R Bachelard

Ph W Courteille

R Kaiser

N Piovella

Resonances in Mie scattering by an inhomogeneous atomic cloud

Keywords: 42.50.Ct -Quantum description of interaction of light and matter, related experiments 42.25.Fx -Diffraction and scattering 42.25.Gy -Edge and boundary effects, reflection and refraction

Despite the quantum nature of the process, collective scattering by dense cold samples of two-level atoms can be interpreted classically describing the sample as a macroscopic object with a complex refractive index. We demonstrate that resonances in Mie theory can be easily observable in the cooperative scattering by tuning the frequency of the incident laser field or the atomic number. The solution of the scattering problem is obtained for spherical atomic clouds who have the parabolic density characteristic of BECs, and the cooperative radiation pressure force calculated exhibits resonances in the cloud displacement for dense clouds. At odds from uniform clouds which show a complex structure including narrow peaks, these densities show resonances, yet only under the form of quite regular and contrasted oscillations.

Introduction. -Mie theory is the well-known solution of Maxwell's equations for the scattering of electromagnetic radiation by spherical objects [START_REF] Mie | Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen[END_REF][START_REF] Lorenz | [END_REF]. Via calculation of the electric and magnetic fields inside and outside the object the theory predicts the total optical cross section, which determines the amount of scattered light, and the form factor, which characterizes the far-field radiation pattern [START_REF] Van De Hulst | Light scattering by small particles[END_REF][START_REF] Bohren | Absorption and scattering of light by small particles[END_REF]. Simple solutions are available in regimes where the object size R differs very much from the radiation wavelength λ, or when the refractive index m is close to unity. For example, for small phase-shifts |m -1|R/λ ≪ 1 in optically dilute media |m -1| ≪ 1, one enters the Rayleigh-Debye-Gans regime whereas for small particles and small phase-shifts, one obtains Rayleigh scattering by point-like objects.

For objects whose size is of the order of the radiation wavelength (e.g. water droplets in the atmosphere or in emulsions), Mie's full theory has to be used to find the scattering pattern. Mie scattering differs from Rayleigh scattering in several respects. While the intensity of Rayleigh-scattered radiation scales with the object size as R 6 and is identical in forward and backward direction, the intensity of Mie-scattered radiation is roughly independent of wavelength, and it is larger in forward than in backward direction. The greater the particle size, the more light is Mie-scattered into forward direction. The hallmark of Mie scattering, however, are the Mie resonances: those are sets of parameters (size, refraction index, wavelength), where Mie scattering is particularly strong or particularly weak. The sharpness of some Mie resonances make them useful for measuring unknown parameters such as particles' size.

Recently, a series of papers demonstrated how collectivities of point-like Rayleigh-scattering particles may cooperate [START_REF] Scully | [END_REF][6][7][8][9][10] in scattering radiation into the forward direction and the relationship to Mie scattering was pointed out [11,12]. Here, we show that the theory of collective scattering by smooth distributions of point-like scatterers is equivalent to Mie scattering by demonstrating that the premisses of both models are identical. Hence, we may apply the Mie scattering technique to atomic clouds as long as their granularity, as well as collisions and nonresonant atomic interactions, can be neglected.

In Mie theory, boundary conditions of the scattering object assume a fundamental role: these are generally sharp since dielectric spheres typically have homogeneous densities, while atomic clouds, in general, have parabolic or Gaussian density distributions and smooth boundaries. Within the framework of the Mie theory, we compare calculations for homogeneous and parabolic densities, and identify the impact of sharp or smooth boundaries on the occurrence and shape of Mie resonances. Our study reveal that some resonances will persist, although not the sharpest: this means that a jump in the refractive index is not required, as one could expect from classical cavities where smooth gradients of index also allow for mode propagation and selection (e.g. graded-index fibers).

Finally, we predict experimental regimes where resonances in Mie scattering can be observed in atomic clouds by monitoring the radiation pressure force acting on the cloud's center-of-mass. To this aim, we calculate the resonances as a function of atom number and pump laser detuning. A major advantage of using resonant atoms as scatters is that the refraction index can be varied over very large ranges by changing the cloud's density and volume, or simply by tuning the light frequency. This greatly facilitates the detection of the resonances.

In this Letter, we first show the equivalence of the cooperative scattering by two-level atoms and the Mie scattering, assuming a scalar photon model and reducing the scattering equation for excitation probability amplitude to a differential Helmholtz equation with complex refraction index. Then, we solve the Mie problem and calculate the radiation pressure force, before discussing the presence of Mie resonances in spherical clouds with parabolic densities. Finally, we draw our conclusions.

From collective atomic scattering to Mie scattering. -In his seminal paper [START_REF] Mie | Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen[END_REF], Gustav Mie proposed an analytical solution to the scattering of light over extended objects under the form of infinite series. This theory was originally developed for homogeneous media of refractive index m 0 , in which the elementary solutions to the wave equation are known to be of the form ψ nl = j n (m 0 k 0 r)Y nl (θ, ϕ) (where j n is the spherical Bessel function, and Y nl the spherical harmonics), while outside they simply read j n (k 0 r)Y nl (θ, ϕ). The continuity of the fields at the boundaries eventually determines the scattering coefficient for each (n, l) mode.

On the other hand, the cooperative scattering by a sample of N two-level atoms (with random position r j , transition frequency ω a and linewidth Γ = d 2 ω 3 a /2πhǫ 0 c 3 , where d is the electric dipole matrix element), illuminated by a resonant uniform field is described by the following coupled equations [7][8][9]

dβ j dt = i∆ 0 - Γ 2 β j - i 2 d h E i (r j ) (1)
- Γ 2 m =j exp(ik 0 |r j -r m |) ik 0 |r j -r m | β m ,
where j = 1, . . . , N and β j is the probability amplitude of excitation of the jth atom, E i (r) = E 0 e ik0•r is the electric field of the incident laser and ∆ 0 = ω 0 -ω a its detuning with respect to the atomic transition, where ω 0 = ck 0 . In this approach, short-range dipole terms and polarization effects are neglected [13]. Neglecting granularity effects, the cloud can be described by a continuous field β(r, t), whose steady-state regime is given by

e ik0•r = (2δ + i) β(r) (2)
+ dr ′ ρ(r ′ ) exp(ik 0 |r -r ′ |) k 0 |r -r ′ | β(r ′ ),
where ρ(r) is the atomic density, δ = ∆ 0 /Γ and we have set

β(r) = dE 0 hΓ β(r). (3)
Let us remark that the kernel of Eq.( 2) is the Green function for the Helmholtz equation, that is

(∇ 2 + k 2 0 ) exp(ik 0 |r -r ′ |) |r -r ′ | = -4πδ(r -r ′ ) (4)
and that (∇ 2), we obtain that β(r) satisfies the Helmholtz equation [14,15] [

2 + k 2 0 ) exp(ik 0 • r) = 0. Then, applying (∇ 2 + k 2 0 ) on Eq.(
∇ 2 + k 2 0 m 2 (r)] β(r) = 0, (5)
where m(r), the cloud refractive index, is given by

m 2 (r) = 1 - 4πρ(r) k 3 0 (2δ + i) . (6)
Hence, the cloud of cold atoms acts on the light as a "classical" medium of index m(r), whose imaginary part originates in the single-atom decay term [16]: it is here responsible for the absorbing nature of the cloud, and vanishes only in the limit of far-detuned incident laser. For a cloud with spherical symmetry m(r), the solutions of the wave equation can be decomposed along the orthogonal basis of the spherical harmonics [START_REF] Bohren | Absorption and scattering of light by small particles[END_REF] as

∞ n=0 n s=-n u n (r)Y ns (θ, ϕ)
where the radial modes u n satisfy

u ′′ n (r) + 2 u ′ n (r) r + m 2 (r) - n(n + 1) r 2 u n = 0. ( 7
)
We focus on the axi-symmetric problem, where only the s = 0 modes are relevant, and the solution write

β(r) = √ 4π ∞ n=0 √ 2n + 1i n β n u n (r)Y n0 (r). ( 8
)
where r is a unit vector in the direction of r. As for the incident wave, it decomposes as

exp(ik 0 • r) = √ 4π ∞ n=0 √ 2n + 1i n j n (k 0 r)Y n0 (r). ( 9
)
Since the exponential kernel is diagonal in the basis of the spherical harmonics

exp(ik 0 |r -r ′ |) k 0 |r -r ′ | = 4πi ∞ n=0 n s=-n Y ns (r)Y * ns (r ′ ) × j n (k 0 r ′ )h (1)
n (k 0 r) for r > r ′ j n (k 0 r)h (1) n (k 0 r ′ ) for r ≤ r ′ , (10)
where h

(1) n (z) are the spherical Hankel functions. We then introduce f n (r) such that

dr ′ ρ(r ′ ) exp(i|r -r ′ |) k 0 |r -r ′ | u n (r ′ )Y n0 (r ′ ) = f n (r)Y n0 (r).
(11) so that, assuming ρ(r) = 0 for r > R f n (R) = 4πih (1) n

(k 0 R) R 0 ρ(r ′ )r ′2 u n (r ′ )j n (k 0 r ′ )dr ′ .
(12) Then, the projection of Eq.( 2) on mode n leads to

j n (k 0 r) = (2δ + i)β n u n (r) + β n f n (r). ( 13
)
Using Eq.( 12) and defining

λ n = 4π R 0 ρ(r)r 2 u n (r)j n (k 0 r)dr (14)
we obtain

β n = j n (k 0 R) (2δ + i)u n (R) + iλ n h (1) n (k 0 R) . ( 15
)
Thus, the exact solution to the scattering problem is obtained under the form of infinite series over the spherical modes, as is reminiscent of Mie's solution [START_REF] Mie | Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen[END_REF][START_REF] Bohren | Absorption and scattering of light by small particles[END_REF]. In the latter approach, the coefficients of scattering for each mode are determined by using the continuity of the tangential electric and magnetic fields, and of their derivatives, at the cloud boundaries. In our problem, we considered a scalar electric field only, and assuming it is orthoradial, it reads [9] E s (r) = -

dk 2 0 4πǫ 0 dr ′ ρ(r ′ ) exp(ik 0 |r -r ′ |) |r -r ′ | β(r ′ ). (16)
Now, the electric field expresses in two different ways inside (r < R) and outside (r ≥ R) the cloud

E (out) s (r) = E 0 2i ∞ n=0 (2n + 1)i n β n λ n h (1) n (k 0 r)P n (cos θ) (17)
E (in) s (r) = E 0 2i ∞ n=0 (2n + 1)i n β n P n (cos θ)4π × r 0 dr ′ r ′2 u n (r ′ )j n (k 0 r ′ ) h (1) n (k 0 r) + R r dr ′ r ′2 u n (r ′ )h (1) n (k 0 r ′ ) j n (k 0 r) . (18)
A straightforward calculation shows that E s (r) and ∂E s (r)/∂r as given by eqs.( 17) and ( 18) are continuous at the cloud boundary r = R. This allows to conclude that our solution is the same as the one proposed by Mie, although we did not make a direct use of continuity hypotheses for the electromagnetic fields. Let us also remark that our solution holds for any radial solution u n (r) of the Helmholtz equation, i.e. it applies to any spherical cloud with finite boundary at r = R.

