Application of the linear sampling method to retrieve cracks with impedance boundary conditions Résumé : Nous utilisons dans ce travail la Linear Sampling Method pour reconstruire une fissure impédante dans le vide. Ce papier est une extension du travail de Cakoni et Colton [START_REF] Cakoni | The linear sampling method for cracks[END_REF] qui applique la Linear Sampling Method pour reconstruire une fissure avec une condition mixte et ceci en utilisant plusieurs mesures de champs lointains associés à différentes ondes incidentes planes. Nous donnons dans ce travail une validation théorique et numérique de notre méthode.

Mots-clés :

Problème inverse en électromagnétisme, méthode d'échantillonnage, condition d'impédance, détection de fissure 1 Introduction We are concerned in this work by the inverse scattering problem from a crack with an impedance boundary condition. More precisely we shall investigate the application of the Linear Sampling Method (LSM) to retrieve the geometry of the crack from multi-static far field data at a given frequency. As compared to previous works on this type of problem and methods [START_REF] Cakoni | The linear sampling method for cracks[END_REF][START_REF] Delbary | Identifications de fissures par des ondes acoustiques[END_REF][START_REF] Kirsh | A linear sampling method for inverse scattering from an open arc[END_REF], the originality of the present one relies on considering the case of the impedance boundary conditions on both sides of the crack. This induces more technical difficulties in the justification of the method and also on the numerical side. For instance the algorithm proposed in [START_REF] Cakoni | The linear sampling method for cracks[END_REF] does not work if one of the impedance values is not infinite. The main difficulty relies on the choice of the orientation of the probing " small " crack. Based on the theoretical justification of LSM we propose a minimization procedure that enforces the correct choice of this orientation. Prior to considering the inverse problem, we propose a study of the forward problem using an integral equation approach. This part is then used in designing a numerical method to solve this problem. It is used for instance, to generate the synthetic data needed for the inverse scheme. The theoretical developments of this first part are inspired by [START_REF] Cakoni | The linear sampling method for cracks[END_REF]. We also quote the work of [START_REF] Kress | Integral equation methods for scattering from an impedance crack[END_REF] for the study of the direct problem using an integral equation approach in Hölder spaces and the work of [START_REF] Liu | Reconstruction of cracks of different types from far-field measurements[END_REF] for the study of the inverse problem using a different sampling method. The extension of the current work to the factorization method [START_REF] Kirsh | The Factorization method for inverse problem[END_REF] and to the reciprocity gap sampling method [START_REF] Colton | An application of the reciprocity gap functionnal to inverse scattering theory[END_REF] is under preparation.

The report is organized as follows. In section 2, the direct problem is introduced along with a study of the uniqueness and existence of the solution using an integral equation method. The validation of the numerical implementation of this problem is then presented by comparing the results to those given by a volumetric method (using FreeFem++ [START_REF] Hecht | FreeFem++,Third edition, version 3.10-2[END_REF]). In section 3, we introduce the inverse problem and we describe our formulation of the LSM along with the mathematical justifications. Lastly, we propose and test different sampling schemes associated with section 3, for different shapes of the cracks and different values of the impedances.

The direct scattering problem for a crack with impedance boundary conditions

Let σ ⊂ R m , m = 2, 3, be a smooth nonintersecting open arc. For further considerations, we assume that σ can be extended to an arbitrary smooth, closed curve ∂Ω enclosing a bounded domain Ω in R m . The normal vector ν on σ coincides with the outward normal vector to ∂Ω, with simply connected complement. Impedance type boundary conditions on σ lead to the following problem

∆u + k 2 u = 0 in R m \ σ ∂ ν u ± ± λ ± u ± = 0 on σ (1)
where the wave number k is positive and λ ± ∈ L ∞ (σ) are the given (complex-valued) impedance functions with non-negative imaginary part and λ + + λ -= 0. Notice that u ± (x) = lim h→0 + u(x ± hν) and

∂ ν u ± = lim h→0 + ν • ∇u(x ± hν) for x ∈ σ.
The total field u = u i + u s is decomposed into the given incident plane wave u i (x, d) = e ikd•x with unitary direction d and the unknown scattered field u s which is required to satisfy the Sommerfeld radiation condition lim r=|x|→+∞ r m-1

2 (∂ r u s -iku s ) = 0, (2)
uniformly in all directions x = x |x| . In order to formulate the scattering problem more precisely we need to define the trace spaces on σ. If H 1 loc (R m \ σ), H 1/2 (∂Ω) and H -1/2 (∂Ω) denote the usual Sobolev spaces we define the following spaces (see [START_REF] Mclean | Strongly Elliptic Systems and Boundary Integral Equations[END_REF]) :

H 1/2 (σ) = u |σ : u ∈ H 1/2 (∂Ω) H1/2 (σ) = u ∈ H 1/2 (σ) : supp(u) ⊂ σ .
Now we denote by H -1/2 (σ) and H-1/2 (σ) the dual spaces of H1/2 (σ) and H 1/2 (σ) respectively. Using the notation

g ± = -(∂ ν u i ± λ ± u i ) (3)
the problem (1)-( 2) is a special case of the

Impedance crack problem (ICP) : Given g ± ∈ H -1/2 (∂Ω) and (λ ± ) ∈ L ∞ (σ) with Im(λ ± ) ≥ 0 and λ + = -λ -, find u s ∈ H 1 loc (R m \ σ) satisfying the Sommerfeld radiation condition (2) and ∆u s + k 2 u s = 0 in R m \ σ (4)
∂ ν u s ± ± λ ± u s ± = g ± on σ. (5)
The aim of this section is to prove the existence of the solution of the (ICP) problem, since the following uniqueness result can be found in [START_REF] Kress | Integral equation methods for scattering from an impedance crack[END_REF] and [START_REF] Delbary | Identifications de fissures par des ondes acoustiques[END_REF] Theorem 1 The (ICP) problem has at most one solution.

To prove the existence of the solution of the (ICP), we use a boundary integral equations approach (see [START_REF] Delbary | Identifications de fissures par des ondes acoustiques[END_REF] for a sound soft crack and a sound hard crack, [START_REF] Cakoni | The linear sampling method for cracks[END_REF] for a partially coated crack and [START_REF] Kress | Integral equation methods for scattering from an impedance crack[END_REF] for a 2D impedance crack when λ + = λ -is Hölder continuous). We define [u] := u +u -and [∂ ν u] := ∂ ν u + -∂ ν u -, the jump of u and ∂ ν u respectively, across the crack σ.

Lemma 1 Further assume that (λ ± ) -1 ∈ L ∞ (σ). The scattered field u s is solution of the (ICP) if and only if the vector ([u], [∂ ν u])
T solves the system of integral equations

A σ   [u] [∂ ν u]   =   -(λ + + λ -)∂ ν u i -(λ + + λ -)u i   (6)
where the matrix operator A σ : H1/2 (σ) × H-1/2 (σ) → H -1/2 (σ) × H -1/2 (σ) is given by

A σ :=     λ + λ -I + (λ + + λ -)T σ - 1 2 (λ + -λ -)I -(λ + + λ -)K ′ σ 1 2 (λ + -λ -)I + (λ + + λ -)K σ I -(λ + + λ -)S σ     (7) Proof If u s is a solution of the (ICP) then [u s ] ∈ H1/2 (σ) and [∂ ν u s ] ∈ H-1/2 (σ) (see Lemma 2.2 in [1]).
Using the Green representation formula, the following relation holds for z ∈ R m \ σ (see theorem 2.24 in [START_REF] Delbary | Identifications de fissures par des ondes acoustiques[END_REF])

u s (z, d) = σ [u(y, d)] ∂Φ(z, y) ∂ν(y) - ∂u(y, d) ∂ν(y) Φ(z, y) ds(y), ( 8
)
where Φ is the fundamental solution to the Helmholtz equation defined by

Φ(x, y) :=      1 4π e iκ|x-y| |x -y| m = 3 i 4 H (1) 0 (κ|x -y|) m = 2
with x = y and H

(1) 0

being the Hankel function of the first kind of order zero. Next by making use of the jump relations of the single and double layer potentials across ∂Ω (see [START_REF] Mclean | Strongly Elliptic Systems and Boundary Integral Equations[END_REF]), we obtain from ( 8)

u s ± = K σ [u] -S σ [∂ ν u] ± 1 2 [u] (9)
∂ ν u s ± = T σ [u] -K ′ σ [∂ ν u] ± 1 2 [∂ ν u] (10)
where S ∂Ω , K ∂Ω , K ′ ∂Ω , T ∂Ω are the boundary integral operators

S ∂Ω : H -1/2 (∂Ω) → H 1/2 (∂Ω) K ∂Ω : H 1/2 (∂Ω) → H 1/2 (∂Ω) K ′ ∂Ω : H -1/2 (∂Ω) → H -1/2 (∂Ω) T ∂Ω : H 1/2 (∂Ω) → H -1/2 (∂Ω)
defined by

S ∂Ω ψ(x) := ∂Ω ψ(y)Φ(x, y)ds(y), K ∂Ω ϕ(x) := ∂Ω ϕ(y) ∂Φ(x, y) ∂ν(y) ds(y), K ′ ∂Ω ψ(x) := ∂Ω ψ(y) ∂Φ(x, y) ∂ν(x) ds(y), T ∂Ω ϕ(x) := ∂ ∂ν(x) ∂Ω ϕ(y) ∂Φ(x, y) ∂ν(y) ds(y)
and S σ , K σ , K ′ σ , T σ are the corresponding operators defined on σ. These restricted operators have the following mapping properties (see [START_REF] Mclean | Strongly Elliptic Systems and Boundary Integral Equations[END_REF])

S σ : H -1/2 (σ) → H 1/2 (σ) K σ : H 1/2 (σ) → H 1/2 (σ) K ′ σ : H -1/2 (σ) → H -1/2 (σ) T σ : H 1/2 (σ) → H -1/2 (σ).
Moreover, from the boundary conditions ( 5) and (3), we have

[∂ ν u] + λ + u s + + λ -u s -= -(λ + + λ -)u i , λ + λ -[u] + λ -∂ ν u s + + λ + ∂ ν u s -= -(λ + + λ -)∂ ν u i . (11)
Finally, by combining the relations ( 9), ( 10) and ( 11), we show that ([u], [∂ ν u]) T solves the system of equations [START_REF] Hecht | FreeFem++,Third edition, version 3.10-2[END_REF].

