practicalities[END_REF] shows an example of one of the scanned documents.

The aim is to test our method on real documents with an unknown model of noise twelve scanned images are tested and evaluated using the Structural similarity (SSIM) index. SSIM [START_REF] Wang | Image quality assessment: From error visibility to structural similarity[END_REF] is also a method for measuring the similarity that is designed to improve MSE, which have proved to be inconsistent with human eye perception. SSIM metric is computed on various windows of images. The distance between two windows si and qi is calculated following the equation [START_REF] Kanungo | Global and local document degradation models[END_REF] where si,qi are taken from the same location of the noisy image and reconstructed image.

S(si,qi)=( 2µs

i µq i + c1 µ 2 s i + µ 2 q i + c1 )( 2σs i σq i + c2 σ 2 s i + σ 2 q i + c2 ) ( 12
)
Original image level 1 level 2 µs i ,µq i ,σs i ,σq i are the average local and the sample standard deviations of si and qi, respectively; c1 =(k1L) 2 ,c2 = (k2L) 2 are two variables to stabilize the division with weak denominator; L is the dynamic range of the pixel-values; and k1,k2 are two constants. The resultant SSIM index is a decimal with a value 1 in the case of two identical sets of data. In this paper, SSIM index is calculated on a window size 8 × 8, and the values of L, k1,k2 are respectively 100, 0.01 and 0.03. Table [START_REF] Choi | A survey of binary similarity and distance measures[END_REF] presents the results in comparison with other methods. We can see that in each case the performance of our approach is good compared to the other.

The last experiment is done on the DIBCO 2009 dataset. This dataset contains images that range from grayscale to color and from real to synthetic. The value of r for this experiment equals 0.7321 and is calculated as the same way as described above. As the experiment before, the SSIM measure is used with the purpose of comparison. Table [START_REF] Daubechies | Iteratively reweighted least squares minimization for sparse recovery[END_REF] presents the results for the 5 handwritten images shown in Figure 7 and we can see that in each case our approach provides a good result. Tables ( 4) and ( 5) present the difference results (the last rows) using also a paired Wilcoxon signed test with a significance level of 5%. We can observe that the difference between our approach performance compared to the other methods per- To conclude this section, all evaluation measures have indicated that our method performs better than all of the other methods in most of the cases.

CONCLUSIONS

A novel algorithm for denoising document images by using learning dictionary based on sparse representation has been presented in this paper. Learning method starts by building a training database from corrupted images, and constructing an empirically learned dictionary by using sparse represen- tation. This dictionary can be used as a fixed dictionary to find the solution of the basis pursuit denoising problem. In addition, we provide a way to define the best value of the tolerance error (ǫ) based on a measure of fidelity between two images. The efficiency of ǫ has been also approved experimentally on different datasets for different resolutions and different kinds of noise. All experimental results show that our method outperforms existing ones in most of the cases.

Figure ( 1 )Figure 1 :

 11 Figure 1: (a): Original binary symbol; from (b) to (g) examples of six levels of Kanungo noise of the GREC 2005 dataset.

  A with the sparse representation model in the purpose of denoising image, following: a. Find the solution of the optimization problem (6) for each patch hj xj =argmin xj 1 subject to Axj -hj 2 ≤ ǫ, b. Compute the denoised version of each patch hj by ĥj = A xj, c. Merge the denoised patches ĥj to get the denoised image ŷ.

Figure 2 :

 2 Figure 2: Normalized cross-correlation between two images. Level 1 2 3 4 5 6 r 0.9133 0.4412 0.5629 0.3698 0.4413 0.2006

  926 (-) 0.937 (-) 0.963 (=) 0.963 Level 2 0.630 (-) 0.177 (-) 0.565 (-) 0.655 Level 3 0.428 (-) 0.787 (-) 0.389 (-) 0.826 Level 4 0.088 (-) 0.001 (-) 0.459 (+) 0.135 Level 5 0.261 (-) 0.268 (-) 0.263 (-) 0.284 Level 6 0.001 (-) 0.000 (-) 0.059 (=) 0.060

Figure 3 :Figure 4 :

 34 Figure 3: Results of denoising the noisy images with Kanungo model at levels 1 and 2 of degredation. Columns 2 and 3 are the denoised images following each method. Columns 4 and 5 are the binarized denoised images of columns 2 and 3, respectively. For the median and OC, images are already binarized in columns 2 and 3.

Figure ( 6 )

 6 presents one learning dictionary build by corrupted patches with size 8 × 8o nD I B C O images. Figure (7a) gives the documents in DIBCO dataset and its denoised versions got by our approach (Figures (7b) and (7c)).

Figure 5 :

 5 Figure 5: An example of a scanned documents.

Figure 6 :

 6 Figure 6: The trained dictionary on DIBCO images with w =8.

Table 1 :

 1 The value of r at six level of noise.value ri =0 .4106; and table[START_REF] Aharon | An algorithm for designing overcomplete dictionaries for sparse representation[END_REF] presents the best values of r corresponding to each level of degradation.

Table 2 :

 2 Average value gained by Jaccard's similarity measure.

	Median	OC	Curvelet Proposed method
	Level 1 2.896 (-) 2.454 (-) 1.403 (=)	1.423
	Level 2 21.015(-) 30.865 (-) 28.421 (-)	19.687
	Level 3 53.690 (-) 8.899 (-) 57.771 (-)	8.008
	Level 4 33.599 (-) 36.819 (-) 19.815 (+)	31.881
	Level 5111.632 (-)107.358 (-) 110.286 (-)	98.212
	Level 6 37.863 (-) 37.915 (-) 35.632 (=)	35.772

Table 3 :

 3 Average values gained by MSE.

Table 4 :

 4 The obtained results when comparing proposed method with structural similarity SSIM index.formance is statistically significant even in the case where the noise model is unknown.

	Images Median	OC	Curvelet Proposed method
	1	0.5374	0.4640	0.4693	0.6834
	2	0.6906	0.5165	0.6184	0.7534
	3	0.5749	0.5158	0.5057	0.7421
	4	0.6482	0.4879	0.5666	0.7454
	5	0.6008	0.5056	0.5186	0.7458
	6	0.6310	0.4409	0.5320	0.7145
	7	0.6018	0.5030	0.5244	0.7580
	8	0.6066	0.4814	0.5199	0.7381
	9	0.6487	0.4662	0.5638	0.7456
	10	0.6203	0.4603	0.5377	0.7317
	11	0.6946	0.4832	0.5987	0.7685
	12	0.7264	0.4494	0.6274	0.7733
	Average 0.6320 (-) 0.4812 (-) 0.5485 (-)	0.7416
	Images Median	OC	Curvelet Ours approach
	1	0.6420	0.6926	0.7054	0.9528
	2	0.4628	0.5155	0.5156	0.9784
	3	0.5115	0.5315	0.5064	0.8648
	4	0.4595	0.4946	0.4692	0.8933
	5	0.6953	0.7283	0.7441	0.9416
	Average 0.5542 (-) 0.5925 (-) 0.5881 (-)	0.9261

Table 5 :

 5 The obtained results with DIBCO 2009 dataset using SSIM measure.

x

( M i=1 |xi| 2 ) 1/2 ,w i t hx ∈ R M

ACKNOWLEDGEMENTS

Oriol Ramos Terrades, has been partially supported by the Spanish project TIN2012-37475-C02-02.