Mie resonances in nonuniform resonant media. -The treatment of scattering in nonuniform media is difficult due to the lack of explicit solutions to the wave equation in that case, so numerical approaches are the most common to tackle with this problem [START_REF] Bohren | Absorption and scattering of light by small particles[END_REF]. We shall here treat the case of homogeneous samples, and of those with a quadratic dependence of the atomic density over the radius.

In the first case, the atomic density of a cloud of radius R is ρ 0 = 3N/4πR 3 , whereas its refractive index is constant m 0 = 1 -3N/[(k 0 R) 3 (2δ + i)] and the solution of the wave equation inside the cloud is simply u n (r) = j n (k 0 m 0 r). Thanks to properties of the Bessel functions [START_REF] Gradshteyn | Ryzhik Table of Integrals[END_REF], λ n can be calculated explicitely as

λ n = (2δ + i)(k 0 R) 2 [m 0 j n-1 (k 0 m 0 R)j n (k 0 R) -j n-1 (k 0 R)j n (k 0 m 0 R)] (19)
As for clouds of size R, with a parabolic density ρ(r) = (5N/2V )[1 -r 2 /R 2 ] and volume V = 4πR 3 /3, their spatially-dependent refraction index reads

m 2 q (r) = m 2 c + γ 2 r 2 , (20)
with

m c = 1 -(15/2)N/[(k 0 R) 3 (2δ + i)]
the index in the core of the sample, and

γ 2 = (15/2)N/[(k 0 R) 5 (2δ +i)].
Remark that when k 0 R → ∞, but at constant central density N/V and at fixed r, Eq.( 7) tends toward the equation for the spherical Bessel function j n (k 0 m c r): the homogeneous medium limit is then recovered, as expected since the center of the cloud is then locally homogeneous. Using the substitution [19] u n (r) = r -3/2 w n (x), with x = γr 2 /2, we obtain a Coulomb wave equation for w n (x), well known in nuclear physics [START_REF] Biedenharn | Coulomb Excitation[END_REF] w

′′ n (x) + 1 + m 2 c 2γx - n(n + 1) 4 - 3 16 1 x 2 w n (x) = 0. ( 21
) Its solutions are the so-called Coulomb wave functions [START_REF] Abramowitz | Handbook of Mathematical Functions[END_REF] F n/2-1/4 (-m 2 c /4γ, x). The irregular Coulomb wave function G is discarded because of its 1/r divergence at the origin, just as the spherical Hankel function is discarded to describe the field inside homogeneous media. Thus, we get the following solution for the radial component of the field

u n (r) = 1 r 3/2 F n 2 -1 4 - m 2 c 4γ , γr 2 2 . ( 22
)
Let us introduce the structure factor s(k) = s(θ, ϕ) for a direction k = k 0 (sin θ cos ϕ, sin θ sin ϕ, cos θ), defined as

s(k) = 1 N dr ′ ρ(r ′ ) β(r ′ )e -ik•r ′ . ( 23
)
After integration over space, it turns into

s(k) = ∞ n=0 (2n + 1)β n λ n P n (cos θ), (24)
where we have made use of

2π 0 dϕ ′ sin ϕ ′ e -i(a cos ϕ ′ +b sin ϕ ′ ) = 2πJ 0 a 2 + b 2 1 -1 dy J 0 (r ′ sin θ 1 -y 2 )P n (y)e -iyr ′ cos θ dy = 2i -n j n (r ′ )P n (cos θ). (25)
Then, in the far-field limit (r ≫ R), the scattered intensity simply reads

I scat (r, θ, ϕ) = E 2 0 4(k 0 r) 2 |s(θ, ϕ)| 2 . ( 26
)
Using the analytic expressions obtained for the homogeneous and the parabolic density, we now turn to studying the resonances in Mie scattering for atomic clouds. Simulations of the intensity scattered along the illuminating axis reveal that homogeneous samples exhibit some spiky irregular structures, whereas inhomogeneous (parabolic) samples are characterized by more regular oscillations (see Fig. 1). This suggests not only that the resonances do not only originate in the sharp boundaries of homogeneous obstacles, but on the contrary that they may even be easier to observe in media with smooth densities. Actually, the numerous spikes in homogeneous media can be interpreted as "whispering gallery modes" [START_REF] Rayleigh | The theory of sound[END_REF][START_REF] Nussenzveig | Diffraction Effects in Semiclassical Scattering[END_REF][START_REF] Oraevsky | [END_REF], that propagate along the surface of the cloud, within a local minimum of the effective radial potential created by the jump of the medium's index. On the other hand, assimilating the cloud to a cavity, one can interpret the regular oscillations as cavity modes which survive despite the smoothening of the boundaries of the potential. Thus, the smoothening of the index leads to a modification of the local minimum for the effective potential which supports the whispering gallery modes [START_REF] Nussenzveig | Diffraction Effects in Semiclassical Scattering[END_REF], so that the latter may be attenuated or broadened (see Fig. 1), or even disappear (see Fig. 2). Finally, the simulations confirm that the theory for continuous media is in good agreements with N -body samples (see Fig. 1), although discrepancies appear for smaller values of detuning, and thus larger refractive index. However, simulations where N is tuned reveal that this effect is due to the small number of particles that one can consider numerically: since finding the stationary solution of (1) requires the inversion of a N × N matrix problem, simulations are typically limited to systems of at most N ∼ 2.10 3 particles. Thus, the finite-N effects observed in Fig. 1 are expected to vanish for larger number of particles that one can typically find in a cold atomic cloud (that is N = 10 5 -10 7 ). More specifically, one can show that N/σ 2 r ≫ 1 is required to neglect granularity [9,10]. This highlights the relevance of the continuous approach to model the scattering process of large-N clouds, on which we shall now focus. a cloud of cold atoms is the displacement of its centerof-mass (e.g. measured by time-of-flight imaging techniques [10,12]) due to the radiation pressure force, which can be interpreted microscopically by a sequence of absorption and spontaneous emission cycles. More specifically, the atom absorbs a laser photon with k = k 0 ẑ and reemits it in a direction k = k 0 (sin θ cos ϕ, sin θ sin ϕ, cos θ). The resulting radiation pressure force reads [9]

Resonances in

F z = -hk 0 Ω 2 0 Γ Im(s(k 0 )) - hk 0 Ω 2 0 N 8πΓ dk cos θ|s(k)| 2 (27)
where Ω 0 = dE 0 /h is the Rabi frequency of the incident field. In the framework of Mie scattering, the absorption part of the force reads

F (a) z = -hk 0 Ω 2 0 ΓN ∞ n=0 (2n + 1)Im(λ n β n ). (28)
As for the emission part, using the orthogonality of Legendre polynomials, as well as the relation

1 -1 xP n (x)P n ′ (x)dx = 2 (nδ n,n ′ +1 + n ′ δ n ′ ,n+1 ) (2n + 1)(2n ′ + 1) , (29)
it reads

F (e) z = -hk 0 2Ω 2 0 ΓN ∞ n=0 (n + 1)Re(λ n λ * n+1 β n β * n+1 ). ( 30
)
These analytical expressions then allow to investigate numerically the presence of resonances in Mie theory in large-N samples [10,[START_REF] Bux | [END_REF], such as those produced in Tübingen [12], where clouds of size σ r = k 0 r 2 ≈ 10 and N ∼ 10 6 were studied. The simulations reveal that the radiation pressure force actually exhibits resonances when the number of atoms N is tuned, at fixed detuning δ (see Fig. 2a). Their contrast is significantly higher at (negative) larger values of δ yet they require larger number of atoms. Plotting the force oscillations as a function of the phase shift of on-axis light rays

φ = k 0 [Re(m(0, 0, z)) -1] dz (31)
makes clear that φ is the relevant variable to characterize the resonances in clouds with quadratic densities, i.e. cavity modes. Note that since the absorption and emission forces compensate at small φ, Mie oscillations actually appear in the radiative force at values of φ higher than for the scattered light (typically 4π). The period of these oscillations can be qualitatively retrieved using the formulae for the homogeneous media (19). If one considers a large cloud k 0 R ≫ 1, then j n (x) ≈ sin(x -nπ/2)/x (for x ≫ 1) leads to the approximate expression

λ n ≈ 2δ + i m 0 m 0 + 1 2 sin((m 0 -1)k 0 R) (32) + m 0 -1 2 sin((m 0 + 1)k 0 R -nπ) , n 2 < k 0 R.
This explains why the radiation pressure force will oscillate with the value of the phase-shift (φ = 2Re(m 0 -1)k 0 R for homogeneous clouds, and (4/3)Re(m 0 -1)k 0 R for samples with quadratic densities). Note that in the case of a dilute cloud such that |m 0 -1| ≪ 1 (yet |m 0 -1|k 0 R can be large), one has φ ≈ -(2/3)b 0 δ/(4δ 2 +1), which highlights the role of the optical thickness [7] b 0 = 6π k 2 0 ρ(0, 0, z)dz (33) instead of the atomic density in the collective scattering problem [10]. Since optical resonances will emerge for N/δσ 2 r ∼ 1, the conditions for the observation reads N > δσ 2 r . Finally, remark that when the spatial density N/σ 3 r becomes significant, other effects such as near field dipole dipole interactions have to be accounted for [11,13].