Conversely, assume that ([u], [∂ ν u])

T satisfies ( 6)-( 7), so

λ -T σ [u] -K ′ σ [∂ ν u] + [∂ ν u] 2 + λ + T σ [u] -K ′ σ [∂ ν u] - [∂ ν u] 2 + λ + λ -[u] = -(λ + + λ -)∂ ν u i (12)
and

λ + K σ [u] -S σ [∂ ν u] + [u] 2 + λ -K σ [u] -S σ [∂ ν u] - [u] 2 + [∂ ν u] = -(λ + + λ -)u i (13)
The potential u s , defined by [START_REF] Kirsh | A linear sampling method for inverse scattering from an open arc[END_REF], belongs to H 1 loc (R m \ σ) and satisfies the Helmholtz equation in R m \ σ and the Sömmerfeld radiation condition. It remains to show that u s satisfies the boundary condition [START_REF] Hamdi | Formulation variationnelle par équations intégrales pour le calcul de champs acoustiques linéaire proches et lointains[END_REF]. To this end, we first inject [START_REF] Liu | Reconstruction of cracks of different types from far-field measurements[END_REF] in ( 12) and ( 9) in (13). We obtain

λ -∂ ν u s + + λ + ∂ ν u s -+ λ + λ -[u] = -(λ + + λ -)∂ ν u i (14) λ + u s + + λ -u s -+ [∂ ν u] = -(λ + + λ -)u i (15)
Multiplying (15) by λ ± and adding the results to (14), we show that

∂ ν u s ± -λ ± u s ± = -(∂ ν u i ± λ ± u i ) on σ.
Therefore, the field u s given by ( 8) is a solution of the (ICP).

Lemma 2

The operator A σ given by [START_REF] Kirsh | The Factorization method for inverse problem[END_REF] has a trivial kernel.

Proof. Let ζ = (α, β) T ∈ H1/2 (σ) × H-1/2 (σ) satisfying A σ ζ = 0. Define the potential v = σ ∂Φ(x, y) ∂n(y) α(y)ds(y) - σ Φ(x, y)β(y)ds(y); x ∈ R m \ σ.
This potential belongs to H 1 loc (R m \ σ) and satisfies the Helmholtz equation in R m \ σ and the Sommerfeld radiation condition. Moreover, using the jump relations of the single and double layer potentials across σ, we get

v ± = K σ (α) -S σ (β) ± α 2 ∂ ν v ± = T σ (α) -K ′ σ (β) ± β 2 Hence, [v] = α and [∂ ν v] = β. (16)
Therefore, as in the proof of Lemma 1, we show that

v ∈ H 1 loc (R 2 \ σ) satisfies ∆v + k 2 v = 0 in R 2 \ σ, ∂ ν v ± -λ ± v ± = 0 on σ
and the Sömmerfeld radiation condition. Then, from the uniqueness of the solution of this system, we have v = 0 and we conclude, by ( 16), that α = β = 0.

Lemma 3 The operator

A σ : H1/2 (σ) × H-1/2 (σ) → H -1/2 (σ) × H -1/2 (σ)
given by ( 7) has a bounded inverse.

Proof.

Let ϕ ∈ H 1/2 (∂Ω) and ψ ∈ H -1/2 (∂Ω) be the extension by zero to ∂Ω of [u] ∈ H1/2 (σ) and [∂ ν u] ∈ H-1/2 (σ) respectively. We denote by T 0 the boundary integral operator corresponding to the Laplace operator, defined as T ∂Ω by replacing the kernel Φ(x, y) by

Φ 0 (x, y) :=      1 4π 1 |x -y| m = 3 - 1 2π ln |x -y| m = 2
The operator -T 0 is coercive on ∂Ω. Hence, we have for all ϕ ∈ H 1/2 (∂Ω)

Re( -T 0 ϕ, ϕ H -1/2 (∂Ω),H 1/2 (∂Ω) ) ≥ ϕ H 1/2 (∂Ω) . Then Re( -T 0 [u], [u] H -1/2 (σ), H1/2 (σ) ) ≥ [u] H1/2 (σ) ,
which leads to the coercivity of the operator -T 0 on σ whence the operator

A 0,σ =   (λ + + λ -)T 0 -1 2 (λ + + λ -)I 0 I   is invertible. Consider now the operator A c =    λ + λ -I + (λ + + λ -)T c -(λ + + λ -)K ′ ∂Ω λ + -λ - 2 I + (λ + + λ -)K ∂Ω -(λ + + λ -)S ∂Ω .   
The operator T c := T ∂Ω -T 0 is compact since it has a continuous kernal. Since the injection from H 1 2 (∂Ω) to H -1 2 (∂Ω) is compact and λ ± ∈ L ∞ (∂Ω), the operators (λ +λ -)I and λ + λ -I are also compact from H 1 2 (∂Ω) to H -1 2 (∂Ω). In the other hand, K ′ ∂Ω is a compact operator from H -1 2 (∂Ω) to H -1 2 (∂Ω). The operator S ∂Ω is continuous from

H -1 2 (∂Ω) to H 1 2 (∂Ω). Thus (λ + + λ -)S ∂Ω is compact from H -1 2 (∂Ω) to H -1 2 (∂Ω) . The operator K ∂Ω is continuous from H 1 2 (∂Ω) to H 1 2 (∂Ω), which implies that (λ + + INRIA λ -)K ∂Ω is compact from H 1 2 (∂Ω) to H -1 2 (∂Ω). We conclude that A c : H1/2 (∂Ω) × H-1/2 (∂Ω) → H -1/2 (∂Ω) × H -1/2 (∂Ω) is a compact operator. Thus the restriction of A c to H1/2 (σ) × H-1/2 (σ) denoted by A c,σ is also compact. Finally A σ = A 0,σ + A c,σ
is Fredholm with index zero and since by Lemma 2 A σ is injective we conclude that it has a bounded inverse. As a consequence of these lemmas, we have the following result Theorem 2 The (ICP) problem has a unique solution given by

u s (z, d) = σ [u(y, d)] ∂Φ(z, y) ∂ν(y) - ∂u(y, d) ∂ν(y) Φ(z, y) ds(y), z ∈ R m \ σ (17)
where,

([u], ∂ ν u])
T is the unique solution of the system of integral equations ( 6).

Numerical solution of the (ICP) problem

Let A σ : H1/2 (σ) × H-1/2 (σ) → H -1/2 (σ) × H -1/2 (
σ) be the matrix operator given by [START_REF] Kirsh | The Factorization method for inverse problem[END_REF]. We use a Galerkin method to solve numerically the system of integral equations

A σ ϕ ψ = f 1 f 2 . ( 18
)
We multiply the two equations in (18) by test functions α and β ∈ H 1 2 (σ) respectively and we integer over σ (in the sense of the duality pairing between H -1/2 (σ) and

H 1 2 (σ)) σ λ + λ -αϕ + σ C 1 αT σ ϕ - σ C 2 αψ - σ C 1 αK ′ σ ψ = σ αf 1 (19) σ C 1 βϕ + σ C 2 βK σ ϕ + σ λ + λ -βψ - σ C 1 βS σ ψ = σ βf 2 (20)
where we simplified the notations by denoting

C 1 = (λ + + λ -), C 2 = 1 2 (λ + -λ -).
Then we discretize the space H -1/2 (σ) × H1/2 (σ) using finite elements P 1 × P 1 . To compute the integral involving the operator T σ we use the following formula (see [START_REF] Hamdi | Formulation variationnelle par équations intégrales pour le calcul de champs acoustiques linéaire proches et lointains[END_REF])

σ α(x) ∂ ∂ν(x) σ ϕ(y) ∂Φ(x, y) ∂ν(y) ds(y) ds(x) = κ 2 σ σ Φ(x, y)ν(x) • ν(y)α(x)ϕ(y) ds(x) ds(y) - σ σ Φ(x, y)(∇α(x) × ν(x)) • (∇ϕ(y) × ν(y)) ds(x) ds(y).
We end the study of the direct problem by some numerical tests obtained by the scheme described above. To validate our algorithm, we use also FreeFem++ to solve (ICP) problem using FEM method. To this end, we discretize the disk of radius 10 times the wavelength by P 1 elements. We compare the modulus of the far field pattern (see ( 21)), computed by the two methods (the scheme based on integral equations and that using FEM) for 100 observation points, different crack shapes and different incident plane waves u i (x) = exp(ik(x 1 cos(θ) + x 2 sin(θ))) with angle θ. We keep the wave number constant k = 2π for all the tests and we choose two different values of the impedances λ -= λ + = 5 + i and λ -= λ + = 1 + i.

We recall that the solution of the (ICP) problem has the asymptotic behaviour of an outgoing spherical wave (see [START_REF] Colton | Inverse acoustic and electromagnetic scattering theory[END_REF])

u s (x, d) = r 1-m 2 e ikr u ∞ (x, d) + O(r -m+1 2 ), (21)
where u ∞ is the far field pattern of the scattered wave, x ∈ R m \σ, x = x/|x| and r = |x|. We can clearly see that the results (compare Figure 1 to 6) obtained by the two methods are very close which validates our scheme. Note that the integral equation method is much faster than the finite element method since the size of the linear system is much smaller (even though, the matrix is not sparse).