Here, due to the special role of the wavelength of the illuminating laser on resonant atoms, the detuning δ can also be used as a control parameter to generate Mie oscillations: at odds from dielectric droplets experiments where it changes the effective size of the cloud k laser R, it here modifies only the index of the cloud. Fig. 3a depicts this phenomenon when the incident wavelength is tuned, and one recovers Mie oscillations in that case as well. As discussed previously, clouds with smooth (quadratic) densities exhibit more regular and more contrasted oscillations. Remark that using δ as a control parameter, the curves F z (φ) or I scat [θ = 0](φ) are extremely similar to those where N is tuned (see Fig. 2b), since only the refractive index is important, and the phase-shift φ captures this information. Finally, we conclude by pointing out that the resonances in Mie theory can be observed as well at positive detunings: As can be observed in Fig. 3b, the scattered intensity is rather symmetric in δ, which can for example be understood by the fact that the phase-shift φ ≈ -(2/3)b 0 δ/(4δ 2 + 1) is antisymmetric in δ (yet the imaginary part of the refractive index is not, thus the difference in amplitudes). The symmetry decreases with δ, when the refractive index deviates significantly from unity, yet model (1) then loses its validity, and a more exhaustive model is required to study the light scattering [13].

Conclusion. -We studied resonances in Mie theory in resonant media with parabolic densities, a case most appropriate for cold atomic clouds. It was shown that the Mie oscillations were more regular in the case of smooth (parabolic) densities, where only cavity modes survive, and that they can be observed as well in the scattered intensity as in the radiation pressure force, even though for the lowest phase shifts the radiation pressure force is a less sensitive measurement. The parameters of Figs.2a and 3a correspond to the experimental case of [12], and it suggests that the resonances can be detected in the available range of parameters. Note that the main difference with this experiment is that the cloud was then cigar-shaped, and the density assumed to be Gaussian; yet parabolic densities are definitely more realistic than homogeneous ones to describe the Gaussian case.

Furthermore, the contrast of the resonances is connected to the imaginary term of the cloud's index (6), as discussed in [15]. More specifically, the smaller the single-atom decay term is (compared to the detuning), the larger the contrast. Hence, large-N and far-detuned configurations will generate sharper resonances, for they minimize the single-atom decay contribution.

A more thorough treatment would of course require considering vectorial fields [15]. Nevertheless, simulations reveal that the scalar approach provides an accurate description of resonances in Mie scattering for large and dilute atomic clouds.

Finally, we note that tuning the illuminating laser toward the blue (δ > 0) can also allow for the observation of the resonances. However, when 4πρ/(2δk 3 0 ) > 1, the sample becomes strongly dissipative since Re(m 2 ) becomes negative (see Eq.( 6)): radiation trapping will then emerge [25], that will damp the scattering process and eclipse the resonances. The occurence of this phenomenon in cold atomic clouds will be the object of future works.

Fig. 1 :

 1 Fig.1: Intensity scattered by the sample along the illuminating axis for homogeneous (thin black line) and quadratic (thick blue line) continuous densities, and for a N -atom homogeneous cloud with random atomic distribution (black dots). Simulations realized for N = 2.10 3 and σr = k0 r 2 = 1 (and thus σr = 3/5k0R for an homogeneous density, and σr = 3/7k0R for a quadratic one).

Fig. 2 :

 2 Fig. 2: (a) Radiation pressure force as a function of the number of atoms N in the sample and for different detuning, namely δ = -667 (black lines) and δ = -50 (blue lines). (b) Radiation pressure force as a function of the phase-shift φ for δ = -667 (blue curves) and square of the structure factor |s(θ = 0)| 2 (black curves), proportional to the far-field scattered light in the forward direction. For both figures, the thin curves correspond to homogeneous samples, the thick ones to quadratic densities. Simulations realized for σr = 10.

Fig. 3 :

 3 Fig. 3: (a) Radiation pressure force for a σr = 10 cloud as a function of the detuning δ and for different numbers of particles, namely N = 10 6 (black line) and N = 10 5 (blue line). The thin curves correspond to homogeneous samples, the thick ones to quadratic densities. (b) Intensity scattered along the illuminating axis |s(θ = 0)| 2 (black thin curve) as a function of the detuning, as well as real (plain blue line) and imaginary (dash-dotted blue line) parts of refractive index m0 of the cloud, for a cloud of size σr = 100 and homogeneously filled with N = 2.10 6 particles.


Pierre Fleckinger
email: pierre.fleckinger@univ-paris1.fr.

Thierry Lafay
email: tlafay@univ-paris1.fr.

Product Flexibility and Price Competition in Hotelling's Duopoly *

Keywords: horizontal differentiation, catalog competition, sequential moves. JEL Classification: L13 D43 M3

In a Hotelling's duopoly with a general transportation cost function, we study competition through catalog: each firm chooses at the same time a price and a location. With simultaneous catalog offer, there is no equilibrium in pure strategies for high valuations of the consumers, while a Stackelberg equilibrium exists under mild conditions. The follower is better off than the leader, whose price is smaller: the location preemption effect is weaker than the price leadership effect. We obtain closed-form solutions for the linear and quadratic costs cases. Using these results, we discuss the nature of competition depending on the relative flexibility of products and prices.

Introduction

On most markets, the product choice is less flexible than the price policy. This explains why a reasonable modeling assumption is that price competition occurs only after the choice of products. But this is not always the case for at least two reasons. A first reason is related to the nature of the product:

In some economic sectors, firms enjoy a large adaptability of their product.

Standardization of intermediary inputs makes production highly flexible.

For example, computer stores can change the characteristics of the computers they sell within one day. The service sector also usually exhibits extreme adaptability: for instance, online travel agencies' offers change virtually in a continuous fashion, both with regards to price and products offered. As a consequence, on such markets, the choice of product and the choice of price are simultaneous. A second reason is related to price rigidity: some firms appear as leaders (in time) concerning both product choice and prices, as they do not revise their price offer when other competitors enter the market. For example, each shop of a chain stores may face different independent markets but the firm chooses the same type of product on all markets and a unique pricing policy. Local competitors reacts to these offers while the converse is obviously not true. Agency situations often explain this inertia: an employee that work as a seller is not allowed to lower the price of the product it sells but must stick to the announced price. The use of national advertisement, too, partly explains the price stickiness of chains. In this pa-per we focus on these horizontally differentiated markets where competition takes place through catalog, a product-price couple (rigid) announcement 1 .

In his pioneering paper, Hotelling's main motivation was to construct a model in which price competition does not take a "winner-takes-all" form, but in which small changes in price only affects smoothly the quantity sold by a competitor. In his setting, the author shows that if firms choose first their locations, their will be a tendency towards homogeneity of the product-the so-called principle of minimum differentiation. It is somehow paradoxical that the idea of 'stability' in competition put forth by Hotelling is actually invalid in his model. Indeed, as pointed out by d'Aspremont et al.

(1979), a pure strategy equilibrium does not exist because a price equilibrium fails to exist when products are too similar. D'Aspremont et al. show in turn that with quadratic transportation costs an equilibrium exists 2 . In this modified setting, the firms locate at the extreme points of the market and differentiation is maximal. Many authors have subsequently found mitigated results about differentiation in equilibrium [START_REF] Economides | The Principle of minimum differentiation revisited[END_REF] Hinloopen and van Marrevijk, 1999, among others). In our model, we obtain that differentiation is generically intermediate, and this does not depend on the shape of costs. 1 We refer here to the informative dimension of catalog. Indeed each firm produces only one kind of product, and it is the information on that product and its price that matters in the catalog. Recently, [START_REF] Monteiro | Catalog competition and Nash equilibrium in nonlinear pricing games[END_REF] have used the term catalog to refer to product line. They study equilibrium existence in a nonlinear pricing game.

2 As pointed out by [START_REF] Macleod | On the non-existence of equilibria in differentiated product models[END_REF], this implies that the mass of indifferent consumers is zero, which restores equilibrium existence in price, thanks to continuity of the payoffs.

Another direction to restore equilibrium existence is to allow for mixed strategy in price (see Dasgupta and Maskin (1986a,b) for a general treatment and applications), which can be interpreted as occurrence of sales [START_REF] Osborne | Equilibrium in Hotelling's Model of Spatial Competition[END_REF].

While models of horizontal differentiation have been the subject of a vast literature, we focus on a topic that has not received enough attention:

the sequentiality of product and price choices (see [START_REF] Prescott | Sequential location among firms with foresight[END_REF] for an early contribution). In most of the related literature, it is usually assumed that firms choose location first (possibly sequentially as in Tabuchi and Thisse (1995)) once and for all, then compete simultaneously in price [START_REF] Neven | Endogenous sequential entry in a spatial model[END_REF][START_REF] Lambertini | Equilibrium locations in a spatial model with sequential entry in real time[END_REF][START_REF] Götz | Endogenous sequential entry in a spatial model revisited[END_REF][START_REF] Economides | Does it Pay to be the First? Sequential Locational Choice and Foreclosure[END_REF]. This is consistent with the view that the location is a geographical parameter, or with the view that the design of the product is far less flexible than its price. As far as we know, only few papers modify the assumption of last stage simultaneous price competition. [START_REF] Anderson | Spatial competition and price leadership[END_REF] studies a price leadership configuration: the firms choose sequentially the locations, then a Stackelberg competition in price takes place. His aim is to determine which firm chooses to be the price leader, the answer is that the second entrant is induced to be price leader, given the location chosen by the first entrant. [START_REF] Lambertini | Unicity of equilibrium in the unconstrained Hotelling model[END_REF] In this paper, we investigate situations where location and prices are necessarily chosen together. The first result concerning this "catalog" competition is that in general there does not exist an equilibrium in pure strategies when competition is simultaneous. However, in a sequential catalog competition, we are able to obtain the existence of subgame perfect equilibria with strictly positive profits for both firms under some assumptions.

Contrary to traditional entry models, it is the follower that decides whether to exclude the incumbent or to accommodate, since only the follower firm has the ability to undercut its rival. The subgame perfect equilibrium is such that the leader (incumbent) always earns less than the follower3 . This introduces a tension between the temptation of attrition -waiting to be the follower, which leads to an unstable configuration, and the will to guarantee a strictly positive profit.

Section 1 introduces the model and its assumptions. Section 2 gives the main insights concerning the general model and characterizes sequential catalog competition when a pure strategy subgame perfect equilibrium exists.

We solve then the cases in which costs are linear (as in Hotelling's original model) and in which costs are quadratic. We also study the catalog competition in the circular model of [START_REF] Salop | Monopolostic Competition with Outside Goods[END_REF]. Section 3 contains a discussion of the model about sequentiality assumptions, examples are given where the results seem to have some predictive power. The concluding section sums up the results.

The model

We study a market where a mass 1 of consumers is distributed along the unit interval according to the density function f . A product choice corresponds to a point x ∈ [0, 1], the ideal product of the consumer located at x. We are interested in competition through catalog in the following sense: each firm chooses once and for all a product in [0, 1] and a price in R + . Firm A chooses (a, p) and firm B chooses (b, q). For convenience, we choose a and b as the distance of firms to zero4 .

A consumer x chooses to purchase a product according to the utility maximization: Max {v -C(x -a) -p, v -C(b -x) -q, 0}. The reservation price is v and a consumer buys a product only if the perceived price is less than his reservation price. The cost function C is even, strictly increasing on the right of 0, convex and twice continuously differentiable (except maybe at zero, where it admits a left and a right derivative). In

Hotelling's original model, the cost function is simply the distance and consumers always consume the good, so that they minimize their disutility Min (C(x -a) + p, C(b -x) + q). We mainly focus on the case in which v is infinite. With or without a reservation price, consumer x prefers firm B whenever C(x -a) + p ≥ C(b -x) + q. We define the indifferent consumer, when unique, as y such that p + C(a -y) = q + C(b -y).

When a consumer is indifferent between firm A and B, we assume that it chooses the closest firm. Wherever this last criterion does not break indifference (when the distances to each firm are equal), we assume that some consumers choose A while others choose B. Formally, there exists some predetermined λ with 0 < λ < 1 such that firm A gets a share λ and firm B a share (1 -λ) of those indifferent consumers. We first study the game where firms choose their catalog simultaneously, second we assume that firm A is a catalog leader. In this second setting, firm A first chooses its strategy (a, p), then firm B chooses (b, q) after having observed A's moves. Hence, we look for the subgame perfect equilibrium of the perfect information extensive game depicted in figure 1.

We may remark that if firm B chooses the same location as firm A, then its best response must be such that q ≤ p otherwise it gets no customer.

In turn, by setting a slightly smaller price than firm A, firm B obtains the whole market. Thus if both firms choose the same location in equilibrium before choosing their price simultaneously, prices will be driven down to zero, "undercutting" strategies. In the sequential setting, as the first firm posts a price before the other enters, it cannot use such an undercutting strategy. On the contrary, firm B has a powerful threat with these undercutting strategies. The question is thus whether the leader can induce the follower not to undercut, thereby securing a positive profit.

General results

Combining the results of Hotelling and d'Aspremont et al., one gets the insights of the following result concerning catalog competition:

Proposition 1 For any distribution of the consumers', any convex cost function and an infinite willingness to pay, simultaneous catalog competition has no equilibrium in pure strategies.

A direct proof is rather intuitive 5 . First we notice that whatever the other firm's strategy, a firm can always get a strictly positive profit. Second, we remark that there cannot be an equilibrium where both firms choose the same product because then prices would be driven to zero. Let us therefore assume that one firm is pricing less than the other. This firm has an incentive to choose the same product as its competitor, but then the other firm would make zero profit, thus it cannot be an equilibrium. The same argument holds when firms use the same price (except that firm will price slightly less than its competitor). Thus there is no Nash equilibrium in pure strategies when v is infinite. However it is obvious that if v is small, there are Nash equilibria as firms can enjoy local monopoly. This continues to hold for small enough v, in which case firms compete for some customers, but there is some finite level of v beyond which pure strategy equilibrium does not exist any more6 .

We now study the extensive game with sequential catalog issuing. We are able to state the following observation.

Proposition 2 For any distribution of the consumers', any cost function and any willingness to pay, the follower makes a (weakly) higher profit than the leader in any equilibrium of the sequential catalog competition.

This proposition is proved in appendix. It is of course straightforward if the leader cannot prevent undercutting7 , since the leader makes then zero profit. This proposition shows that there is always a second mover advantage in this setting. 8 . In other words, the price followership effect is stronger than the location leadership effect in our model. This statement departs from the result in [START_REF] Anderson | Spatial competition and price leadership[END_REF]. He studies the case in which firms sequentially choose their location before sequentially choosing their prices. With that different timing structure he shows that a leader in location and price earns more than its follower.

This proposition also underlines that even when there exists an equilibrium in simultaneous catalog competition (which requires a finite v as already mentioned), both firms are better off with sequential catalog competition. Indeed, the leader can always get as much profit as in the simultaneous game by choosing the equilibrium product and price of the simultaneous game. Formally, let Π S be the largest profit among the simultaneous Nash equilibria in pure strategies 9 then Π B ≥ Π A ≥ Π S . However, even if both firms earn more in the sequential setting, they both prefer to be the follower.

In other words, they can agree to play a catalog competition sequentially but none wants to publish its catalog first, creating an attrition temptation.

From now on, feminine pronouns will refer to the leader and masculine ones to the follower. For a better understanding of the general shape of equilibrium (when it exists), a classification of the follower's strategies is in order. In a sequential catalog competition, he has two kinds of strategies.

Either he chooses an exclusion strategy, which means that he sets a catalog such that he serves the whole market. Or he chooses an accommodation strategy such that the leader keeps a strictly positive market share. Among the possible exclusion strategies, the best one is clearly to choose the same product as the leader but with a slightly lower price, p -ǫ, with ǫ as small as desired. In order to secure a positive market share, the leader must thus guarantee a profit at least equal to p to the follower when using an accommodation strategy. We may note that contrary to the classical literature 9 When they exist.

about entry, in a catalog competition it is the follower who chooses between accommodation and exclusion.

Consider the point of view of the follower, for a given strategy of the leader (a, p), with a < 1 2 as it is pictured in Figure 2. The case a ≥ 1 2 happens to be similar, with symmetric considerations each time. For any price q < p, i.e. in zones I and IV, it is clear that the best reply is to undercut the leader, by setting b = a and q = p -ǫ. Any location in zone II is dominated by the point in zone III such that the location is symmetric with respect to a, keeping the same price, because then the market share is strictly bigger.

It also holds for any catalog (b, q) in II' such that the follower has a positive market share. The cases where the follower has zero market shares is obviously uninteresting. Overall, to find the best-reply of the follower, we can restrict our attention on one hand to the subspace where b ≥ a and q > p (accommodation), and on the other hand to the best undercutting strategy (exclusion). Now, in zone III and III', for any price q, the follower is better off choosing a catalog along the perceived price curve of the leader (x, C (a -x) + p) , because it maximizes his market share. To see this, just notice that by definition of y, the market share of the follower is the mass of consumers to the right of y. Thus, for a constant price q, the goal of the follower is to minimize y. This is formally expressed in the next lemma which is proved in appendix. Here the first mover is also the 'leader' in price (as she posts the lowest price). Indeed, the leader defines a price schedule and the follower uses this price umbrella to fix its price. It also departs from Bertrand competition where the follower in time is the leader in price.

We now investigate equilibrium existence. The main difficulty is related to continuity. First, even the existence results of Dasgupta and Maskin (1986a,b) for mixed strategy equilibrium do not apply because of a lack of upper hemi-continuity of the follower profit function 10 . Second, even though we now know that the follower chooses a catalog (b, q) such that the indifferent consumer is b, the best-reply b * (a, p) may be a multivalued correspondence 11 . Classically, this non uniqueness of best reply would imply a multiplicity of subgame perfect equilibria depending on the anticipation made by the leader regarding the way the follower will react. Third, the profit of firm A may not be a continuous function of (a, p) 12 . To rule out these situations, we impose the following sufficient condition for uniqueness of the best reply of firm B.

Condition 1 C ′′ (x) < 2C ′ (x) for all x > 0.
This condition imposes an upper bound on the relative curvature of the cost function. We are now in position to state the central result of this section.

Proposition 3 With uniform consumers' distribution and, under condition

1, there exists a subgame perfect equilibrium in a sequential catalog competition such that the leader makes a positive profit.

10 For example the function ΠB is such that lim q→p q<p ΠB(a, q) = p > (1 -λ)p = ΠB(a, p). To overcome this issue, one can think of setting λ = 0 but it would only shift the continuity problem to the leader's optimization.

11 For example, consider the following convex, twice differentiable function:

C(x) =    x 1-x if x ≤ 1 2 8x 2 -4x + 1 otherwise
. For a = 0 and p = 1 the accommodation best reply is not uniquely defined as any b ∈ [0, 1 2 ] (associated with the corresponding q = 1 + C(b)) gives the same profit to the follower.

12 Consider again the example of footnote 11, and fix a = 0. For p < 1, the follower's best reply is some b ≥ 1 2 , for p = 1, there is a continuum of best-replies, for p > 1, exclusion is the unique best-reply. Therefore the profit of firm A is not continuous at (a, p) = (0, 1).