3 The inverse problem

Settings and theoretical justification of the LSM

In this section, we adapt the LSM for scattering by a partially coated crack (see [START_REF] Cakoni | The linear sampling method for cracks[END_REF]) to the following inverse problem :

Inverse scattering by an Impedance Crack (IIC). Given the far field pattern u ∞ (•, •) on S m-1 × S m-1 of the solution to (ICP), reconstruct the crack σ.

To solve the (IIC) by the LSM method, we first define the far field operator

F : L 2 (S m-1 ) → L 2 (S m-1 ) g → S m-1 u ∞ (•, y)g(y)ds(y) (22)
and consider the far field equation

F (g L )(x) = Φ ∞ L (x) for all x ∈ S m-1 (23)
where

Φ ∞ L ∈ L 2 (S m-1 ) is given by Φ ∞ L (x) = γ L α L (y) ∂e -iκx•y ∂ν(y) + β L (y)e -iκx•y ds(y), (24)
with γ =      e iπ/4 √ 8πκ in 2D 1 4π in 3D (25)
Notice that Φ ∞ L is the far field pattern of the potential Φ L defined by

Φ L (x) := L α L (y) ∂Φ(x, y) ∂ν(y) + β L (y)Φ(x, y) ds(y) (26)
with densities α L ∈ H 1 2 (L) and β L ∈ H-1 2 (L), for any smooth non intersecting open arc L ∈ R m-1 . We will characterize the crack σ by the behaviour of an approximate solution g L of the far field equation (23).

To prove the existence of an approximate solution of (23), we factorize the operator F as F = BH where B :

H -1/2 (σ) × H -1/2 (σ) → L 2 (S m-1
) maps the boundary data (g + , g -) to the far field pattern of the solution of (ICP) and H is the trace operator defined by

H : L 2 (S m-1 ) → H -1/2 (σ) × H -1/2 (σ) g → ((∂ ν -λ -)v g , (∂ ν + λ + )v g )
where v g is the Herglotz wave function of kernal g ∈ L 2 (S m-1 ),

v g (x) := S m-1 e iκx•d g(d) ds(d), x ∈ R m . ( 27
)
We will show that the traces of the solution of the (ICP) on both sides of σ can be approximated by the appropriate traces of the Herglotz wave function v g . We shall keep the same hypothesis on λ ± as in section 2 and further assume that (λ + + λ -)(x) = 0 for a.e x ∈ σ.

Lemma 4

The operator H is injective and has a dense range.

Proof. Let g ∈ L 2 (S m-1 ) be an element of the kernal of H. Then,

(∂ ν ± λ ± )v g = 0 on σ.
Since by assumption (λ + + λ -) = 0 then v g = 0 and ∂ ν v g = 0 on σ 0 ⊂ σ. From the unique continuation principle it implies that v g = 0 in R m and therefore g = 0 which proves that H is one to one.

The main idea of the proof of the second part of the theorem is to show that H * , the adjoint operator of H, is injective.

To this end, we determine first the expression of H * . Let g ∈ L 2 (S m-1 ) and α, β ∈ H1/2 (σ), then

H * : H1/2 (σ) × H1/2 (σ) → L 2 (S m-1
) satisfies

H(g), (α, β) H * ,H = g, H * (α, β) L 2 (S m-1 ),L 2 (S m-1 )
where H := H1/2 (σ) × H1/2 (σ). By changing the order of integration and using (27),

H(g), (α, β) H * ,H = S m-1 g(d) σ α(y) ∂ ∂ν(y) -λ -+ β(y) ∂ ∂ν(y) + λ + e -iκd•y ds(y) ds(d).
Therefore, for d ∈ S m-1 , the operator H * is given by

H * (α, β)(d) = σ α(y) ∂ ∂ν(y) -λ -e -iκd•y + β(y) ∂ ∂ν(y)
+ λ + e -iκd•y ds(y).

We observe that H * is the far field pattern of the potential

γ -1 V (z) = σ (α(y) ∂ ∂ν(y) -λ -Φ(z, y) + β(y) ∂ ∂ν(y) + λ + Φ(z, y))ds(y),
for z ∈ R m \ σ. This function is well defined in R m \ σ since α and β can be extended by zero to functions in H 1/2 (σ). Moreover, V ∈ H 1 loc (R m \ σ) satisfies the Helmholtz equation and the sommerfeld radiation condition. Therefore, if H * (α, β) = 0, the far field pattern of V is zero and from Rellich's lemma and the unique continuation principle we conclude that V = 0 in R m \ σ. Then, by the jump relations of the layer potentials, we have

[V ] = γ(α + β) and [∂ ν V ] = γ(λ -α -λ + β). This implies that α + β = 0 and λ -α -λ + β = 0.
Finally, since by assumption λ + (x) = -λ -(x) for a.e x ∈ σ, α = β = 0 and the operator H * is injective.

In the next step, we provide some properties of the operator B. We need to introduce the operators F : H1/2 (σ) × H-1/2 (σ) → L 2 (S m-1 ) given by

F(ϕ, ψ)(x) := γ σ ϕ(y)
∂e -iκxy ∂ν(y) + ψ(y)e -iκxy ds(y)

and

M : H1/2 (σ) × H-1/2 (σ) → H -1/2 (σ) × H -1/2 (σ) defined by M := T + λ + (K + 1 2 I) -λ + S -K ′ + 1 2 I T -λ -(K -1 2 I) λ -S -K ′ -1 2 I . ( 29
(∂ ν + λ + )P + (ϕ, ψ) (∂ ν -λ -)P -(ϕ, ψ) = M ϕ ψ
where

M : H1/2 (σ) × H-1/2 (σ) → H -1/2 (σ) × H -1/2 (σ)
is given by (29). This operator is related to the matrix A σ defined by [START_REF] Kirsh | The Factorization method for inverse problem[END_REF]. More precisely,

A σ = λ -I λ + I I -I M
and by Lemma 3, M -1 :

H -1/2 (σ) × H -1/2 (σ) → H1/2 (σ) × H-1/2 (σ)
exists and is bounded.

Lemma 6

The operator

F : H1/2 (σ) × H-1/2 (σ) → L 2 (S m-1
) is injective and has a dense range in L 2 (S m-1 ).

Proof. The injectivity of F can be proved in the same way as in Lemma 4, by replacing the potential V by P . Proceeding again as in the proof of Lemma 4, let g ∈ L 2 (S m-1 ) and (α, Therefore,

β) ∈ H1/2 (σ) × H-1/2 (σ), F(α, β), g = S m-1 g(d)F(α, β)(d)ds(d) = γ S m-1 g(d)
F * (g) = γ(∂ ν v g , v g ). Now, if F * (g) = 0 then v g = ∂ ν v g = 0.
Thus, as in Lemma 4, g = 0 which proves the density of the range of F. Summarizing the previous results, the operator F defined by ( 22) is factorized as F = FM -1 H. Hence, the range of F is included in the range of F. Therefore, thanks to the following Lemma, there exist an approximated solution of the equation (23).

Lemma 7 For any smooth non intersecting arc L and functions

α L ∈ H 1 2 (L), β L ∈ H-1 2 (L), the operator Φ ∞
L given by ( 24) belongs to R(F) the range of F if and only L ⊂ σ. Proof. First assume that L ⊂ σ. Since H± 1 2 (L) ⊂ H± 1 2 (σ), it follows from (28) that Φ ∞ L (x) ∈ R(F). Now let L ⊂ σ and assume, on the contrary, that Φ ∞ L ∈ R(F). Hence, there exists ϕ ∈

H 1 2 (σ) and ψ ∈ H-1 2 (σ) such that Φ ∞ L (x) = γ L ϕ(y) ∂e -iκx•y ∂ν(y) + ψ(y)e -iκx•y (y) ds(y).
Thus Φ ∞ L is the far field pattern of the potential

P (x) = σ ϕ(y) ∂Φ(x, y) ∂ν(y) + ψ(y)Φ(x, y) ds(y), x ∈ R m-1 \ σ.

INRIA

Since by definition Φ ∞ L is also the far field pattern of the potential Φ L given by (26) then using the Rellich lemma and the unique continuation principle, the potentials Φ L and P coincide in R m-1 \ (σ ∪ L). Let x 0 ∈ L\σ and B ǫ a small neighborhood of x 0 with B ǫ ∩ σ = ∅. Then, P is analytic in B ǫ while Φ L has a singularity at x 0 which is a contradiction. This proves that Φ ∞ L / ∈ R(F).

Theorem 3 We assume that L is a nonintersecting smooth open arc. The following is true:

1. If L ⊂ σ; there exists a sequence (g n ) n∈N on L 2 (S m-1 ) such that lim n→+∞ F (g n ) -Φ ∞ L L 2 (S m-1 ) = 0 and lim n→+∞ ||v gn || * < ∞,
where

v gn * := ∂ ν v gn + λ + v gn H -1 2 (σ) + ∂ ν v gn -λ -v gn H -1 2 (σ)
.

Otherwise, for any sequence

(g n ) n∈N ⊂ L 2 (S n-1 ) that satisfies lim n→+∞ F (g n ) -Φ ∞ L L 2 (S m-1 ) = 0 we have lim n→+∞ ||v gn || * = +∞.
Proof.