The proof proceeds in four steps, and the sketch is as follows. First, we obtain the best reply of the follower if he uses an accommodation strategy. The leader has to choose her catalog in order to prevent an exclusion strategy. In turn, this implies that p must be bounded. Finally, the leader maximizes a continuous function over a compact, which guarantees the existence of a solution. Of course, there might be equilibria even when condition 1 is not fulfilled as we will show: the quadratic model possesses a subgame perfect equilibrium obtained by backward induction.

Linear cost (à la Hotelling)

Following Hotelling's original assumption, we consider linear costs: C(x) = t|x|, and look for the subgame perfect equilibrium of sequential catalog competition. Note that the conditions for equilibrium existence of proposition 2 are met. Hence, we follow the lines of the preceding proof to obtain the equilibrium. The optimal limit strategy of firm B if he decides to exclude the leader is such that b Ex , q Ex = (a, p), implying Π Ex B (a, p) = p. As the problem is symmetric regarding locations we compute the equilibrium such that a ≤ 1 2 . We begin by computing firm B's best reply if he decides to accommodate.

From lemma 1, the optimal accommodation strategy is such that q

= p + t(b -a). Therefore, firm B maximizes Π Ac B = (p + t(b -a)) (1 -b) s.t. 0 ≤ a ≤ b ≤ 1
and p ≥ 0, q ≥ 0 (these constraints, that define the valid domain, are omitted in the following). This profit is a concave function, and the first order condition implies: b Ac , q Ac = 1+a-p/t 2 , 1-a+p/t For any p, Π Ac A is increasing while Π Ac B is decreasing in a, thus the constraint must be binding. Substituting in the objective function, A's program boils down to maximizing p t 1 -p t for p > 0. The value of the equilibrium parameters are in turn fully determined by using binding constraint and the expression of the best-reply. This allows us to state:

Proposition 4
The subgame perfect equilibrium of the linear cost game is such that (up to geometrical symmetry):

(a * , p * , b * , q * ) = 1 9 , 4t 9 , 1 3 , 2t 3
Π * A = 4t 27 and Π * B = 4t 9
We may notice that the distance between the firms is such that no equilibrium with competitive price exists. It violates the existence condition given by d'Aspremont et al. (1979). In other words, if the firms were able to revise (only) their prices, profits would be driven to zero by a price war. A price commitment of the leader is therefore valuable for both firms, because it allows to avoid an unstable configuration leading to zero profits. The locations and prices at equilibrium are depicted in figure 3. Of course the symmetric equilibrium (a * , p * , b * , q * ) = 8 9 , 4t 9 , 2 3 , 2t 3 is also possible.

Quadratic cost (à la d'Aspremont et al.)

In the quadratic cost case, if firms first choose locations (either sequentially or simultaneously) prior to prices, there is maximal differentiation With quadratic costs, condition (1) of section 2 is not satisfied, and we cannot apply proposition 2 to claim existence of an equilibrium. However, we demonstrate in appendix that an equilibrium still exists by constructing it, and obtain:

Proposition 5
The subgame perfect equilibrium of the catalog competition in the quadratic cost game is such that (up to geometrical symmetry):

(a * , p * , b * , q * ) = 0, t 4 , 1 2 , t 2
Π * A = t 8 and Π * B = t 4 p * = 1 4 q * = 1 2 a * = 0 1 b * = 1 2 Π * A = 1 8 Π * B = 1 4

Consumers disutility

Figure 4: Equilibrium with quadratic costs for t = 1 Thus, differentiation is more important with quadratic costs than linear ones. The leader chooses a niche, while the follower chooses a generic product. The locations and prices at equilibrium are depicted in figure 4.

Catalog competition in Salop's model

We investigate here how the catalog competition operates on a circular market such as [START_REF] Salop | Monopolostic Competition with Outside Goods[END_REF]. There is a unit mass of consumers uniformly distributed along a circle. Indeed, it is easy to show that simultaneous catalog competition exhibits no equilibrium in pure strategies. The sequential catalog competition is such that the follower earns more than the leader and prices higher. However, contrary to Hotelling's model, in Salop's model there exists an equilibrium if both firms choose first a location (either simultaneously or sequentially 13 ) then compete simultaneously in price. This equilibrium is such that firms choose opposite locations and price at t, so that their profits are t/2. The question arises whether a sequential catalog competition is likely to appear. In order to answer this question, we investigate the equilibrium when costs are linear.

It is clear that either the follower excludes the leader by choosing the same location and pricing at p -ǫ, or he accommodates with an opposite kind of product. There is thus maximal differentiation in any equilibrium where the leader makes a strictly positive profit. In this case, the circular market can be seen as two separate linear markets with firms located at the extreme points. The indifferent consumer is such that: p + ty = q + t(1 -y),

and thus y = q-p+t 2t . Moreover it should be the case that y > 0 if the leader makes a profit, thus one should have: p < q + t.

The follower maximizes q(1 -y) = q t+p-q 2t . The first order condition implies: q = p+t 2 , and consequently Π B = (p+t) 2 8t . Firm A maximizes p 3t-p 4t under the constraints p < 3t (positive market share) and p ≤ (p+t) 2 8t (the follower prefers accommodation to exclusion).

The second constraint is the binding one and we obtain the equilibrium

values p = 3 -2 √ 2 t and q = 2 - √ 2 t. Profits are: Π A = 3 √ 2 2 -2 t and Π B = √ 2Π A .
We may notice that whereas there is still maximal differentiation, prices are lower with catalog competition than in the standard timing of Salop 13 Let us point out that, as long as both firms locate at the opposite, any location couple is possible. In addition the simultaneous choice of location exhibits a strong coordination problem.

(1979). In fact, in a linear city, the leader can use both price and product characteristic to prevent exclusion -he can locate away from the center of the market in addition to pricing low. In turn, there is no niche on a circular market -the only way to discourage undercutting is to post a low price. This implies that while locations are the same as in the two-stage (locations, then prices) Nash equilibrium, the price of the leader is smaller.

Overall, in equilibrium, the leader is thus using a low price -high market share (more than a half) strategy, while the follower is using a high pricelow quantity strategy.

3 Discussion

The relative flexibility of prices

A feature of the model that deserves attention concerns the relative flexibility of location and price. While in a geographical interpretation, the location choice is a long-term decision relatively to the price decision, one can think of markets where both variables are equally flexible. The catering industry provides a good example of this. The product lifetime is very short (one day, if the cooks behave properly), and the choice of product can be a daily decision for independent restaurants, making product choice and price choice equally flexible. In contrast, for chains, uniform pricing and centralized decisions imply more inertia. In other words, chains make their price public by advertisement and national catalogs, while independent stores react to these leaders. The converse is obviously not true: local competition by a small competitor do not lead chains to revise their catalog. Going deeper into this example, one can remark that the model also fits the observation that most chain restaurants sell rather specialized products (like pizzas or Tex-mex food 14 ), at lower prices than small competitors.

Travel agencies on internet provide an illustration of the predicted results in the quadratic costs case. Typically common travel agencies were rather Of course, the conclusions would be affected when considering multiproduct firms. It would clearly be worth inquiring the full catalog competition problem with product line differentiation (see [START_REF] Monteiro | Catalog competition and Nash equilibrium in nonlinear pricing games[END_REF] for the most recent contribution on that topic).

A business strategy interpretation

In our sequential setting we proved that the follower always prices higher and locates exactly where the indifferent consumer is in equilibrium. On linear market, the leader's strategy appears therefore as a low price strategy a low market share strategy. This type of strategies is described in [START_REF] Gelman | Judo Economics: Capacity Limitation and Coupon Competition[END_REF]. In their model, the strategic choices are also bidimensional but instead of choosing a product/price couple firms choose a capacity/price couple. Leaders choose a low capacity/price couple giving 14 Hamburgers also are considered specialized food in Europe.

rise to low market share. This is what Gelman and Salop (1983) call a 'judo economics' effect. In our model, leaders commit to be small and hence limit the loss of market share due to competition by choosing a specific product exactly as leaders would choose a low capacity in order to not threaten future competition. Moreover they choose a low price in order to make its market unattractive to competition and to limit the followers' willingness to exclude leaders. 15Fudenberg and Tirole (1984) call this behavior the 'puppy dog' behavior. In their analysis, firms also face a bi-dimensional competition involving an investment choice followed by a strategic choice, such as a price. Here the price decisions are strategic complements and a central location choice appears as tough. The same conclusion therefore holds: the leader uses a 'puppy dog' behavior by choosing a specific product and in order to protect its own market it uses a low price strategy.

Interestingly, the equilibrium strategies in the circular setting are different. The leaders cannot anymore use the location strategy to pacify competition: a specialized niche market does not exist. In other words, the leader's only means to appear soft would be to use a high price strategy.

But this strategy precisely makes undercutting at the same location very attractive to the follower. Hence this induces leaders to be actually more aggressive in prices, to make undercutting unprofitable. As a result, in the circular market, the leader's strategy is 'top dog' behavior, as described in Fudenberg and Tirole (1984).

On sequential settings with linear costs

In models of horizontal differentiation, seven possible sequential settings may be considered 16 . We already know that the (three) cases where firms choose prices simultaneously have no equilibrium in pure strategies. Up to our knowledge, [START_REF] Anderson | Spatial competition and price leadership[END_REF] is the closest paper about sequential prices and differentiation. He studies the two cases (2 and 3 in figure 5) where firms first sequentially choose their location and then sequentially choose their prices with linear costs. He proves that the most likely situation is the one with the location leader being subsequently the price follower (case 3).

The explanation is that the location leader can choose a product such that the location follower prefers to be the price leader: the location leader then locates at the center of the market.

The credibility of the equilibrium he favors might be questioned using our timing structure. If the location follower decides to wait for the location leader to post her price before choosing her product (which is our timing structure), he makes a greater profit. Thus if, ex ante, the follower can credibly commit to enter an attrition game until the leader posts a price, then the location leader will post her price before the entry of the follower.

However, this attrition strategy is not credible per se: if the leader chooses her strategy as in Anderson's model (a = 0.5) then the follower has an incentive to choose his product before the leader posts her price. Otherwise, the location leader would have to post a tiny price in order to prevent undercutting, which would spoil the market for both firms.