❼ If L ⊂ σ, it is easy to find a bounded solution of the far field equation ( 23). In fact, we have

H 1 2 (L) ⊂ H 1 2 (σ) and H-1 2 (L) ⊂ H-1 2 (σ). Thus, Φ ∞ L ∈ R(F) there exists a unique (α, β) ∈ H 1 2 (σ) × H-1 2 (σ) such that Φ ∞ L = F(α, β) Therefore by Lemma 5 there exists a unique (α, β) ∈ H 1 2 (σ) × H-1 2 (σ) such that Φ ∞ L = FM -1 (α, β) T .
Moreover by Lemma 4 the range of H is dense on H -1/2 (σ) × H -1/2 (σ), hence there exists (α n , β n ) n∈N ⊂ R(H) such that lim n→+∞ (α n , β n ) = ( α, β). Using the continuity of H, we show the existence of a sequence (g n ) n∈N ⊂ L 2 (S n-1 ) verifying lim n→∞ Hg n = (α, β) which proves that lim n→+∞

F (g n ) -Φ ∞ L L 2 (S m-1 ) = 0.
❼ Let L ⊂ σ and let us assume that v gn * < ∞. Therefore,

∂ ν v gn + λ + v gn H -1 2 (σ) + ∂ ν v gn -λ -v gn H -1 2 (σ) < ∞. (30) Let (α n , β n ) ∈ R(H). Since H is injective there exists a sequence (g n ) n∈N ⊂ L 2 (S n-1 ) such that H(g n ) = (α n , β n ). From (30), the sequence (α n , β n ) is bounded in H -1 2 (σ) × H -1 2 (
σ). Therefore we can extract a subsequence, that we still denote (α n , β n ) which weakly converges to (α, β). The integral operator F is compact since it has a regular kernal and M -1 is a bounded operator, so that the operator FM -1 is a compact operator. Consequently,

lim n→+∞ F (g n ) -FM -1 (α, β) L 2 (S m-1 ) = 0
and by the uniqueness of the limit, we have

FM -1 (α, β) = Φ ∞
L . We deduce that Φ ∞ L ∈ R(F) and L ⊂ σ which is a contradiction.

Numerical schemes and results

Regularization

For the purpose of the numerical experiments, we construct a nearby solution to (23) using a Tikhonov regularization. Therefore we solve the following equation

(ηI + F * F )g η = F * Φ ∞ L
where η is a parameter of regularization. By the singular value decomposition (SVD) we mean a representation of F in the form

F (g) = i σ i (g, f i )l i ,
where (f i ), (l i ) are orthonormal systems in L 2 (S 1 ), and σ i are positive constants, the singular values of F . The adjoint of F is given by

F * (g) = i σ i (g, l i )f i ,
Thus, the operator F * F is given by

F * F (g) = i σ 2 i (g, f i )f i ,
On the other hand, we have

F * (Φ ∞ L ) = i σ i (Φ ∞ L , l i )f i .
Finally, we get the expression of g η , the solution of (23)as

g η = i σ i (Φ ∞ L , l i ) η + σ 2 i f i (31)
The regularization parameter is chosen using the Morozov discrepancy principle. Discretization

The numerical experiments are conducted in a 2 D setting of the problem. We consider n equally distant observation points of the far field (x l ) 1≤l≤n on the unit circle.

F g(x l ) ≃ n j=1 w j u ∞ (x l , xj )g(x j ) ( 32
)
where w j is the arclengh between two adjacent points. Let L be a small segment of center z and with normal ν. Then

Φ ∞ L (x l ) ≃ γ|L|(α(z)e -ikx l z + β(z)(-ikx l .ν)e -ikx l z ) (33)
The discrete equation to solve is then

F g(x l ) ≃ Φ ∞ L (x l ) ∀x l (34)
using the Tikhonov procedure explained above. The sampling procedure will consist then in varying the z and ν in (33). According to Theorem 3 we expect g (where g is a solution to (34)) to be large except when z ∈ σ and ν is the normal to σ at z. We shall consider two types of solutions: the first one denoted by g z corresponds to α(z) = 1 and β(z) = 0, the second one denoted by g z,ν corresponds to α(z) = 0 and β(z) = 1.

First criterion

Let us consider two independent normals for instance ν 1 = (0, 1) t and ν 2 = (1, 0) t . At each sampling point z we compute

z -→ 1 g z + 1 g z,ν1 + 1 g z,ν2
.

This criterion may not be efficient if ν 1 or ν 2 coincide with the exact normal to σ at z, specially if σ is a segment with normal N , N = ν 1 and N = ν 2 (see numerical section).

Second criterion

Let us define the normal ν as follows:

ν = ζν 1 + 1 -ζ 2 ν 2 with 0 ≤ ζ ≤ 1
Therefore, by linearity of the equation ( 34)

g z,ν = ζg z,ν1 + 1 -ζ 2 g z,ν2
.

Based on the theoretical justification, the normal ν to σ at z ∈ σ corresponds with the value ζ that minimizes

g z,ν 2 = ζ 2 g z,ν1 2 + (1 -ζ 2 ) g z,ν2 2 + 2ζ(1 -ζ 2 ) g z,ν1 , g z,ν2 (36)
The proposed criterion will be the determination on each point z of

z -→ 1 g z + 1 g z,ν (37)
where g, g z,ν corresponds with ζ that minimizes (36).

Numerical tests

The efficiency of this approach is tested using both criteria for different shapes, namely arcshaped cracks, L-shaped cracks and the union of two segments having an oblique angle between them and for several ranges of the impedance values. We present in the following figures the isovalues of the right hand side of (35). The red isovalues correspond to the highest values whereas the blue ones represent the smallest ones. This means that the crack corresponds to red isovalues in the figures. In all the numerical tests we use 100 observation points of the far field pattern and the same number of incident plane waves. Both criteria give a good reconstruction of the crack for the case of large impedances (see Figures 7,8). In the case of small impedances, using the first criterion, the result is not satisfactory for all geometries. This is due to the fact that for large impedances we are close to the Dirichlet case, hence the dominant term in (35) is 1 gz . In the case of the small impedances, we are close to the Neumann case and in this case the dominant term is the one that contains the normal derivatives. Figure 9 shows that if ν 1 or ν 2 coincides with the exact normal to the crack we have a good reconstruction otherwise only the part of the crack for which the normal coincides with ν 1 or ν 2 is correctly reconstructed: see Figure 10, 11. Hence having a good approximation of ν is very important for the precision of the result. This problem is fixed by the use of the second criterion as demonstrated by the reconstruction shown in Figures 7,[START_REF] Kirsh | A linear sampling method for inverse scattering from an open arc[END_REF][START_REF] Kress | Integral equation methods for scattering from an impedance crack[END_REF]10 and 11. Let us observe however that the quality of the results slightly deteriorates when the impedance values are between high and small magnitudes: see Figure 12, 13 and 14. To illustrate the importance of the reconstruction of the normals in the precision of the results we visualize the reconstructed ones in Figure 15, 16 and 17. We observe that the best results correspond with the cases where the normals at the crack are the exact ones.

Case of Large impedances

= λ + = 100(1 + i) (left), λ -= λ + = 1000(1 + i) (middle), λ -= λ + = 10 6 (1 + i) (right).

Case of small impedances

-= λ + = 0.1(1 + i) (left), λ -= λ + = 0.01(1 + i) (middle) λ -= λ + = 10 -6 (1 + i) (right).
= λ + = 0.1(-1 + i) (left), λ -= λ + = 0.01(-1 + i) (middle), λ -= λ + = 10 -6 (-1 + i) (right).

BEN HASSEN & BOUKARI & HADDAR

= λ + = 0.1(1 + i) (left), λ -= λ + = 0.01(1 + i) (middle), λ -= λ + = 10 -6 (1 + i) (right).

Case of impedances with "moderate values"

Influence of the frequency

A more relevant physical model for the impedance boundary conditions would be

∂ ν u ± ± k λ± u ± = 0 on σ ±
where now λ ± are dimensionless constants. We shall study the influence of the frequency on the reconstructions for these type of conditions. Figures 18, 19 show the reconstructions of the normals for two different choices of the frequency. We observe that the best orientation for the normals are given by the smaller frequency. This then suggests to use the small frequency case to first reconstruct the normal fields then use these normals to evaluate the criterion (37). This procedure improves the results obtained for the higher frequency: compare

Figure 1 :

 1 Figure 1: The modulus of u ∞ computed by Freefem++ (red) and by the integral equations method (blue) for λ -= λ + = 5 + i. The angle of incidence θ = π/2 (left) and θ = π/6 (right). The crack is the segment [-0.5, 0.5] × {0}.

Figure 2 :

 2 Figure 2: The modulus of u ∞ computed by Freefem++ (red) and by the integral equations method (blue) for λ -= λ + = 1 + i. The angle of incidence θ = π/4 (left) and θ = π/3 (right). The crack is the segment {0} × [-0.5, 0.5].

Figure 3 :

 3 Figure 3: The modulus of u ∞ computed by Freefem++ (red) and by the integral equations method (blue) for λ -= λ + = 5 + i. The angle of incidence θ = π/2 (left) and θ = π/6 (right). The crack is of shape L with peaks (0, 1), (1, 1), (1, 0).

Figure 4 :

 4 Figure 4: The modulus of u ∞ computed by Freefem++ (red) and by the integral equations method (blue) for λ -= λ + = 1 + i, θ = π/4 (left) and θ = π/3 (right). The crack is is of shape L with peaks (0, 1), (1, 1), (1, 0).

Figure 5 :

 5 Figure 5: The modulus of u ∞ computed by Freefem++ (red) and by the integral equations method (blue) for λ -= λ + = 5 + i, θ = π/2 (left) and θ = π/6 (right). The crack is an arc of the unit circle with an angle varying from 0 to 90 degrees.

Figure 6 :

 6 Figure 6: The modulus of u ∞ computed by Freefem++ (red) and by the integral equations method (blue) for λ -= λ + = 1 + i, θ = π/4 (left) and θ = π/3 (right). The crack is an arc of the unit circle with an angle varying from 0 to 90 degrees.