Hence, we can distinguish two cases. If the choice of product is irre- Moreover, there is a value to commit to a product independently of a price only for the leader.

Conclusion

We have investigated the existence and shape of equilibrium in pure strategies when firms choose at the same time a location and a price in Hotelling's duopoly. There is no equilibrium in the simultaneous game, while when firms successively choose their catalog there exist equilibria with positive profits. An equilibrium is such that: 1) the follower always earns more than the leader, whatever the distribution of consumers and the costs structure;

2) the follower charges a higher price than the leader: the leader in time is the leader in price; 3) in both linear and quadratic cases, the leader makes the follower indifferent between excluding him from the market and choosing a location such that each firm has a fraction of the market; 4) Differentiation is never minimal nor maximal in equilibrium, and this is qualitatively independent of the costs' shape; 5) The follower may end up with both higher price and higher market shares than the leader.

In the linear cost case, a sequential catalog competition is likely to stabilize the market (there is no pure strategy equilibrium otherwise), while if costs are quadratic or when the market is circular, catalog competition harms the firms. This might explain why on some markets firms mostly communicate on the characteristics of their products but do not post prices.

Moreover, in the circular version of the model, the firms use opposite strategies: one chooses a high volume strategy, and the other chooses a high price strategy.

There exist many markets where the choice of product is flexible and where consumers get information only through advertisement and catalog, and as such activities are costly, firms do not continuously undertake them.

This suggests that the dynamics of prices should also be analyzed in terms of sequentiality of announces and heterogeneous reactive abilities in horizontal differentiation models.

A Proofs

A.1 Proof of proposition 2:

We assume that consumers are spread over the market according to some given density function f (x). We consider the case of a finite willingness to pay v. The case of infinite willingness to pay is obtained in a similar (and even simpler) way.

Assume that Π B < Π A in a subgame perfect equilibrium, denote the corresponding strategies by (a * , p * ) and (b * , q * ) and set ∆ = Π A -Π B > 0.

We show that firm B is better off by choosing catalog (a * , p * -ǫ).

Let S be the set of consumers served by firm A in such an equilibrium:

S = {x|v -C(x, a * ) -p * ≥ 0 and C(x, a * ) + p * ≤ C(x, b * ) + q * }
and let µ be its weight: µ = S f (x)dx. By definition of S, Π A = p * µ.

Let S ′ be the set of consumers served by firm A if firm B was not here:

S ′ = {x|v -C(x, a * ) -p * ≥ 0} ⊃ S
For any ǫ > 0, by setting (b, q) = (a * , p * -ǫ), firm B clearly attracts all consumers in S ′ . Since S ⊂ S ′ , he gets then a profit at least equal to

(p -ǫ)µ = Π A -ǫµ. For ǫ < ∆ µ , (p -ǫ)µ > Π A -∆ = Π B .
This contradict the fact that (b * , q * ) is an equilibrium strategy. Thus in any equilibrium one has necessarily Π B ≥ Π A .

A.2 Proof of lemma 1

There remains to consider zones III and III' of figure 2. The follower wants to minimize y because its market share consists of the clients to the right of y. By definition, y is implicitly given by: p

+ C(y -a) = q + C(b -y),
for all b such that the solution is in [0, 1]. If such y does not exist, then the follower has no market share (recall q ≥ p in these zones), and we can discard this strictly dominated case. Differentiating the implicit definition of y with respect to b, we have: Since y is continuous in b, we can conclude that it must be the case that b = y at an optimal accommodation strategy. Finally, q = p + C(b -a) > p by substituting in the definition of y.

C ′ (y -a) + C ′ (b -y) ∂y ∂b = C ′ (b -

A.3 Proof of proposition 3:

From lemma 1, we can restrict attention to y = b for accommodation, so that q = p + C(b -a). By symmetry, we assume without loss of generality that 0 ≤ a ≤ 1 2 , and clearly a ≤ b. We consider a uniform distribution of consumers, i.e. f (x) = 1 for all x.

We study the game backwards: the first three steps analyze the follower's program and the last step concerns the leader's optimization. The accommodation profit of the follower is:

Π Ac B = q(1 -y) = [p + C(b -a)] (1 -b)
The first and second order conditions are:

(F OC) p + C(b -a) = (1 -b)C ′ (b -a) (SOC) (1 -b)C ′′ (b -a) -2C ′ (b -a) ≤ 0
Step 1: Uniqueness of the interior solution in case of existence.

From condition 1, the SOC is always satisfied. Moreover, the accommodation profit is strictly concave for b = a. Therefore, for any catalog of the leader, there is at most one b that satisfies the first-order condition.

Step 2: Boundedness of p at an interior solution.

Now we need to check that the optimal p is necessarily finite when there is an interior solution (otherwise it could be the case that there does not exist a best strategy of the leader). We know that a best accommodation strategy necessarily satisfies (FOC), thus

p ≤ p + C(b -a) = (1 -b)C ′ (b -a) ≤ C ′ (1)
. Necessarily p ≤ C ′ (1) for accommodation.

Step 3: The exclusion constraint implies the existence of an interior solution.

We want to show that when the leader prevent exclusion, then necessarily there exists some b Ac a strict interior solution to the follower maximization.

By excluding the leader, the follower can get a profit as close as desired to p. So in order to prevent exclusion, the leader must choose a catalog such that the accommodation strategy yields a profit greater than or equal to p: Thus the accommodation profit is (1-λ)p, which is lower than the exclusion profit since exclusion yields as close as desired to p. Therefore, under the exclusion constraint, there exists a strictly interior optimal accommodation strategy. We know that this accommodation strategy is unique from step 2.

Step 4: The Leader's optimization.

We have seen in step 2 that the corresponding domain is bounded since it is The function Π Ac B is a differentiable function of (b, a, p). We have seen that under the exclusion constraint there exists a unique interior solution for the maximum problem for all (a, p) thus b Ac (a, p) is continuous. Thus Π Ac A is a continuous function. The domain is non empty: consider for example a = 0 and 0 < p < C( 1 2 ), then the follower prefers b = 1 2 to exclusion because p < 1 2 (p + C( 1 2 )) = Π Ac B . This shows that on this domain the leader obtains strictly positive profits. The constraints are not strict and the domain is bounded thus it is a non-empty compact. This ensures that the leader's program has a solution. Therefore, there exists a subgame perfect equilibrium to the sequential catalog game with a strictly positive profit for both firms.

A.4 Proof of proposition 5:

The proof follows the lines of proposition 3, although condition 1 does not apply. The indifferent consumer y is such that p + t(y -a) 2 = q + t(y -b) 2 .

There is a unique solution such that y ∈ [a, b]. Moreover, we know that the best reply of the follower must be such that y = b, thus q = p + t(b -a) 2 .

We first characterize the best accommodation strategy. The follower's accommodation program is:

(P Ac ) max q,b q(1 -b) = max b∈[a,1] p + t(b -a) 2 (1 -b)
When the solution of (P Ac ) is not interior, this means that the follower uses an exclusion strategy. Indeed, a corner solution is either b = 1 but then this would mean zero profit for the follower, and it is therefore impossible, or b = a and q = p, but this is dominated for the follower by exclusion, with b = a and q = p -ǫ for ǫ smaller than λ. Thus the leader should act such that the solution of (P Ac ) is interior to make a profit.

First order condition implies: -3b 2 + b(4a + 2) -p/t -a 2 -2a = 0. The discriminant is ∆ = 4 (a -1) 2 -3p/t , so the leader must use a strategy

  compare different timing and underlines the fact firms should play simultaneously in locations and sequentially in prices in an unconstrained Hotelling model with quadratic costs.

Figure 1 :

 1 Figure 1: Sequential setting

Lemma 1 Figure 2 :

 12 Figure 2: Strategies for the follower

  2

  be interior under the exclusion constraint Π Ex B ≤ Π Ac B since Π Ac B (b = 1) = 0 and Π Ac B (b = a) < Π Ex B . Substituting the interior solution, we obtain Π Ac B (a, p) = (1-a+p/t) 2 4 and Π Ac A (a, p) = 1+a-p/t 2 p/t. Firm A seeks to maximize Π Ac A (a, p) under the constraint Π Ac B (a, p) = (1-a+p/t) 2 4 ≥ p.

( d 'Figure 3 :

 d3 Figure 3: Equilibrium with linear costs for t = 1

  specialized in a niche (as in d'Aspremont et al.(1979)) such as a continent destination or a category ranking of packages. Internet changed the market positioning strategies of firms because the choice of product is more flexible on internet. Indeed software routines can recombine a package bundling hotels and flights almost instantly, and information technologies have at the same time provided access to large databases of offers that companies use dynamically to redesign products. Therefore firms seem now to compete in a way comparable to our model. This may explain that differentiation has been reduced and competition is fiercer, in the spirit of our results.

Figure 5 :

 5 Figure 5: Timing comparison

.

  In zone III, one has a ≤ y ≤ b. Thus C ′ (y -a) + C ′ (b -y) ≥ 0 and C ′ (b -y) ≥ 0, therefore y is increasing in b, and the follower wants to set b as small as possible. In zone III', one has a ≤ b ≤ y, thus y -a ≥ y -b. Then by convexity of C, C ′ (y -a) + C ′ (b -y) ≥ 0. Since b -y ≤ 0, C ′ (b -y) ≤ 0, and overall y is thus decreasing in b. Therefore in zone III' the follower wants to set b as large as possible.

[

  p + C(b -a)] (1 -b) ≥ p for at least one b along the accommodation curve defined in lemma 2. If there does not exist an interior solution then the best accommodation strategy is such that either b is equal to 1 which yields to zero profit or b is equal to a.

  included in [0, 1] × [0, C ′ (1)]. Step 3 tells us that, on this domain and under the exclusion constraint, an interior accommodation strategy always exists and is uniquely defined as a function of (a, p). The program can thus be stated as: max a,pΠ Ac A = pb Ac (a, p)under the constraints:b Ac (a, p) = Arg max a≤b≤1 {Π Ac B (b, a, p) = (1 -b)(p + C(b -a))} (1 -b Ac (a, p))(p + C(b Ac (a, p) -a)) ≥ p 0 ≤ a ≤ 1 2 0 ≤ p ≤ C ′ (1)

Note that if the incumbent is a chain store, this may be true at the local level, but overall, the chain store probably obtains a higher total profit (the sum over local retailers or franchisees) than a single local competitor.

Note that in[START_REF] Hotelling | Stability in competition[END_REF], b is the distance of product B to 1.

The results of[START_REF] Economides | On Nash Equilibrium existence and optimality in Oligopolistic Competition in Prices and Varieties[END_REF] apply to the present setting, and we only sketch here the intuitions.

In a related paper, we studied this point in details, see[START_REF] Fleckinger | Horizontal Differentiation and Price Competition with Sequential Entry[END_REF].

For example, when the distribution boils down to a single atom, Bertand competition for this atom obtains, and there is a continuum of equilibria in which the leader makes zero profit.

For a full analysis of first versus second mover advantage in the case of unidimensional strategies, see[START_REF] Gal-Or | First Mover and Second Mover Advantages[END_REF].

[START_REF] Gelman | Judo Economics: Capacity Limitation and Coupon Competition[END_REF] note p.319 that the judo economics story could also hold with differentiated products: choosing a specialty product is akin to choosing a low capacity according to them. We give a precise formal meaning to their claim.

We do not consider unlikely cases in which firms choose their prices before the locations.

such that p ≤ t (a-1) 2 3 in order not to be excluded by the follower.

We have two solutions in b, and the second order condition is 2(2a + 1 -3b): The only maximum is b * (a, p) = 1 3 1 + 2a + (a -1) 2 -3p/t . However, the leader must still guarantee a profit of p to the follower in order to prevent an exclusion strategy.

The leader maximizes: max

under the constraint that the follower accommodates:

Let us ignore the other constraints (0 ≤ a ≤ 1 2 and a ≤ b ≤ 1) and check them at the end. Maximizing first in a and then in p yields: Finally, we obtain p * = t 4 and a * = 0 and the optimal strategy of the follower is: (b * , q * ) = 1 2 , t 2 , this solution respects all the constraints. Tabuchi, T., Thisse, J.F., 1995. Asymmetric equilibria in spatial competition. International Journal of Industrial Organization 13, 213-227. 4


F Chatelain

A Costard

O J J Michel

A BAYESIAN MARKED POINT PROCESS FOR OBJECT DETECTION. APPLICATION TO MUSE HYPERSPECTRAL DATA

Keywords: Hierarchical Bayesian Models, Point Processes, Hyperspectral Data, Galaxy Detection

Marked point processes have received a great attention in the recent years, for their ability to extract objects in large data sets as those obtained in e.g. biological studies or hyperspectral remote sensing frameworks. This paper focuses on an original Bayesian point process estimation for the detection of galaxies from the hyperspectral MUSE data 'cube'. It is shown that this approach allows to obtain a a synthetic representation of the detection problem and circumvent the computational complexity inherent to high dimensional pixel based approaches. The reversible jump Monte Carlo Markov Chain (RJ-MCMC) implemented to sample the parameters is detailed, and the results obtained on benchmark data mimicking the real instrument are provided.

INTRODUCTION, DATA DESCRIPTION

The overwhelming growth in size and complexity of experimental data from research fields such as e.g. biology, remote sensing, astrophysics requires to develop new approaches to warrant optimal scientific exploitation of the instrument. This paper focuses on the automated object detection task, formulated in a Bayesian marked point process detection framework. A marked point process is a stochastic process whose realizations are random configurations of objects. Thus an object is modeled as a point, associated with some marks that accounts for geometrical or spectral features. These processes yield a natural sparse representation for a configuration of a random number of objects in an image. The number of parameters actually depends on the effective number of objects in the current configuration, allowing one to obtain a synthetic representation of the problem. It appears to be a useful tool to circumvent the computational complexity due to high dimensional space problems inherent to pixel-wise approaches. These processes have received a great attention during the last several years for object extraction from biological ( [START_REF] Rue | Bayesian object recognition with Baddeley's delta loss[END_REF]) or remotely sensed images (see e.g. [START_REF] Descamps | Automatic flamingo detection using a multiple birth and death process[END_REF], [START_REF] Ortner | A marked point process of rectangles and segments for automatic analysis of digital elevation models[END_REF]) A known drawback of previously developed approaches resides in their high sensitivity to some hyper-parameters that controls the prior and data energy of the process. Consequently, these latter parameters are customary tuned by hand, to some data specific value. The Bayesian approach proposed in this paper overcomes this difficulty by introducing non-informative priors, possibly with

The first author would like to thank Xavier Descombes and Josiane Zerubia for fruitful discussion about point processes during his post-doctoral fellowship at INRIA Sophia Antipolis. This work was supported by ANR DAHLIA, and P.P.F. ISSO, University of Nice Sophia Antipolis, France. a second level of hierarchy within the Bayesian model. It results in a so-called hierarchical Bayesian model. Finally, the posterior distribution is sampled thanks to a Reversible Jump Markov chain Monte Carlo (RJMCMC) method. This hierarchical Bayesian model leads to a general and robust algorithm where the hyperparameters are estimated thanks to a full data-driven criterion [START_REF] Dobigeon | Bayesian orthogonal component analysis for sparse representation[END_REF]. This is, to our knowledge, the first time that a fully Bayesian marked point process is developed for object recognition. Moreover, for the purpose of galaxy detection in the MUSE data cube, the proposed model is designed to handle the large number of monochromatic images (data cube slices). MUSE (for Multi unit Spectroscopic Explorer, see [START_REF]Official website of the MUSE project[END_REF]) project aims at providing the astronomers with a new generation of optical instrument, capable of simultaneously imaging the sky (in 2D) and measuring the optical spectra of the light received at a given position on the sky. MUSE is planed to be installed on the VLT and to be operational in 2012, and its performances are expected to allow observation of far galaxies up to 100 times fainter than those presently detectable. MUSE will deliver 3D 'data-cube' made of a stack of images recorded at 3578 different wavelengths over the range 465-930 nm. Each 'monochromatic' image represents a field of view of 60X60 arcsec, recorded with a spatial sampling of 0.2 arcsec 0.2 arcsec. Each record results in a data cube of size 1570 MB encoding 3578 images of 300X300 pixels, possibly containing thousands of objects (galaxies) existing over different subsets of wavelengths! In the literature [START_REF] Simard | The deep groth strip survey. ii. hubble space telescope structural parameters of galaxies in the groth strip[END_REF][START_REF] Peng | Detailed Structural Decomposition of Galaxy Images[END_REF], the galaxy morphology is classically assumed to result from elliptical components. Following this assumption, in this paper, a galaxy is approximated by an elliptic object. The marks correspond to the ellipse parameters and to the intensity vector of the galaxy. Galaxy detection is then performed jointly on the different slices of the hyperspectral cube. Throughout this study, we analyze simulation data mimicking the expected hyperspectral record (in terms of SNR, density of objects, spectral content,...), as provided by CRAL [START_REF]Official website of the CRAL laboratory[END_REF] within the ANR Dahlia [START_REF]Official website of the Dahlia project[END_REF] collaborative project. This paper is organized as follows. The Bayesian model is described in section 2. The RJMCMC sampler is derived in section 3. Simulations conducted on the MUSE cube are reported in section 4. Conclusions are drawn in section 5.

BAYESIAN MODEL

Let y be a M × 1 vector corresponding to a vectorized image (at a given wavelength λ) counting M pixels, while u denotes a configuration of objects. Finally, n(u) denotes the number of objects in the configuration u.

Monochromatic Observation model

y = s + 1m + n,
where m > 0 is the mean of the astrophysical background intensity, 1 = (1 . . . 1) T is the M × 1 matrix with unit elements, n is a M × 1 centered white noise vector with component variance σ 2 , and s is a M × 1 vector that models the device response to a galaxy configuration u in the observed scene. The signal s is expressed as:

s = n(u) X i=1 wixi,
where xi stands for the the response of imaging device to an ellipse ui with unitary intensity ; wi is the intensity associated with ui. An equivalent matrix formulation yields:

s = X w, where X = ˆx1 . . . x n(u) ˜∈ R M ×n(u) and w = (w1, . . . , w n(u) ) T ∈ R n(u) .

Hyperspectral Likelihood

From the observation and noise models above, under the assumption that different spectral bands or slices yield independent observations, the hyperspectral likelihood is obtained as the product of the monochromatic (at λ) likelihoods

f (y λ |u, m λ , σ 2 λ ) = " 1 2πσ 2 λ « M 2 exp " - Z T λ Z λ 2σ 2 λ « , (1)
where Z λ = y λ -1m λ -X λ w λ , and all parameters but u depend on λ. This stresses on the fact that a configuration is shared by all slices (wavelengths) in a given data cube. In the remainder of the paper, λ will be omitted for sake of simplicity. Although the independence of the observations from different slices may seem oversimplified, this holds for sufficiently separated (in the wavelength domain) slices. Possible spectral correlations will not be tackled in this paper.

Parameter and Hyperparameter Priors

A noninformative improper prior is chosen for the pair of parameters (m, σ 2 ):

p(m, σ 2 ) = 1 σ 2 1 (0,+∞) 2 (m, σ 2 ).
A Gaussian prior is introduced for the intensity vector:

w|u, σ 2 ∼ N " 0, η 2 σ 2 (X T X ) -1 " ,
Note that all these priors satisfy a principle of invariance. Any affine transformation of the measurements y, or of the signal basis X , will transform the posterior distribution, and thus our inferences, in the same way. Moreover, the hyperparameter η 2 represents the signalto-noise ratio (SNR) of the observed scene since it is straightforward to show that

η 2 = E[s T s] E[n T n] .
Finally, a Bayesian hierarchical structure is proposed, by introducing a hyperprior distribution for the hyperparameter η 2 . This hyperparameter is modeled as a conjugate inverse-Gamma distribution IG(α0, α1/2):

p(η 2 ) ∝ " 1 η 2 « α 0 +1 e - " α 1 2η 2 « .
The parameters α0 and α1 are chosen sufficiently small to obtain a vague prior (basically, α0 = α1 = 10 -3 ).

Posterior distribution conditionally on a galaxy configuration u

The resulting posterior distribution, conditionally on a configuration u, is obtained as:

p(w, m, σ 2 , η 2 |u, y) ∝ " 1 2πσ 2 « M +n(u) 2 +1 1 η 2 n(u) 2 × exp - Z T Z + 1 η 2 w T X T X w 2σ 2 ! × exp(-α 1 2η 2 ) (η 2 ) α 0 +1 1 (0,+∞) 2 (m, σ 2 ) (2)
Based on (2), one can see that the conditional posterior distribution of the intensity vector w given m, σ 2 , η 2 , u, y is Gaussian with mean and covariance matrix respectively:

µ = " 1 + 1 η 2 « -1 (X T X ) -1 h X T (y -1m) i , Σ = σ 2 η 2 1 + η 2 " X T X " -1 .
It allows one to integrate out the parameter vector w to obtain the following marginalized posterior distribution:

p(m, σ 2 , η 2 |y, u) ∝ " 1 σ 2 « M 2 +1 " 1 1 + η 2 « n(u) 2 × e - (y-1m) T " I- " η 2 1+η 2 « X(X T X) -1 X T « (y-1m) 2σ 2 × exp(-α 1 2η 2 ) (η 2 ) α 0 +1 1 (0,+∞) 2 (m, σ 2 ).
(3)

Priors on the configuration

The galaxy configuration u = {u1, u2, . . .} is modeled as a marked point process (see [START_REF] Stoyan | Stochastic Geometry and Its Applications[END_REF] for more details about point processes). Each marked point ui corresponds to an ellipse ui = (pi, ai, bi, ωi), where pi is the center, ai, bi denotes respectively the first and second axis values and ωi is the orientation of the first axis. This process is defined by its density with respect to the measure µ(•) of a unitary Poisson process defined on W × K, where W ⊂ R 2 is the coordinate space of the observed scene and K = [rmin, rmax]×[rmin, rmax]× [0, π[ is the mark space. In order to regularize the solution and to avoid multiple detections, a hard core penalization is introduced to prevent overlapping. Let r(ui, uj) = max

P s∈u i ∩u j xi,s P s∈u i xi,s , P s∈u i ∩u j xi,s P s∈u j xi,s ! ∈ [0, 1]
be the overlapping ratio between the responses of two objects ui and uj, where xi,s stands for the object response xi evaluated at location s. It yields the following hard core density

h(u) = ( 0 if it exists i = j such that r(ui, uj) > t, 1 otherwise,
where the threshold t is set to the deterministic value t = 1/2. Finally, the number of galaxy n(u) is governed a priori by a Poisson distribution with mean β. A second level of hierarchy is introduced with the choice of a Jeffreys noninformative hyperprior on this hyperparameter: p(β) = 1/ √ β. The resulting prior on the configuration is obtained as:

p(u, β) ∝ β n(u)-1/2 e -β h(u)µ(u).
Integrating out β, it yields the following marginalized prior:

p(u) ∝ Γ (n(u) + 1/2) h(u)µ(u), (4)
Γ(a) = R +∞ 0 t a-1 e -t dt being the classical Gamma function.

Posterior distribution

The joint posterior distribution on both the configuration u and the parameter vector θ = (m, σ 2 , η) expresses now as:

p(u, θ|y) ∝ p(θ|u, y)p(u) (5)
where p(θ|u, y) and p(u) are defined respectively in ( 3) and (4).

RJMCMC SAMPLER

Several kinds of movements are considered in the RJMCMC dynamics [START_REF] Peter | Reversible Jump Markov chain Monte Carlo computation and Bayesian model determination[END_REF]. Simple moves just update some model parameters, or object features (location, marks). More complex moves allow for dimensional changes in the model, such that the birth of the death of an object in the current configuration. The different moves defined below are finally randomly selected in the RJMCMC dynamic.

Sampling the parameters

One can deduce from (3) the conditional posterior distributions for each parameter given the other ones. These posterior distributions can be used as proposals, leading to an acceptance ratio equal to 1. These moves reduce to classical Gibbs moves. Note that the different moves to update the parameters are performed sequentially in the RJMCMC dynamic. Let ν = M -1, and

W = I - " η 2 1 + η 2 « X (X T X ) -1 X T , δ 2 = (1 T W 1) -1 , m = δ 2 1 T W y, s 2 = ν -1 δ 2 h y T W y -δ 2 (1 T W y) 2 i . (6)
Based on ( 3) and ( 6), straightforward computations lead to the following Inverse-Gamma density for the variance parameter:

σ 2 |(m, λ 2 , y, u) ∼ IG `M 2 , 1 2δ 2 (νs 2 + (m -m) 2 )
´. By integrating out σ 2 , the marginal posterior for m becomes:

p(m|λ 2 , y, u) ∝ ν + " m - m s « 2 ! -ν+1 2 1 (0,+∞) (m).
Thus the marginal posterior for m is a positive truncated Student distribution with degree of freedom ν, location parameter m, and scale parameter s: m|(λ 2 , y, u) ∼ T + (ν, m, s).

Finally, sampling from the bivariate posterior for (m, σ 2 ), given λ 2 , y and u, is performed in two-steps:

1. draw m|(λ 2 , y, u) ∼ T + (ν, m, s),

2. draw σ 2 |(m, λ 2 , y, u) ∼ IG( M 2 , 1 2δ 2 [νs 2 + (m -m) 2 ]).

Sampling the RSB parameter η 2

From (3), the posterior of η 2 is

p(η 2 |m, σ 2 , y, u) ∝ " 1 1 + η 2 « n(u) 2 e » - " 1 1+η 2 « β 2σ 2 - e - α 1 2η 2 (η 2 ) α 0 +1 ,
where β = (y -1m) T X (X T X )X T (y -1m). This distribution is not standard and it is not possible to sample according to this distribution directly. However, the generation of η 2 |(m, σ 2 , y, u) can be achieved thanks to a standard Metropolis-Hasting step, by introducing an adequate proposal distribution. It leads to a "so-called" Metropolis-within-Gibbs sampler.

Birth and Death moves

For a given configuration u, pB(u) denotes the probability to select the birth move, while pD(u) = 1 -pB(u) is the probability to choose the reversible move, i.e. the death move (basically, pB(u) = pD(u) = 1/2).

For a birth move, a new object v is drawn independently from the configuration u, according to the reference Poisson process. In the death case, the object ui ∈ u that is proposed to be removed is uniformly selected, thus with probability pS(ui|u) = 1/n(u).

Finally, in the birth case, the proposed configuration is v = u ∪ {v} and the following Metropolis-Hastings-Green ratio is obtained:

r(u, v) = pD(v) pB(u)
p(v, θ|y) p(u, θ|y) pS(v|v).

In the death case, the proposed configuration is v = u\{ui} and the Metropolis-Hastings-Green ratio is:

r(u, v) = pB(v) pD(u) p(v, θ|y) p(u, θ|y) 1 pS(ui|u)
.

Finally, each one of this moves is accepted with probability α = min (1, r(u, v)).

Updating an object of the configuration

This move can be viewed as the composition of a death and a birth move. In fact, an object ui ∈ u is uniformly selected with probability pS(ui|u) = 1/n(u). A new object v is drawn according to the reference Poisson process. We propose to replace the object ui by the new one v leading to the new configuration v = u\{ui} ∪ {v}.

The following Metropolis-Hasting ratio is obtained:

r(u, v) = pS(v|v) pS(ui|u) p(v, θ|y) p(u, θ|y) ,
and the acceptance rate is α = min (1, r(u, v)).

RESULTS

The algorithm was applied to a partially integrated (over λ) the MUSE data cube: instead of considering all 3578 wavelengths, 'thick' slices integrating 8 consecutive wavelengths are considered. This lead to a new data set of size 300 × 300 × 448, where images at some average λ are assumed independent. The intensity shape xi associated with the ellipsis ui is modeled as a bivariate Gaussian density f2D for which ui is a α iso-probability contour defined by R (x,y)∈u i f2D(x, y)dxdy = 1α. Probabilities to select the parameter update PParameter (see section 3.1), the birth/death PB/D (see 3.2), the object update move PObject (see 3.3) are set to arbitrary values. Note that a fusion/split move (not discussed in this paper) was added for the computation whose results are shown below. The number of MCMC iterations that has been done is 1.1 × 10 7 . It takes about 250 minutes on a 2.40GHz (single core) CPU. Figure 1 shows the obtained parameters vs the wavelengths. These parameters are estimated by taking the MMSE estimate over the last 50000 samples provided by the MCMC. The problem which appears around λ = 850nm was already noticed by the astronomers who prepared the data. It is noteworthy that the shape of the variance receives some physical interpretation, as it matches the expected behavior of the spectrum when water is present in the atmosphere. This was not given as a prior. A projection (integrated over λ) of the detected galaxy configuration is shown on figure 2.

CONCLUSIONS

A fully Bayesian marked point process has been studied to automatically extract the galaxies in the MUSE data cube. Future works include the definition of new moves in the RJMCMC dynamic in order to improve the mixing properties of the Markov Chain. The Fig. 2. Projection of the 384 detected galaxies (in negative grayscale colors) and contours (in red) extension to more complex intensity shape models, taking into account the point spread function of the imaging device must be investigated. Some improvements are also necessary in order to account for very large dynamical variations of the intensity detected for different classes of astrophysical objects.

3. 1 . 1 .

 11 Sampling the background parameters (m, σ 2 )

Fig. 1 .

 1 Fig. 1. Estimated background mean m (a), logarithm of the estimated background variance σ 2 (b), estimated SNR η (c) as functions of the central wavelength (in nm) for each slice of the data cube. (d): log-posterior density at the current parameter values vs the MCMC iteration.