) Lemma 5

 5 The operator M has a bounded inverse and B = γFM -1 .Proof. For a given (ϕ, ψ) ∈ H1/2 (σ) × H-1/2 (σ), the function F(ϕ, ψ)(x) is the far field pattern of the potential P (ϕ, ψ)(x) )Φ(x, y)ds(y).The function P ∈ H 1 loc (R m \ σ) satisfies the Helmholtz equation in R m \ σ and the Sommerfeld radiation condition. In addition, following the proof of Lemma 2, we show that ϕ = [P ], ψ = -[∂ ν P ] and

  )e -iκd•y ds(y) ds(d) ∂e -iκd•y ∂ν(y) ds(d)ds(y) + γ σ β(y) S m-1 g(d)e -iκd•y ds(d)ds(y).

  Reconstruction using the second criterion

Figure 7 :Figure 8 :

 78 Figure 7: Reconstruction of an L-shaped crack with peaks (0.75, 0), (0, 0), (0, 0.75) for λ -= λ + = 100(1 + i) (left), λ -= λ + = 1000(1 + i) (middle), λ -= λ + = 10 6 (1 + i) (right)

  Reconstruction using the second criterion

Figure 9 :

 9 Figure9: Reconstruction of an L-shaped crack with peaks (0, 0.75), (0, 0), (0.75, 0) for λ -= λ + = 0.1(1 + i) (left), λ -= λ + = 0.01(1 + i) (middle) λ -= λ + = 10 -6 (1 + i) (right).

Figure 10 :

 10 Figure10: Reconstruction of an arc-shaped crack with center (-0.5, -0.5), radius 0.8 and angle varying from 0 to 90 degrees for λ -= λ + = 0.1(-1 + i) (left), λ -= λ + = 0.01(-1 + i) (middle), λ -= λ + = 10 -6 (-1 + i) (right).

Figure 11 :

 11 Figure 11: Reconstruction of two segments {0} × [0, 0.8] and [0, 0.4] × [0, -0.8] for λ -= λ + = 0.1(1 + i) (left), λ -= λ + = 0.01(1 + i) (middle), λ -= λ + = 10 -6 (1 + i) (right).

  Figure12: Reconstruction of an L-shaped crack with peaks (0, 0.75), (0, 0), (0.75, 0) for λ -= 1.5+1 and λ + = 2 + 1.2i (left), λ -= λ + = 5(1 + i) (middle), λ -= λ + = 10(1 + i) (right).

  Figure13: Reconstruction of an arc-shaped crack with center (-0.5, -0.5), radius 0.8 and angle varying from 0 to 90 degrees for λ -= 1.5 + i and λ + = 2 + 1.2i (left), λ -= λ + = 5 + 5i (middle), λ -= λ + = 10(1 + i) (right).

  Figure 14: Reconstruction of two segments {0} × [0, 0.8] and [0, 0.4] × [0, -0.8] for λ -= 1.5 + i and λ + = 2 + 1.2i (left), λ -= λ + = 5 + 5i (middle), λ -= λ + = 10(1 + i) (right).

Figure 15 :

 15 Figure 15: Reconstruction of the normals for λ -= λ + = 0.001 + 0.001i (left), λ -= λ + = 5 + 5i (middle), λ -= λ + = 10(1 + i) (right).

Figure 16 :

 16 Figure 16: Reconstruction of the normals for λ -= λ + = 0.001 + 0.001i (left), λ -= λ + = 5 + 5i (middle), λ -= λ + = 10(1 + i) (right).

Figure 17 :

 17 Figure 17: Reconstruction of the normals for λ -= λ + = 0.001 + 0.001i (left), λ -= λ + = 5 + 5i (middle), λ -= λ + = 10(1 + i) (right).

FiguresFigure 18 :

 18 Figure 18: Reconstruction of the normals for λ-= λ+ = 0.79(1 + i), k = π/2.

Figure 19 :

 19 Figure 19: Reconstruction of the normals for λ-= λ+ = 0.79(1 + i), k = 4π.

Figure 20 :Figure 21 :Figure 22 :

 202122 Figure 20: Reconstruction of an L-shaped crack with peaks (0, 0.75), (0, 0), (0.75, 0) for λ+ = 0.23 + 0.15i λ-= 0.31 + 0.19i (left) λ-= λ+ = 0.79(1 + i) (middle) λ-= λ+ = 1.58(1 + i) (right) with k = 2π.

RR n➦ 7478


Valentin Emiya

Bertrand David

Roland Badeau

A PARAMETRIC METHOD FOR PITCH ESTIMATION OF PIANO TONES

Keywords:

The efficiency of most pitch estimation methods declines when the analyzed frame is shortened and/or when a wide fundamental frequency (F0) range is targeted. The technique proposed herein jointly uses a periodicity analysis and a spectral matching process to improve the F0 estimation performance in such an adverse context: a 60ms-long data frame together with the whole, 7 1 /4-octaves, piano tessitura. The enhancements are obtained thanks to a parametric approach which, among other things, models the inharmonicity of piano tones. The performance of the algorithm is assessed, is compared to the results obtained from other estimators and is discussed in order to characterize their behavior and typical misestimations.

INTRODUCTION

Numerous methods dedicated to fundamental frequency (F0) estimation of periodic signals try to extract the signal self-similarities by maximizing a function of time or frequency. In this manner, they measure a degree of internal resemblance in the waveform (ACF [START_REF] Rabiner | On the use of autocorrelation analysis for pitch detection[END_REF][START_REF] Meddis | Virtual pitch and phase sensitivity of a computer model of the auditory periphery[END_REF], AMDF [START_REF] Ross | Average magnitude difference function pitch extractor[END_REF][START_REF] De Cheveigne | YIN, a fundamental frequency estimator for speech and music[END_REF], cepstrum [START_REF] Noll | Cepstrum pitch determination[END_REF]) or in the spectrum [START_REF] Klapuri | Multiple fundamental frequency estimation based on harmonicity and spectral smoothness[END_REF]. When processing real world musical sounds, these techniques are confronted to deviations from the theoretical model, such as the presence of noise, which can be both stationary and non stationary, or the possibly non-uniform distribution of the harmonics.

The development and applications of the quoted methods often deal with an extension to subband processing [START_REF] Meddis | Virtual pitch and phase sensitivity of a computer model of the auditory periphery[END_REF][START_REF] Klapuri | A perceptually motivated multiple-f0 estimation method[END_REF], to an optimization of the main function [START_REF] De Cheveigne | YIN, a fundamental frequency estimator for speech and music[END_REF][START_REF] Klapuri | A perceptually motivated multiple-f0 estimation method[END_REF] or to the joint use of both time and frequency domains [START_REF] Peeters | Music pitch representation by periodicity measures based on combined temporal and spectral representations[END_REF]. Typical errors that usually occur give a general idea of the difficulties the F0 estimation task must cope with. Temporal or spectral methods tend to make sub-octave or octave errors respectively. Both of them come up against difficulties like a large F0 search range (e.g. 27-4200 Hz for the piano), non-regular spectral envelopes and inharmonic deviations of the frequency components [START_REF] Klapuri | Multiple fundamental frequency estimation based on harmonicity and spectral smoothness[END_REF][START_REF] Godsill | Bayesian computational models for inharmonicity in musical instruments[END_REF]. In addition, a short analysis frame prevents spectral methods from resolving components for low F0 values whereas the uniformely-distributed discrete time scale used by temporal methods makes the estimation fail above some F0 limit.

The new F0 estimation algorithm we describe aims at enhancing F0 estimation results in the case of a short analysis window and a large F0 search range. We will focus on piano sounds since they present all the listed difficulties and usually cause one of the worst estimation error rates per instrument (e.g. see [START_REF] Peeters | Music pitch representation by periodicity measures based on combined temporal and spectral representations[END_REF]). The pitch of a harmonic or quasi-harmonic sound is an attribute that only depends on the sinusoidal components of the signal. Thus a F0 estimator only requires the parameters of components such as frequency, amplitude,

The research leading to this paper was supported by the French GIP ANR under contract ANR-06-JCJC-0027-01, Décomposition en Éléments Sonores et Applications Musicales -DESAM damping factor and initial phase. So far, the other part of the sound, including the ambient noise, transients, etc. has not been used in the F0 estimation task, as far as the authors know. Therefore, the preliminary task in the F0 estimation method we present consists in extracting the parameters of components. The F0 estimator then includes a spectral function and a temporal function. The parametric approach allows to take into account the inharmonicity of sounds both in time and frequency domains and to optimize the precision of the F0 numeric estimation. The F0 estimation system is described in section 2. Evaluation results and comparisons with other algorithms are then detailed in section 3 and conclusions are finally presented in section 4.

PITCH ESTIMATION SYSTEM

High Resolution analysis

The Na-length analyzed waveform is modeled by:

s(t) = K k=1 α k z t k + w(t) (1)
defined for t ∈ 0, Na -1 and composed of a sum of K exponentially-modulated sinusoids α k z t k , k ∈ 1, K with complex amplitudes α k = A k e iΦ k ∈ C * , (A k being the real, positive amplitude and Φ k the initial phase), and distinct poles z k = e d k +i2πf k (f k being the frequency and d k the damping factor), plus an additive colored noise w(t). This section details how the signal is preprocessed, how poles z k are then estimated via the ESPRIT (Estimation of Signal Parameters via Rotational Invariance Techniques) algorithm [START_REF] Roy | ESPRIT-a subspace rotation approach to estimation of parameters of cisoids in noise[END_REF], and how amplitudes α k are finally extracted.

Preprocessing. A two-step preprocessing stage is applied to the signal sampled at kHz:

1. The cubic computational cost of the ESPRIT algorithm is reduced when the number of poles to be estimated is low. This is achieved by using a filter bank. The signal is splitted into D = 32 subbands with width 500-Hz by using cosinemodulated filters [START_REF] Vaidyanathan | Multirate systems and filter banks[END_REF]. The order of magnitude of the computational cost drops from N 3 a to N 3 a /D 2 (N 3 a /D 3 per band) leading to a satisfactory processing time for the analysis bloc. 2. Components of piano sounds are particularly well represented by the exponential sinusoidal plus noise model introduced in (1). However, the ESPRIT algorithm only applies to the restrictive case of white noise. Thus, the second preprocessing step consists in whitening the noise in each subband thanks to an AR filter estimated on the smoothed spectrum of the signal. ESPRIT algorithm. The signal in each preprocessed subband is a sum of exponentially-modulated sinusoids plus white noise. Assuming the number of poles is known, the ESPRIT algorithm [START_REF] Roy | ESPRIT-a subspace rotation approach to estimation of parameters of cisoids in noise[END_REF] gives an estimation of those poles. The method is based on a subspace projection on the so-called signal subspace and benefits from the rotational invariance property of this signal subspace.

Estimation of the number of poles. In the current application, the number of poles in each subband is not known a priori and thus must be estimated. The ESTER [START_REF] Badeau | A new perturbation analysis for signal enumeration in rotational invariance techniques[END_REF] algorithm establishes a criterion J(p) that provides an estimation of the number of poles as argmax p∈P (J(p) > δJ ), P being the set of candidates for the number of poles and δJ a threshold tuned to δJ = 10 in the current study. The result obtained by this method is either correctly estimated, or slightly over-estimated. As shown in [START_REF] Badeau | A new perturbation analysis for signal enumeration in rotational invariance techniques[END_REF], the latter case is not disturbing for the ESPRIT analysis, and weak amplitudes are estimated for the spurious poles.

Estimation of amplitudes. Once the poles extracted, amplitudes are estimated by a least squares algorithm applied to the subband signal. The effects of the preprocessing stage on the amplitudes in each subband are corrected by applying the inverse filters of the various preprocessing steps -whitening, filter bank and preemphasis filter series -, leading to the estimation of the amplitudes α k , k ∈ 1, K .

Pitch estimation

A temporal method and a spectral method are first introduced. Although each one could account for a F0 estimator, they are jointly used in the same manner as in [START_REF] Peeters | Music pitch representation by periodicity measures based on combined temporal and spectral representations[END_REF] to obtain the whole, more efficient estimator detailed in the last part.

Temporal

Periodicity is often analyzed by assuming the signal is an observation of a real, wide-sense stationary (WSS) process y and by estimating its autocovariance function Ry(τ ) = E [y(t)y(t + τ )]. When the signal is periodic, the maxima of Ry(τ ) are located at τ = 0 and at every multiple of the period. Let us consider a real, WSS process y composed of K undamped sinusoids with frequencies ν k , real amplitudes 2a k , initial phases ϕ k , which are assumed to be independant and uniformely distributed along [0, 2π[. The autocovariance function of y is Ry(τ ) = K k=1 2a 2 k cos (2πν k τ )+δ(τ )σ 2 wy . Therefore we can define a temporal function R(τ ) for F0 estimation from the parameters estimated by the high resolution analysis:

R(τ ) = K k=1 p k cos (2πf k τ ) (2)
p k = |α k | 2 if |z k | = 1 |α k | 2 Na 1-|z k | 2Na 1-|z k | 2 otherwise ( 3
)
where τ > 0,

f k = arg(z k ) 2π
is the normalized frequency of component k, and the instantaneous power p k is an estimate of coefficient 2 k over the analysis frame.

In the case of a slightly inharmonic sound, the frequency deviation weakens or even removes the maxima of R(τ ) at the multiples of the period. The inharmonicity law [START_REF] Fletcher | The Physics of Musical Instruments[END_REF] for a piano tone of fundamental frequency causes partial h to be located at frequency hf0 but at hf0 1 + β(h 2 -1), β being the inharmonicity coefficient of the note. As illustrated in fig. 1, this frequency stretching may be inversed by remapping the set of estimated frequencies {f k , k ∈ 1, K } to a set of frequencies {g f 0 ,k , k ∈ 1, K }: where β (f0) is an approximative inharmonicity coefficient for fundamental frequency f0 averaged from the results presented in [13, pp. 365]. The assumed partial order h (f0, f k ) associated to frequency f k is extracted from the inharmonicity law:

g f 0 ,k = f k 1 + β (f0) (h 2 (f0, f k ) -1) (4)
2 (f0, f k ) = (1 -β (f0)) 2 + 4β (f0) f 2 k f 2 0 -1 + β (f0) 2β (f0) (5)
As the remapping process causes the remapped frequencies g f 0 ,k of the partials to be perfect multiples of the actual fundamental frequency f0, we replace k with g 1 τ ,k in (2) to obtain a temporal function Rinh(τ ) for piano tones which is maximum for τ = 1 f 0 :

Rinh(τ ) = K k=1 p k cos 2πg 1 τ ,k τ (6)

Spectral method

A parametric amplitude spectrum is designed from the estimates of frequencies f k and energies E k of components k ∈ 1, K . It is composed of a sum of K gaussian curves centered in f k with constandard deviation σ, weighted by the square root of the component energies as average amplitudes:

S(f ) = K k=1 √ E k √ 2πσ e - (f-f k ) 2 2σ 2 (7)
σ is set to f0min/4 where f0min is the lower bound of the F0 search range in order to prevent overlap between successive partials. Our spectral estimator U (f ) relies on maximizing a scalar product between the parametric amplitude spectrum and theoretical harmonic unitary patterns of F0 candidates:

U (f ) = H f h=1 w f,h S (hf ) (8)
where H f is the maximum number of partials possible for fundamental frequency f and {w f,h , h ∈ 1, H f } is the pattern associated to f . The choice of the pattern is based on an approximative logarithmic spectral decrease of components. The slope p of a linear regression between log( √ E k ) and f k is extracted and weights w f,h are then defined as:

w f,h = w0e phf (9)
where w0 =

H f h=1 e 2phf -1 2
is a normalizing term such that

H f h=1 w 2 f,h = 1.
The spectral estimator is then adapted to piano tones by selecting the values of the spectrum on an inharmonic stretched scale instead of a harmonic scale:

Uinh(f ) = H f h=1 w f,h S hf 1 + β(f ) (h 2 -1) (10)
Finally, the estimator efficiency can be improved by ignoring all frequencies and weights below a cut-off frequency of 100 Hz since the impedance at the piano bridge [START_REF] Fletcher | The Physics of Musical Instruments[END_REF] causes a significant deviation of low frequencies from the inharmonicity law and the highest weigths w f,h of patterns are allocated to those frequencies.

Pitch estimator

As mentioned in the introduction, sub-harmonic and harmonic error trends are opposed in temporal and spectral methods. A way to benefit from this phenomenon is described in [START_REF] Peeters | Music pitch representation by periodicity measures based on combined temporal and spectral representations[END_REF]. It consists in multiplying a temporal and a spectral function on a common F0 scale in order to preserve common peaks from both functions and to remove or attenuate other peaks (see fig. 2). Thus, the pitch is estimated by maximizing the product of the methods Rinh 1 f and Uinh(f ):

f0 = argmax f Rinh 1 f Uinh (f ) (11)
Thanks to the analytic expressions ( 6) and ( 10), Rinh 1 f and Uinh(f ) can be directly evaluated for any f value. As the F0 distribution of an equal-tempered musical scale is logarithmic, the F0-scale support is set to N f points logarithmically spaced in the F0-search range. This unconstrained choice is a key advantage of the method since the logarithmic F0 distribution is not offered by many methods (see [START_REF] De Cheveigne | YIN, a fundamental frequency estimator for speech and music[END_REF][START_REF] Peeters | Music pitch representation by periodicity measures based on combined temporal and spectral representations[END_REF]). Actually, temporal methods have a linearly distributed time scale, which results in a lack of precision in high frequency and too much resolution in low frequency, whereas Fourier-based spectral methods have a linear F0 distribution. In those cases, the estifunction must often be interpolated to achieve the required precision and may still suffer from this.

In a Matlab implementation on a 2.4GHz-CPU, the overall processing of a 60ms frame averages 6.5s. About 1s is necessary for the analysis. About 95% of the remaining time is required by the spectral F0 estimator and may be optimized and written in C for a computationally-efficient implementation.

EVALUATION

The algorithm has been evaluated on isolated piano tones from various sources: 3168 notes from three pianos of RWC database [START_REF] Nishimura | RWC music database: Music genre database and musical instrument sound database[END_REF], 270 notes from five pianos of a PROSONUS database and 264 notes from a Yamaha upright piano of a private database. All recordings include several takes of all the 88 notes of piano range (except PROSONUS in which notes are spaced by fourth) with a varying loudness. RWC samples also offer various play modes (normal, staccato, with pedal). The F0 search scale is composed of N f = 8192 values logarithmically distributed between f0min = 26.73 Hz and f0max = 4310 Hz. The estimation is performed after the analysis of a single ms or 93 ms frame: ms is quite a challenging frame length since it is below twice the period of lowest notes while ms is a well spread duration for this kind of evaluation. Each estimated F0 is associated to the closest note in the equal tempered scale with A4 tuned to 440 Hz. Errors are then defined as incorrect note estimations. The method is compared to two estimators. The first one is as similar to our estimator as possible, replacing the ESPRIT analysis stage with a classical analysis: the ACF is estimated from the signal by the formula r(τ ) = Na Na-τ DFT -1 |DFT [s]| 2 , the factor Na Na-τ being a correction of the bias; the spectral estimator Uinh(f0) computed by replacing the parametric spectrum with the modulus of the DFT of the signal, using a zero-padding on 8N f points; r(τ ) is mapped to the frequency scale by interpolation as described in [START_REF] Peeters | Music pitch representation by periodicity measures based on combined temporal and spectral representations[END_REF]; the pitch is finally estimated by maximizing the product between the spectral function and the remapped r(τ ). The second method is the YIN algorithm [START_REF] De Cheveigne | YIN, a fundamental frequency estimator for speech and music[END_REF] which is considered as a very efficient monopitch estimator. We used the code available on the authors' website.

Evaluation results are reported in fig. 3. At the target window length of 60 ms, the global error rate of our estimator is around 4.4% which is at least twice better than the other estimators. This is due to a low error rate on a large F0 range (1.1% in the F0 range 65 -2000 and slowly increasing values at the very bass and treble limits. In comparison, the non-ESPRIT based estimator achieves

Fig. 1 .

 1 Fig.1. At any given F0, the frequencies f k are remapped to g f 0 ,k , leading to a harmonic distribution for the actual F0. One theoretical partial over 5 is represented with f0 = 27.5Hz and β = 2.54e -4.

Fig. 2 .

 2 Fig. 2. From top to bottom, on a logarithmic frequency scale: parametric spectrum, spectral estimation function Uinh(f ), remapped temporal estimation function Rinh 1 f , joint F0 estimation function. Functions result from the 60 ms analysis of a D3 piano note.

Octave-averaged error rates (analysis on 93 ms) Parametric f0 estimator (mean: 2.4%) Non-parametric f0 estimator (mean: 3.0%) YIN estimator (mean: 11.0%) Fig. 3. Octave-averaged error rates per note with two different frame lengths, for the parametric F0 estimator and two other methods: a similar but non-parametric algorithm and the YIN estimator a 1.1% error rate in the range 240 -2000 Hz. Its low efficiency outside this range shows how the F0 estimation is improved by both the high resolution analysis and the handling of parametric, analytic formulas. The YIN algorithm is slightly less efficient in the medium range than our estimator and has similar results in the bass range (for the first octave both curves should be at the same level, but our estimator results seem to be worse since they include the lowest four note error rates that cannot be estimated by the YIN algorithm with a 60 ms window length). In the high range, it shows a quite high error rate, which is a typical behavior of temporal methods. Global results are improved with a 93 ms frame length. Nevertheless, the high resolution analysis does not enhance significatively the F0 estimation even if its error rate remains the lowest.

Typical errors are now discussed, in the 60 ms analysis case. As expected, usual errors are under-estimations of high f0s overestimations of low f0s. Around 18% of errors made by each algorithm are octave and suboctave errors. In the case of our algorithm, the remaining error intervals are of all kinds, with only 5% that are half-tone errors, whereas this rate reaches 10% for the other two algorithms. The YIN algorithm makes a high proportion of sub-harmonic errors (13% are sub-octaves, 8% are subnineteenth). Thus, even if our makes a reduced number of harmonic/subharmonic errors, those errors remain difficult to avoid. Half-tone error rates show the efficiency of our method while the other algorithms suffer from a lack of precision of temporal estimators for high F0. Finally, the management contributes to lower the global error rate, from 4.9 to 4.4% in the 60-ms frame case. As expected, the improvement is localized in the lowest F0 range: the error rate in the MIDI range 21, 37 decreases from 16.6 to 14.1%.

CONCLUSIONS

The F0 estimator designed in this paper allows to address typical error trends in a short frame analysis and a wide F0-range context. It is based on a preliminary extraction of the parameters of components and on the design of temporal and spectral parametric function. Satisfying performances have been obtained and a large part was allocated to the discussion on typical errors and the way to avoid them.


Valentin Emiya

Nancy Bertin

Bertrand David

Roland Badeau

† V Emiya

‡ B David

MAPS -A piano database for multipitch estimation and automatic transcription of music *

Keywords: Audio, database, piano, pitch, multipitch, transcription, music, MAPS Audio, base de données, piano, fréquence fondamentale, transcription, musique, MAPS

MAPS -standing for MIDI Aligned Piano Sounds -is a database of MIDI-annotated piano recordings. MAPS has been designed in order to be released in the music information retrieval research community, especially for the development and the evaluation of algorithms for single-pitch or multipitch estimation and automatic transcription of music. It is composed by isolated notes, random-pitch chords, usual musical chords and pieces of music. The database provides a large amount of sounds obtained in various recording conditions.

1 Introduction

In the field of multipitch estimation (MPE) and automatic transcription of music (ATM), annotated sound databases are needed both to develop and to evaluate the algorithms. Public databases are useful for individual works while private databases are used for contests like MIREX [START_REF]International Music Information Retrieval Systems Evaluation Laboratory[END_REF]. In the former case addressed here, a number of issues are commonly faced: a little amount of sounds is available due to production, copyright or distribution reasons; the ground truth is often generated a posteriori, with some inaccurate or erroneous values of pitch or onset and offset times. Thus, few databases are currently available (e.g. [START_REF] Opolko | Mcgill university master samples[END_REF][START_REF] Goto | RWC music database: Music genre database and musical instrument sound database[END_REF][START_REF]The University of Iowa Musical Instrument Samples[END_REF]). They are usually made up of isolated tones from various musical instruments and/or musical recordings. Then, when necessary, isolated tones may be added by the user to generate chords to be analyzed. These databases provide a large quantity of sounds and were generally obtained after considerable efforts, but may still suffer from some of the drawbacks previously mentioned. In particular, the annotation process is time-consuming when dealing with numerous events. Several strategies may be adopted: manual annotation of the recordings [START_REF] Goto | AIST annotation for the RWC Music Database[END_REF], semi-automatic annotation [START_REF] Gillet | ENST-Drums: an extensive audio-visual database for drum signals processing[END_REF][START_REF] Yeh | Synthesized polyphonic music database with verifiable ground truth for multiple f0 estimation[END_REF] or entertaining systems [START_REF] Turnbull | A game-based approach for collecting semantic annotations of music[END_REF]. In this work, we use a reverse process in which the ground truth is first created as standard MIDI files and then generated in an automatic way, somehow similar to [START_REF] Yeh | Synthesized polyphonic music database with verifiable ground truth for multiple f0 estimation[END_REF], resulting in a fully-automatic and reliable annotation.

In this documentation, we describe the contents and the generation of the new database called MAPS (standing for MIDI Aligned Piano Sounds). The main features provided in MAPS are described in Section 2. The contents of the database are then detailed in Section 3. In Section 4, the recording devices and processes are explained. Instructions on how to get and cite MAPS are finally given in Sections 5 and 6.

Main features of MAPS

MAPS provides recordings with CD quality (16-bit, 44-kHz sampled stereo audio) and the related aligned MIDI files as ground truth 1 . The overall size of the database is about 40GB, i.e. about 65 hours of audio recordings. The database is available under a Creative Commons license.

A large amount of sounds and a reliable ground truth are provided thanks to some automatic generation processes, consisting in the audio synthesis from MIDI files. The use of a Disklavier (MIDIfied piano) and of high quality synthesis software based on libraries of samples permitted a satisfying tradeoff between the quality of the sounds and the time consumption needed to produce such a quantity of annotated sounds.

In order to favor generalization to many audio scenes, several grand pianos and upright pianos have been played in various recording conditions, including various rooms and close/ambient takes. Table 1 details each of the nine configurations in terms of instrument, recording conditions and code reference. It also specifies the origin of the recording, which may be high quality synthesis software based on sample libraries or a Disklavier. For each of these configurations, similar but not equal contents have been produced and can be stored in one 4.7GB DVD.

The contents of MAPS is divided in four sets, which are detailed in section 3:

• the ISOL set: isolated notes and monophonic excerpts;

• the RAND set: chords with random pitch notes;

• the UCHO set: usual chords from Western music;

• the MUS set: pieces of piano music.

3 Detailed contents

ISOL: isolated notes and monophonic excerpts

The ISOL set specifically provides monophonic excerpts. It thus aims at testing single-pitch estimation algorithms or at training multipitch algorithms when isolated tones are required. Each sound file is characterized by a playing style ps, by a loudness i0, by the use/no use of the sustain pedal s and by the pitch m. The related file is named

MAPS_ISOL_ps_i0_Ss_Mm_instrName.wav

The playing style ps can be:

• NO: 2-second long notes played normally;

• LG: long notes (the duration varies from 3 seconds for the highest-pitch notes to 20 seconds for the lowest-pitch notes);

• ST: staccato;

• RE: repeated note, faster and faster, from about 1.4 to 13.5 notes per second;

• CHd: chromatic ascending and descending scales, with various note duration indexed by d;

• TRi: trills, faster and faster, up to a half tone (i = 1) or to one tone (i = 2), from about 2.8 to 32 notes per second.

The loudness i0 can be: P (piano), M (mezzo-forte), F (forte). The sustain pedal is pressed in half of the cases, as specified by the binary variable s (s= 1 when the pedal is pressed). When it is used (50% of the cases), the pedal is pressed 300ms before the beginning of the sequence and released 300ms after the end2 . The field instrName is a code defined in Table 1.

Except for chromatic scales, the pitch is coded as a MIDI code m ∈ 21; 108 , each note of the piano scale 21; 108 being recorded.

RAND: random chords

The RAND set provides chords composed of randomly-chosen notes. It was designed in order to evaluate the algorithms in an objective way, without any a priori musical knowledge, which is commonly performed in the papers on multipitch estimation. The generation process is: Algorithm 1 RAND-set MIDI-file generation process for each polyphony level x do for each pitch range m1-m2 do for a number of outcomes indexed by n do randomly choose x notes in the pitch range m1-m2 randomly and individually choose their loudness in the range i1-i2 randomly choose the chord duration and the use/no use of the sustain pedal generate the resulting MIDI file end for end for end for Each chord is stored in a file named MAPS_RAND_Px_Mm1-m2_Ii1-i2_Ss_nn_instrName.wav, where • the polyphony level x varies from 2 to 7;

• the pitch range m1-m2 can be 21 -108 or 36 -95; the former range is the full, 7 1 / 4 -octave piano range while the latter spreads over the centered 5 octaves and is commonly used to evaluate multipitch algorithms;

• the loudness is chosen, independently for each note, in two possible ranges: 60 -68 (mezzo-forte, which may represent a typical chord situation with similar note intensities) or 32 -96 (from piano to forte, which may reflect the polyphonic contents when several tracks/melodic lines are played, resulting in heterogeneous loudnesses);

• s denotes the use/no use of the sustain pedal, as in the ISOL set (see section 3.1);

• n denotes the outcome index;

For a given configuration of the parameters, 50 outcomes are actually generated. For instance, the database provides 50 random 3-notes chords for which pitches are chosen between 36 (C 2 ) and 95 (B 6 ), with a mezzo-forte loudness, around half of the chords being played using the sustain pedal.

UCHO: usual chords

The UCHO set provides usual chords from Western music such as jazz or classical music. Thus, these chords are useful to assess the performances with an a priori knowledge and are made with notes that are harmonically related.

The 2-note "chords" are all the intervals from 1 to 12 semitones, plus the 13 th (fifth at the upper octave) and the 16 th (two octaves), as detailed in Table 2. In polyphony 3, the database provides major, minor, diminished and augmented triads. The seven usual 7 th chords are available in polyphony 4, while the ten usual 9 th chords are recorded in polyphony 5. In polyphony 3, 4 and 5, all inversions are provided as detailed in Tables 3, 4 and 5 respectively. In a given chord, each note is coded according to the distance in semitones from the root of the chord. For instance, a major triad is coded by 0 -4 -7.

A chord with p notes is stored in a file named • i1-i2 is the pitch range, as in the RAND set (see section 3.2);

• s denotes the use/no use of the sustain pedal, as in the ISOL and RAND sets (see section 3.1);

• n is the outcome index, the root of the chord being randomly and uniformly chosen among the possible notes (e.g. between 21 and 101 for the major triad);

• additionally, the chord duration is set to one second.

For a given configuration of the parameters, 10 outcomes with different roots are actually generated and are indexed by n∈ 1; 10 . For chords with 4 notes and more, only 5 outcomes are generated.

Interval

Interval minor 2 nd 0-1 minor 6 th 0-8 major 2 nd 0-2 major 6 th 0-9 minor 3 rd 0-3 major 7 th 0-11 major 3 rd 0-4 perfect 8 ve 0-12 perfect 4 th 0-5 perfect 13 th 0-19 diminished 5 th 0-6 two octaves 0-24 perfect 5 th 0-7

Table 2: Intervals.

Triads

Root position Inversion 1 Inversion 2 major 0-4-7 0-3-8 0-5-9 minor 0-3-7 0-4-9 0-5-8 diminished 0-3-6 0-3-9 0-6-9 augmented 0-4-8 0-4-8 0-4-8

Table 3: Three-note chords: triads and related codes.

7 th chords Root position Inversion 1 Inversion 2 Inversion 3 major 7 th 0-4-7-11 0-3-7-8 0-4-5-9 0-1-5-8 minor 7 th 0-3-7-10 0-4-7-9 0-3-5-8 0-2-5-9 dominant 7 th 0-4-7-10 0-3-6-8 0-3-5-9 0-2-6-9 half diminished 7 th 0-3-6-10 0-3-7-9 0-4-6-9 0-2-5-8 diminished 7 th 0-3-6-9 0-3-6-9 0-3-6-9 0-3-6-9 minor major 7 th 0-3-7-11 0-4-8-9 0-4-5-8 0-1-4-8 augmented major 7 th 0-4-8-11 0-4-7-8 0-3-4-8 0-1-5-9

Table 4: Four-note chords: tetrads and related codes.

MUS: pieces of music

The MUS set provides pieces of music generated from standard MIDI files available on the Internet3 under a Creative Commons license. These high quality files have been carefully hand-written in order to obtain a kind of musical interpretation as a MIDI file. The note location, duration and loudness have thus been adjusted by hand by the creator of the MIDI database. About 238 pieces of classical and traditional music were actually available when MAPS was created.

For each set of recording conditions (i.e. each line in Table 1), 30 pieces of music are randomly chosen and recorded. The database thus provides a number of different musical pieces, some of them being available several times in various recording conditions.

Each file is named using a description of the musical piece as

MAPS_MUS_description_instrName.wav

9 th chords Root position Inversion 1 Inversion 2 Inversion 3 Inversion 4 dominant 7 th and major 9 th 0-4-7-10-14 0-3-6-8-10 0-3-5-7-9 0-2-4-6-9 0-2-5-8-10 dominant 7 th and minor 9 th 0-4-7-10-13 0-3-6-8-9 0-3-5-6-9 0-2-3-6-9 0-3-6-9-11 minor 7 th and major 9 th 0-3-7-10-14 0-4-7-9-11 0-3-5-7-8 0-2-4-5-9 0-1-5-8-10 minor 7 th and minor 9 th 0-3-7-10-13 0-4-7-9-10 0-3-5-6-8 0-2-3-5-9 0-2-6-9-11 half diminished 7 th and minor 9 th 0-3-6-10-13 0-3-7-9-10 0-4-6-7-9 0-2-3-5-8 0-2-5-9-11 major 7 th and major 9 th 0-4-7-11-14 0-3-7-8-10 0-4-5-7-9 0-1-3-5-8 0-2-5-9-10 major 7 th and augmented 9 th 0-4-7-11-15 0-3-7-8-11 0-4-5-8-9 0-1-4-5-8 0-1-4-8-9 diminished 7 th and minor 9 th 0-3-6-9-13 0-3-6-9-10 0-3-6-7-9 0-3-4-6-9 0-2-5-8-11 minor major 7 th and major 9 th 0-3-7-11-14 0-4-8-9-11 0-4-5-7-8 0-1-3-4-8 0-1-5-9-10 augmented 7 th and major 9 th 0-4-8-11-14 0-4-7-8-10 0-3-4-6-8 0-1-3-5-9 0-2-6-9-10

Table 5: Five-note chords and related codes.

Recording devices

Two procedures were used to record the database: a software-based sound generation and the recording of a Disklavier piano (see Table 1). In both cases, the MIDI files had been created beforehand and were automatically "performed" by one of the devices. The software-based generation was performed using three steps:

1. concatenating the numerous MIDI files into a low number of large files;

6 4devices 7 5 8 6

 678 : isolated notes and monophonic excerpts . . . . . . . . . . . . . . . . . . . . . . . . 3 3.2 RAND: random chords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.3 UCHO: usual chords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.4 MUS: pieces of music . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recording How to get MAPS? How to cite MAPS? 8

MAPS_UCHO_Cc 1 -

 1 . . .-c p _Ii1-i2_Ss_nn_instrName.wav, where • c 1 -. . .-c p denotes the contents of the chord: for 1 ≤ k ≤ p, c k is an integer related to distance in semitones from the root of the chord and note k.

2 .

 2 generating the audio using a sequencer (Steinberg's Cubase SX); 3. segmenting the large audio files into individual files related to the original MIDI files4 . Picture of the "close" configuration.

Figure 1 :

 1 Figure 1: Disklavier recording device: MIDI files are sent from the sound card to the MIDI input of the Disklavier. The generated audio and MIDI signal are recorded using the same sound card.The Disklavier recording device is illustrated in Figure1. The room is a studio with a rectangular shape and dimensions equal to about 4 × 5 meters. It has been designed to perform recordings and its walls are covered with wood and absorbent panels. The distance between the piano and the microphones is about 50cm in the "close" position and about 3 -4m in the "ambient" position. Unlike in the previous

Table 1 :

 1 MAPS: instruments and recording conditions.

	Code	Instrument model	Recording conditions	Real instrument or software
	StbgTGd2	Hybrid	Software default	The Grand 2 (Steinberg)
	AkPnBsdf	Boesendorfer 290 Imperial	church	Akoustik Piano (Native Instruments)
	AkPnBcht	Bechstein D 280	concert hall	Akoustik Piano (Native Instruments)
	AkPnCGdD	Concert Grand D	studio	Akoustik Piano (Native Instruments)
	AkPnStgb	Steingraeber 130 (upright)	jazz club	Akoustik Piano (Native Instruments)
	SptkBGAm	Steinway D	"Ambient"	The Black Grand (Sampletekk)
	SptkBGCl	Steinway D	"Close"	The Black Grand (Sampletekk)
	ENSTDkAm	Yamaha Disklavier	"Ambient"	Real piano (Disklavier)
		Mark III (upright)
	ENSTDkCl	Yamaha Disklavier	"Close"	Real piano (Disklavier)
		Mark III (upright)

In order to make the use of MAPS easy in various contexts, the ground truth is also available as text files, including onset times, offset times and pitches.

Although the pedal is not commonly pressed before playing a note in a musical context, this way of playing is chosen here in order to separate the sound effects due to the pedal and to the note.

B. Krueger, "Classical Piano MIDI files", http://www.piano-midi.de.

This three-step process was performed since the sequencer could not be managed by scripts and thus implied a human action for each MIDI file.

http://creativecommons.org/licenses/by-nc-sa/2.0/

software-based process, the individual MIDI files are here sent one by one from the computer sound card (M-Audio FireWire 410) to the Disklavier via a MIDI link using home-made software. The audio is recorded using two omnidirectional Schoeps microphones and the audio input ports of the same sound card. Since the performance of the Disklavier is improved when a 500ms delay is automatically inserted by the instrument, a MIDI link from the Disklavier to the sound card is set up, which provides the audio-synchronized MIDI files.

How to get MAPS?

MAPS is under a Creative Commons License 5 and is freely available. MAPS can be downloaded from: http://www.tsi.telecom-paristech.fr/aao/en/category/database/ 6 How to cite MAPS? Any use of MAPS should be reported by citing one of the following references: