Sylvain Durand
email: sylvain.durand@parisdescartes.fr

Yves-Michel Frapart
email: yves.frapart@parisdescartes.fr

Maud Kerebel
email: maud.kerebel@parisdescartes.fr

Electron paramagnetic resonance image reconstruction with total variation and curvelets regularization

Spatial electron paramagnetic resonance imaging (EPRI) is a recent method to localize and characterize free radicals in vivo or in vitro, leading to applications in material and biomedical sciences. To improve the quality of the reconstruction obtained by EPRI, a variational method is proposed to inverse the image formation model. It is based on a leastsquare data-fidelity term and the total variation and Besov seminorm for the regularization term. To fully comprehend the Besov seminorm, an implementation using the curvelet transform and the L 1 norm enforcing the sparsity is proposed. It allows our model to reconstruct both image where acquisition information are missing and image with details in textured areas, thus opening possibilities to reduce acquisition times. To implement the minimization problem using the algorithm developed by Chambolle and Pock, a thorough analysis of the direct model is undertaken and the latter is inverted while avoiding the use of filtered backprojection (FBP) and of non-uniform Fourier transform. Numerical experiments are carried out on simulated data, where the proposed model outperforms both visually and quantitatively the classical model using deconvolution and FBP. Improved reconstructions on real data, acquired on an irradiated distal phalanx, were successfully obtained.

Introduction

Electron paramagnetic resonance (EPR) is a spectroscopy method that characterizes and quantifies [START_REF] Weil | Electron Paramagnetic Resonance: Elementary Theory and Practical Applications[END_REF] molecules with paramagnetic properties, such as free radicals or transition metals. Fields of application are various and englobe, among others, pharmacy [START_REF] Mader | The application of EPR spectroscopy in the field of pharmacy[END_REF], chemistry [START_REF] Hirata | Free radicals in living systems: In vivo detection of bioradicals with EPR spectroscopy[END_REF], medicine [START_REF] Khan | Clinical applications of in vivo EPR: Rationale and initial results[END_REF] or dermatology [START_REF] Plonka | Electron paramagnetic resonance as a unique tool for skin and hair research[END_REF].

Electron paramagnetic resonance imaging (EPRI) is a technology that maps the spatial distribution of a paramagnetic species [START_REF] Frapart | In vivo electron paramagnetic resonance and imaging in biomedical science[END_REF], making it part of the molecular imaging technologies, by opposition and complementarity to anatomic and functional imaging. Fields of application include oximetry [START_REF] Velan | Electron paramagnetic resonance oxygen mapping (EPROM): direct visualization of oxygen concentration in tissue[END_REF], cardiology [START_REF] Kuppusamy | Cardiac applications of EPR imaging[END_REF], drug release [START_REF] Blank | Electron spin resonance microscopy applied to the study of controlled drug release[END_REF], geology [START_REF] Binet | Potential of EPR imaging to detect traces of primitive life in sedimentary rocks[END_REF], redox [START_REF] Fujii | Mapping of redox status in a brain-disease mouse model by three-dimensional EPR imaging[END_REF] or batteries [START_REF] Sathiya | Electron paramagnetic resonance imaging for real-time monitoring of Li-ion batteries[END_REF]. Similarly to EPR, the sample is submitted to a specific magnetic field that triggers the resonance phenomenon of the electronic spins of the paramagnetic molecule. In order to retrieve the spatial repartition of the latter, a magnetic field gradient is added. Spectroscopy acquisitions are conducted for each orientation of the gradient, yielding a variety of spectrum. The set of raw data obtained by this acquisition process is called a sinogram, and the object of this paper is to retrieve from the sinogram the image featuring the spatial repartition of the paramagnetic species.

The mathematical modelisation of the process turning an image u into a sinogram S is the convolution

S = Ru * h (1)
Quantity h stands for the spectrum of the paramagnetic species and Ru for the Radon transform of u, first introduced a hundred years ago by Johann Radon [START_REF] Radon | Über die Bestimmung von Funktionen durch ihre Integralwerte längsgewisser Mannigfaltigkeiten[END_REF].

Inverting the Radon transform is an ill-posed problem [START_REF]The Mathematics of Computerized Tomography, Ill-posedness and Accuracy[END_REF] and has been the subject of many papers. The classical approach for EPRI reconstruction is the filtered backprojection (FBP), thoroughly explained for computed tomography (CT) [START_REF] Herman | Fundamentals of Computerized Tomography, Image Reconstruction from Projections[END_REF], associated with a deconvolution step. It is the choice made by commercial software for EPRI reconstruction such as Xepr [1]. Even though the FBP is a simple and time-efficient method [START_REF] Hayner | The missing cone problem in computer tomography and a model for interpolation in synthetic aperture radar[END_REF], it admits some major drawbacks [START_REF] Pan | Why do commercial CT scanner still employ traditional, filtered back-projection for image reconstruction? Inverse Problems[END_REF]: FBP requires equally-spaced acquisition angles and is not robust to a low number of acquisition angles. The current development of EPRI goes toward in vivo imaging and clinical applications [START_REF] Bezières | First combined in vivo X-Ray tomography and high-resolution molecular electron paramagnetic resonance (EPR) imaging of the mouse knee joint taking into account the disappearance kinetics of the EPR probe[END_REF] [START_REF] Khan | Clinical applications of in vivo EPR: Rationale and initial results[END_REF]: because acquisition times are significantly long in EPRI, it may be interesting to decrease the sampling, and it is not possible due to the traditional post processing of the data.

Improvements of the FBP have been proposed specifically for EPRI [START_REF] Ahmad | Enhanced resolution for EPR imaging by two-step deblurring[END_REF]. Unfortunately, the issues of backprojection, deconvolution, deblurring and denoising are treated as separate problems, thus leading to approximation errors that add to each other.

To consider the reconstruction as a whole issue instead of the sum of subproblems, variational methods are well suited. Tikhonov and entropy-based regularization have been proposed in [START_REF] Johnson | Maximum entropy reconstruction methods in electron paramagnetic resonance imaging[END_REF] and [START_REF] Tseitlin | Comparison of maximum entropy and filtered back-projection methods to reconstruct rapid-scan EPR images[END_REF], in which the desired image u is retrieved from the data S by solving

arg min u 1 2 ||Cu -S|| 2 2 + λJ(u) (2)
where Cu = Ru * h models the response of an EPRI device and S is the raw data, or sinogram. The second term J(u) corresponds either to the Tikhonov regularization [START_REF] Tseitlin | Regularized optimization (RO) reconstruction for oximetric EPR imaging[END_REF] or to the Shannon entropy [START_REF] Johnson | Maximum entropy reconstruction methods in electron paramagnetic resonance imaging[END_REF], ponderated by a coefficient λ. More recently, [START_REF] Johnson | Compressed sensing of spatial electron paramagnetic resonance imaging[END_REF] has proposed an objective function making most of the sparsity induced by the L 1 norm. It retrieves the image u solution of

arg min u 1 2 ||Cu -S|| 2 2 + λ 1 ||u|| 1 + λ 2 TV(u) (3)
where the second regularization term TV(u) ||∇u|| 1 stands for the total variation of u.

They obtain satisfying numerical results, although their minimization scheme is not formally proven and may lead to false convergence results. Indeed, [START_REF] Johnson | Compressed sensing of spatial electron paramagnetic resonance imaging[END_REF] solves problem (3) by minimizing the objective function with alternatively applying one step of FISTA algorithm to the objective function with λ 2 = 0 and then applying the proximal operator of λ 2 TV.

Another reference [START_REF] Qiao | 3D pulse EPR imaging from sparse-view projections via constrained total variation minimization[END_REF] uses the total variation regularization for EPRI by solving min TV(u) s.t. S = Cu

In this model, the hard constraint on the data consistency does not take the noise created by the device into account. The minimization part is ensured with a gradient descent applied to the objective function smoothed where the L 1 norm is not differentiable. However, huge progress has been made recently in the field of optimization and new algorithms have been developed to solve this kind of variational problems. We propose a new regularization, especially adapted for images with textures, and a thorough analysis of the direct problem which allow us to minimize efficiently the objective function.

Our first objective function is a simpler version of (3) with only the total variation as regularization

arg min u 1 2 ||Cu -S|| 2 2 + λ TV(u) (5)
Because this simple regularization is not subtle enough to deal with images with textures, we propose a second model with another regularization term: a combination of the total variation and a ponderated L 1 norm on the curvelets coefficients arg min

u=u 1 +u 2 ||C(u 1 + u 2 ) -S|| 2 2 + λ 1 TV(u 1 ) + λ 2 ||W u 2 || 1 (6)
where the additional regularization term has the interesting property to retrieve the texture in the reconstructed image.

To efficiently solve problems ( 5) and ( 6), we undergo an analysis of the dataconsistency term which highlights the presence of C * C as a Toeplitz operator [START_REF] Feichtinger | Efficient numerical methods in non-uniform sampling theory[END_REF]. It leads to acceleration properties that allow us to minimize the objective functions with the algorithm recently developed by Chambolle and Pock [START_REF] Chambolle | A first-order primal-dual algorithm for convex problems with applications to imaging[END_REF]. Tests are conducted on both simulated data and real data. Simulations are processed on the Shepp-Logan phantom, both the original phantom and a textured phantom. Real data are acquired on an irradiated distal phalanx. The irradiation map retrieved with EPRI admits details whose existence is confirmed by a comparison with an anatomic map of the bone density performed by a CT-scanner.

Model of spatial EPRI

Direct model

In the case of EPR spectroscopy, the magnetic field B inside the acquisition cavity is spatially uniform and varies over time.

In the case of EPRI, an orientated field gradient µe θ is added inside the cavity, where e θ = (cos θ 1 cos θ 2 , sin θ 1 cos θ 2 , sin θ 2 ) represents the unit vector whose direction is characterized in spherical coordinates by the angles θ = (θ 1 , θ 2 ). At the point x ∈ R 3 , the total magnetic field is the sum of the field B and the field gradient

B + µe θ • x (7)
The addition of a microwave causes the resonance phenomenon of the free electrons. Due to the quantization of energy levels, the resonance property happens where the total magnetic field is equal to the resonance magnetic field B res . However, it is impossible to create a magnetic field with absolute accuracy, and even in that case the precision would theoretically be limited by the uncertainty principle. Therefore the resonance can happen shortly before of after the expected resonance field B res . The repartition of those non-exact resonances has the shape of the distribution g 0 , which is a mixture of gaussian and/or lorentzian distributions and characterizes the paramagnetic species.

To remove some noise from the acquisition data, the energy absorbed by the electronic transitions is measured through the process of amplitude modulation. It implies that the distribution of the inexact resonance has the shape of h 0 the derivative of g 0 . At spatial position x ∈ R 3 , the absorbed energy is directly proportional to the quantity v(x) of the studied paramagnetic species

E abs (x) = v(x)h 0 (B + µe θ • x) (8)
However, the measurement takes place all over the cavity and not only at position x. Therefore the accessible raw data at magnetic field B and field gradient orientation θ is the total amount of absorbed energy, whose sum over every spatial position gives the sinogram

S 0 (B, θ) = R 3 v(x)h 0 (B + µe θ • x)dx (9)
Any spatial position x can be uniquely decomposed into x = ye θ + z with z ∈ H θ where H θ is the hyperplane passing through the point of spherical coordinates (y, θ 1 , θ 2 ) and orthogonal to e θ . It leads to the following expression of the sinogram

S 0 (B, θ) = R H θ v(ye θ + z)dz h 0 (B + µy)dy (10)
because y = e θ • x. Here appears the 3D Radon transform of v at angle θ

R θ v(y) = H θ v(ye θ + z)dz (11)
which yields

S 0 (B, θ) = R R θ v(y)h 0 -µ(-B µ -y) dy (12)
By considering both sinogram and spectrum in spatial position instead of magnetic field by S(x, θ) = S 0 (-µx, θ) and h(x) = h 0 (-µx), equation ( 12) yields the convolution product

S(x, θ) = (R θ v * h)(x) def := R R θ v(y)h(x -y)dy (13)
Similarly to [START_REF] Palmer | Dual-scan acquisition for accelerated continuous-wave EPR oximetry[END_REF], [START_REF] Tseitlin | Regularized optimization (RO) reconstruction for oximetric EPR imaging[END_REF] and [START_REF] Ahmad | Optimization of data acquisition for EPR imaging[END_REF], we assume that the data is corrupted with an additive independant gaussian white noise denoted n, thus yielding the direct model of spatial EPRI acquisition

S(x, θ) = (h * R θ v)(x) + n(x, θ) (14)
However, EPRI acquisitions are not always made in 3D, but also in 1D and in 2D. For 2D acquisition, the angle θ 2 is set to 0, and the field gradient is supported by the vector e θ = (cos θ, sin θ) in the plane θ 2 = 0. The retrieved 2D image u is the projection of the 3D image v on the plane θ 2 = 0. We have indeed

R 2d θ u = R 3d (θ,0) v (15)
where R 2d represents the 2D Radon transform and R 3d represents the 3D Radon transform. All our simulation and experiences are in 2D, that's why the following work considers only the 2D Radon transform, still denoted R θ with θ a single angle, and the 2D image u. The following work can however be generalized to 3D reconstructions.

Data-fidelity term

In order to retrieve the 2D image u from the data S whose relation is described in equation ( 14), we employ the maximum a posteriori method. It requires the definition of a data-difelity term and a regularization term. The L 2 norm is the log-likelihood estimate to remove the independent white gaussian noise from the data, thus yielding the data-fidelity term

F (u) = 1 2 ||h * R θ u -S|| 2 2 . ( 16
)
For simplicity, we will rewrite this equation in terms of the Fourier transform of u which is defined, for u ∈ L 1 (R), by u(ξ) = u(x)e -2iπx•ξ dx. By using the associated Plancherel equality, we obtain

F (u) = 1 2 || h • R θ u -S|| 2 2 . (17)
Projection-slice theorem [START_REF] Deans | The Radon Transform and Some of Its Applications[END_REF] states that, for all ξ ∈ R and θ ∈

[0, π], R θ u(ξ) = u(ξe θ ), (18)
with e θ = (cos θ, sin θ). It means that, for every acquisition angle θ, the 1D Fourier transform of the Radon transform of image u at angle θ is equal to the 2D Fourier transform of image u evaluated on the line carried by e θ . Data-fidelity term [START_REF] Frapart | In vivo electron paramagnetic resonance and imaging in biomedical science[END_REF] has thus become

F (u) = 1 2 ||Au -d|| 2 2 ( 19
)
with d def := S and A being the linear operator in variable u Au(ξ, θ)

def := h(ξ) u(ξe θ ) (20)
The Radon transform is typically defined on the Schwartz space S(R 2 ) and the same applies for A. The spectrum h that is used in its definition is however the derivative of a function g that is a finite linear combination of Gaussian and Lorentzian functions. The Fourier transforms h and g have therefore fast decay. Contrary to the Radon transform alone, the mapping A can therefore be easily extended onto L 2 (R 2 ) by continuity, thus allowing to formally define its adjoint.

Proposition 1

The operator A defined on S(R 2 ) admits a unique continuous extension from

L 2 (R 2 ) to L 2 (R × [0, π]).
Proof. Observe that, since h and g are bounded, there exists c > 0 such that for every

ξ ∈ R | h(ξ)| 2 = 4π 2 ξ 2 | g(ξ)| 2 min(4π 2 ξ 2 g 2 ∞ , h 2 ∞ ) c|ξ|.
Hence, for every

u ∈ S(R n ) π 0 R |Au(ξ, θ)| 2 dξdθ = π 0 R | h(ξ) u(ξe θ )| 2 dξdθ c π 0 R |ξ| | u(ξe θ )| 2 dξdθ = c R 2 | u(ω)| 2 dω = c R 2 |u(x)| 2 dx
The conclusion follows from the density of

S(R 2 ) in L 2 (R 2 ).
The following proposition gives a simple way to compute the operator A * A, where A * stands for the adjoint operator of A. This expression appears in the minimization algorithms proposed thereafter.

Proposition 2 For every u ∈ L 2 (R 2 ) A * Au = k * u ( 21
)
where the point spread function k is defined by

k(x) def := π 0 R | h(ξ)| 2 e 2iπξx•e θ dξdθ, ∀x ∈ R 2 (22)
Proof. Let us first compute the adjoint operator

A * : L 2 (R × [0, π]) → L 2 (R 2
) that is well defined thanks to the continuous extension of proposition 1. For all

u ∈ L 1 (R 2 ) ∩ L 2 (R 2 ) and v ∈ L 2 (R 2 ), we have R 2 π 0 R |u(x) h(ξ) v(ξ, θ)| dξdθdx ≤ u L 1 (R 2 ) √ π h L 2 (R) v L 2 (R×[0,π]) < ∞
We can therefore apply Fubini's theorem in order to show that

Au, v = π 0 R Au(ξ, θ) v(ξ, θ) dξdθ = π 0 R h(ξ) R 2 u(x)e -2iπξx•e θ dx v(ξ, θ)dξdθ = R 2 u(x) π 0 R h(ξ) v(ξ, θ) e -2iπξx•e θ dξdθ dx
where v(ξ, θ) denotes the complex conjugate of v(ξ, θ). Identifying the latter with

u, A * v , for all u ∈ L 1 (R 2 )∩L 2 (R 2 ) which is dense in L 2 (R 2 )
, yields the expression of the adjoint operator

A * v(x) = π 0 R h(ξ) v(ξ, θ) e 2iπξx•e θ dξdθ (23)
and allows to compute A * Au. We have indeed, for all

u ∈ L 1 (R 2 ) ∩ L 2 (R 2 ) (A * Au)(x) = π 0 R h(ξ) Au(ξ, θ) e 2iπξx•e θ dξdθ = π 0 R h(ξ) h(ξ) R 2 u(y)e -2iπξy•e θ dy e 2iπξx•e θ dξdθ = R 2 u(y) π 0 R | h(ξ)| 2 e 2iπξ(x-y)•e θ dξdθ dy = R 2 u(y)k(x -y) dy = (u * k)(x)
where the third equality is a consequence of Fubini's theorem again, since

R 2 π 0 R |u(y) h(ξ) 2 | dξdθdy ≤ π u L 1 (R 2 ) h L 2 (R) < ∞
It remains to show that ( 21) is still satisfied when u ∈ L 2 (R 2 ). Observe that | h| is even because h is real valued. We infer that

k(x) = π -π ∞ 0 | h(ξ)| 2 e 2iπξx•e θ dξdθ = π -π ∞ 0 ξ| g(ξ) h(ξ)|e 2iπξx•e θ dξdθ = R 2 | g(|ω|) h(|ω|)|e 2iπx•ω dω Since ω → | g(|ω|) h(|ω|)| belongs to L 2 (R 2
), its inverse Fourier transform k belongs to L 2 (R 2 ) as well, and the mapping u → k * u can be extended into a continuous operator from L 2 (R 2 ) to L ∞ (R 2 ). On the other hand, A * A is continuous from L 2 (R 2 ) to itself, and we can conclude the proof by a density argument.

TV(u) = sup R 2 u div φ | φ ∈ C ∞ c , |φ(x)| 1 (24)
whose interest regarding computer vision has first been discussed in [START_REF] Rudin | Nonlinear total variation based noise removal algorithms[END_REF]. The expression for a discrete or W 1,1 image u is

TV(u) = |∇u| = ||∇u|| 1 (25)
An accurate inversion of equation ( 14) is obtained by solving

ū = arg min u F (u) + λ TV(u) ( 26
)
where λ is a trade-off parameter to balance the effect of the total variation regularization with the data-fidelity term F , and allows to choose the remaining level of noise.

The main advantages of this regularization choice are the sharpness of the reconstructed images and its ability to reconstruct images from incomplete data.

It also tends to recreate images that are piecewise constant, due to the enforced sparsity of the L 1 norm applied to the gradient. Therefore, the total variation regularization is well adapted to images without oscillations.

However, textured areas suffer from the use of total variation regularization and tends to become homogeneous with this reconstruction. This is why another prior is presented along with it.

Total variation and curvelets

The second regularization model harnesses the idea of [START_REF] Meyer | Wavelets and operators[END_REF] to decompose an image u into its cartoon part u 1 and its textured part u 2 . The reconstructed image u is then the sum of the two images u 1 and u 2

u = u 1 + u 2 ( 27
)
While it is generally acknowledged that the total variation allows to efficiently model cartoon images, several criteria, nearing more or less the dual norm of the total variation, have been proposed in order to characterize the texture component, or more precisely the oscillating component. Among the most notable ones, we can quote div(BM O) [START_REF] Meyer | Wavelets and operators[END_REF], div(L ∞ ) [START_REF] Meyer | Wavelets and operators[END_REF], the homogeneous Besov norm Ḃ-1 ∞,∞ [START_REF] Meyer | Wavelets and operators[END_REF] and the homogeneous Sobolev norm Ḣ-1 [START_REF] Vese | Modeling textures with total variation minimization and oscillating patterns in image processing[END_REF].

Contrarily to [START_REF] Vese | Modeling textures with total variation minimization and oscillating patterns in image processing[END_REF] or [START_REF] Aujol | Dual norms and image decomposition models[END_REF], our objective is not to obtain the best decomposition into a cartoon and a texture part, but to retrieve the sum u 1 + u 2 from the noisy data. We propose therefore to use a criterion that allows to separate efficiently the texture and the noise and that is based, to this end, on a sparse representation of the texture. To our knowledge, the closest model to ours was introduced by Frühauf, Pontow and Scherzer [START_REF] Frühauf | Texture enhancing based on variational image decomposition[END_REF] who characterized the textured part u 2 with the L 1 norm of its weighted curvelet transform W u 2 = (w j u 2 , φ j,k ) j,k , the index j being the scale parameter and k standing for the translation and direction parameters of the curvelet φ j,k . However, unlike [START_REF] Frühauf | Texture enhancing based on variational image decomposition[END_REF], we set w j = 2 -n+2 2 j where n is the dimension of the space on which the image is defined. Hence, for 2D images, we have

W u 2 1 = j,k 4 -j | u 2 , φ j,k | (28)
If the curvelet frame is changed into a wavelet basis, it is equivalent to the homogeneous Besov Ḃ-1 1,1 norm [START_REF] Meyer | Wavelets and operators[END_REF] that is close to the Sobolev Ẇ -1 1 norm. The reconstructed image ū is thus solution of the problem ū = arg min

u=u 1 +u 2 (F (u 1 + u 2 ) + λ 1 TV(u 1 ) + λ 2 W u 2 1 ) (29)
where λ 1 and λ 2 are parameters associated with each prior. They can be adapted to the level of cartoon or textured effect desired in the image ū.

Minimization algorithm

First-order primal-dual algorithm

Chambolle and Pock [START_REF] Chambolle | A first-order primal-dual algorithm for convex problems with applications to imaging[END_REF] give an efficient iterative algorithm to minimize problems of the form arg min

u f (u) + g(Ku) (30)
with f and g proper, positive, lower semi-continuous and convex functions, and K a continuous linear operator. The minimization algorithm consists of the steps

Initialization Initialize σ, τ > 0, u 0 ∈ R n , p 0 ∈ R n × R n and set v 0 = u 0 . Iterations For k 0, iterate (a) p k+1 = (I + σ∂g ) -1 (p k + σKv k ) (b) u k+1 = (I + τ ∂f ) -1 (u k -τ K * p k+1 ) (c) v k+1 = 2u k+1 -u k
where I stands for the identity operator, ∂f the subgradient of f and g the conjugate function of g, according to the Legendre-Fenchel duality

g (u) = sup v u, v -g(v) (31)
Step (a) is a gradient ascent in the dual space composed with the proximal operator of σg . Step (b) is a gradient descent in the primal space composed with the proximal operator of τ f . Step (c) is an acceleration step by extrapolating the point to which apply step (a). Under the condition στ ||K|| 2 < 1, this algorithm converges to a minimizer of problem [START_REF] Meyer | Wavelets and operators[END_REF]. For convergence acceleration, those parameters may be modified during the run of the algorithm, mimicking a new minimization problem initialized with the previous u obtained.

By choosing f (u) = F (u), Ku = λ∇u and g(p) = ||p|| 1 with p = (p 1 , p 2 ) the dual variable of ∇u, the primal-dual algorithm converges to a solution of problem [START_REF] Khan | Clinical applications of in vivo EPR: Rationale and initial results[END_REF]. By choosing

f (u 1 , u 2 ) = F (u 1 + u 2 ), K(u 1 , u 2 ) = (λ 1 ∇u 1 , λ 2 W u 2 )
and g(p, w) = ||p|| 1 + ||w|| 1 with p the adjoint variable of ∇u 1 and w the adjoint variable of W u 2 , it converges to a solution of problem [START_REF] Mader | The application of EPR spectroscopy in the field of pharmacy[END_REF].

Note that by choosing K(u) = (λ 1 ∇u, λ 2 u) instead, the algorithm converges to a solution of (3). This method is a more general and flexible method, and furthermore the convergence of the Chambolle and Pock algorithm is formally proven in [START_REF] Chambolle | A first-order primal-dual algorithm for convex problems with applications to imaging[END_REF].

Implementation

2.2.1. TV model For step (a), the computation of the resolvent (I + σ∂g ) -1 for g(p) = ||p|| 1 applied to p = p k + σλ∇v n is detailed in [START_REF] Chambolle | An algorithm for total variation minimization and applications[END_REF]. The computation yields

p k+1 = (I + σ∂G * ) -1 ( p) ⇐⇒ p k+1 i,j = p i,j max(1, |p i,j |) ( 32
)
with |p i,j | = (p 1 [i, j]) 2 + (p 2 [i, j]) 2
Step (b) involves K * (p) = -λ div(p) the adjoint operator of K(u) = λ∇u. The computation of the resolvent (I + τ ∂F ) -1 for the data-fidelity term

F (u) = 1 2 ||Au -d|| 2 2 applied to u = u k + τ λ div p n+1 is the solution of u k+1 = arg min u 1 2 ||u -ũ|| 2 2 + τ F (u) (33)
which is the minimizer of

H(u) = 1 2 ||u -u|| 2 2 + τ 2 ||Au -d|| 2 2 .
Its derivative is

H (u) = (u -u) + τ A * (Au -d) (34)
and the minimizer u k+1 verify Euler-Lagrange equation

H (u k+1 ) = 0 ⇐⇒ (I + τ A * A)u k+1 = u + τ A * d (35)
Assuming we know how to invert I + τ A * A, an immediate solution of (35) follows. As the computation of section 1.2 can be generalized to the discrete case, A is here a Vandermonde hypermatrix, and thus A * A a Toeplitz hypermatrix. Therefore, the operator A * A is a non-periodic convolution with the point spread function k. This property was first used by Kostov for the inversion of the Radon transform alone [START_REF] Kostov | Toeplitz structure in slant-stack inversion[END_REF] and by Feichtinger, Gröchenig and Strohmer for the restoration of irregularly sampled images [START_REF] Feichtinger | Efficient numerical methods in non-uniform sampling theory[END_REF]. To be able to use the Fast Fourier Transform (FFT), we make here the assumption that the non-periodic convolution can be approximated by a periodic convolution. It is however justified by the physical properties of an EPR acquisition: the sinogram on the border of the acquisition area is an almost null signal.

Thanks to the convolution theorem, equation ( 35) is equivalent in the Fourier space to

u k+1 + τ k u k+1 = u + τ A * d ⇐⇒ 1 + τ f u k+1 = u + τ A * g ⇐⇒ u k+1 = u + τ A * d 1 + τ f 2.2.

TV-Curvelets model

To implement the TV-Curvelets model in step (a) of the minimization algorithm, one needs to adapt the result of equation [START_REF] Pan | Why do commercial CT scanner still employ traditional, filtered back-projection for image reconstruction? Inverse Problems[END_REF] for the dual variable (p, w). The computation of the resolvent (p k+1 , w k+1 ) = (I + σ∂g

) -1 ( p, w) consists of projecting ( p, w) = (p k + σλ 1 ∇v k 1 , w k + σλ 2 W v k 2 ) onto the set (p, w) : (p 1 [i, j]) 2 + (p 2 [i, j]) 2 1 and |w r,s,t | 1 (36)
Step (b) involves

K * (p, w) = (-λ 1 div(p), λ 2 W * u 2 ) which is the adjoint operator of K(u) = (λ 1 ∇u 1 , λ 2 W u 2 )
. Thanks to the tight frame property of the curvelet transform described in [START_REF] Candès | Fast discrete curvelet transforms[END_REF], the adjoint of W is equal to the inverse curvelet transform W -1 , thus yielding K * (p, w) = (-λ 1 div(p), λ 2 W -1 u 2 ).

The computation of the resolvent (I + τ ∂F ) -1 is similar to the resolution of equation [START_REF] Plonka | Electron paramagnetic resonance as a unique tool for skin and hair research[END_REF], except that the variable u is now equal to (u 1 , u 2 ) instead. In this case, the Euler-Lagrange equation is

I + τ A * A τ A * A τ A * A I + τ A * A u 1 u 2 = u 1 + τ A * d u 2 + τ A * d (37)
with u = u k -τ K * (p k+1 , w k+1 ). Again, by considering operator A * A as a convolution with the point spread function k and computing it in the Fourier space, the resolution of system 37 yields

u 1 = (1 + τ k) v 1 -τ k v 2 1 + 2τ k and u 2 = (1 + τ k) v 2 -τ k v 1 1 + 2τ k (38) with v 1 v 2 = u 1 + τ A * d u 2 + τ A * d (39)

Results and discussion

Numerical simulations are processed with Matlab [START_REF]MATLAB and Image Processing Toolbox Release 2015a[END_REF]. Discrete Fourier transforms are computed using FFTW [START_REF] Frigo | The design and implementation of FFTW3[END_REF]. When computation is required on a polar raster, we use the NUFFT [START_REF] Greengard | Accelerating the Nonuniform Fast Fourier Transform[END_REF]. The curvelet part requires the package described in [START_REF] Candès | Fast discrete curvelet transforms[END_REF]. Real data are loaded using Easyspin [START_REF] Stoll | Easyspin, a comprehensive software package for spectral simulation and analysis in EPR[END_REF].

3.1. Simulated data 3.1.1. Robustness to few angle acquisitions First tests and simulations have been conducted on the Shepp-Logan phantom introduced in [START_REF] Shepp | The Fourier reconstruction of a head section[END_REF]. Figure 1a presents the Shepp-Logan phantom. Figure 1b presents the spectrum h, simulated as a derivative of a gaussian distribution. Figure 1c is the simulated sinogram S, created as the Radon transform of the Shepp-Logan transform, implemented with Matlab's function radon with 100 angles and then convoluted with the spectrum h. Both spectrum and sinogram are then corrupted with a gaussian noise of variance σ = 0.03 proportionally to their amplitudes. Our model with a TV regularization is compared to the reconstruction algorithm implemented in Xepr [1], the commercial software provided by Bruker with EPRI devices. It consists of a deconvolution step followed by a backprojection step.

Figure 2 presents the reconstruction results of the two methods applied to an input data admitting a decreasing number of acquisition angles. The TV reconstructions are computed with the parameter λ = 0.05. The Xepr deconvolution is made with a gaussian window of width 0.1 and threshold 0.5. The backprojection step is made with the parameters Reduce = 512 and ImageSize = 512.

The proposed TV model outperforms the Xepr model in figure 2a both visually: the TV reconstruction is sharp whereas the Xepr image is noisy, and quantitatively: the PSNR of the TV reconstruction compared to the original phantom is 18.6, whereas it is only 16.4 for the Xepr image. As the number of angles decreases significantly in figures 2b and 2c, the Xepr reconstruction admits visible artifacts whereas the TV reconstruction is very robust to the lack of information in between acquisition lines. In figure 2c, the PSNR or the TV model is 15.4 where it is only 10.9 for the Xepr image. This numerical experiment thus validates our first model with the total variation as a regularization for simulated sinograms of piecewise-constant images.

Because the acquisition time of an EPRI device is a strong limitation for in vivo measurements, the ability to correctly reconstruct an image from a small number of acquisition lines is a real improvement for small animals and human EPR imaging. For comparison, reconstructions made with Xepr and TV model are shown in figure 9. The parameter λ was set to 10 12 . As shown by the results obtained on simulated data, the TV model outperforms Xepr, especially regarding the noise level, but does not reconstruct the textures enough to take into account some details of the phalanx. Subjective evaluation yields a clear preference for the TV-Curvelets model.

Furthermore, the small details of the texture retrieved by the TV-Curvelets model in figure 9c are coherent with the ones visible on the CT scan of the phalanx in figure 6b. It means that the effects of the curvelet regularization did not create false texture, but instead retrieved existing textures. Because the CT is an anatomic imaging device whereas the EPRI is a molecular imaging device, we can assume that the lasting effects of the irradiation is correlated with the anatomic map of the bones. Those similarities strongly validate our TV-Curvelets model.

1. 3 .

 3 Regularization term 1.3.1. Total variation A first choice for the regularization term is the total variation

Figure 1 :

 1 Figure 1: Simulated EPRI data on Shepp-Logan phantom

Figure 2 :

 2 Figure 2: Reconstructions with a decreasing number of acquisition angles.

3. 1 . 2 .

 12 Retrieval of simulated texturesThe TV model has been validated on acquisitions with very few angles. This section justifies the need for a second choice of regularization for textured data: the TV-Curvelets model. Henceforth, some texture of various kinds has been added to the phantom of Shepp-Logan. Input simulated data are shown in figure3a, along with their reconstruction with the TV model in figure3bwith parameter λ = 10 -2 .

Figure 3 :

 3 Figure 3: Reconstruction of textured Shepp-Logan with TV model

part u 2 Figure 4 :

 24 Figure 4: Reconstruction with TV-Curvelets model

Figure 5 :

 5 Figure 5: Details of the Shepp-Logan phantom with texture

Figure 6 : 7 .

 67 Figure 6: Irradiated phalanx

Figure 7 :

 7 Figure 7: Acquisition data on the phalanx

  Cartoon part u 1 (c) Textured part u 2

Figure 8 :

 8 Figure 8: Reconstruction with TV-Curvelets model

Figure 9 :

 9 Figure 9: Three reconstruction models and zoom

Figure 10 :

 10 Figure 10: Reconstruction with reduced data

Acknowledgments

The EPR data in this manuscript were obtained using equipment supported jointly by the French National Ministry of Research (PPF IRPE), the "Fondation pour la Recherche Médicale" (FRM DGE20061007745), the CNRS (Department of Chemistry and Life Sciences). Imaging was conducted on the Imaging Life Platform (Plateforme d'Imageries du Vivant) of University Paris Descartes and Research Infrastructure CNRS FR 3443 RENARD. The PhD of Maud Kerebel is founded by a grant double-culture by University Sorbonne Paris Cité. The authors would like to thank Philippe Lévèque and Bernard Gallez (REM, Bruxelles) for having provided the phalanx, Rémy Abergel for the discussions regarding the minimization algorithms, and Sonia Lajnef for her constant support regarding the EPRI acquisitions.


Didier Bert

Stéphane Bersac

Josep Juárez-Ruiz

Zoë Hughes

giant heteromorphic ammonites (Early Cretaceous, northwestern Tethyan margin)

Keywords: Ancyloceratidae, Barremian, Vocontian Basin, France, Biostratigraphy, Evolution

Size reduction and ornamental oscillation within a Barremian lineage of

Introduction

The ammonite family Ancyloceratidae Gill, 1871 (see [START_REF] Klein | Fossilium catalogues, I: Animalia -Lower Cretaceous Ammonites III : Bochianitoidea, Protancyloceratoidea, Ancyloceratoidea, Ptychoceratoidea[END_REF] for the generally accepted generic content) ranges from late Barremian to late Aptian (Early Cretaceous). In this family, the genus Toxancyloceras [START_REF] Delanoy | Toxancyloceras gen. nov. (Ammonoidea, Ancyloceratina) un nouveau genre du Barrémien supérieur[END_REF] is the oldest known representative, as it is present at the base of the lower Barremian (Toxancyloceras vandenheckei Zone) with the species T. vandenheckei [START_REF] Astier | Catalogue déscriptif des Ancyloceras appartenant à l'étage Néocomien d'Escragnolles et des Basses-Alpes[END_REF]. Strong morphological convergences link Toxancyloceras to the more recent Aptian Ancyloceras s. str. through a variety of forms, but whose phyletic relationship is still poorly known (Jaubertites [START_REF] Sarkar | Révision des Ammonites déroulées du Crétacé inférieur du Sud-Est de la France[END_REF], Hoheneggericeras Baudouin et al., 2008, etc.).

The origin of the genus Toxancyloceras was discussed by [START_REF] Delanoy | Toxancyloceras gen. nov. (Ammonoidea, Ancyloceratina) un nouveau genre du Barrémien supérieur[END_REF] and Vermeulen (2005). According to Delanoy (2003, p. 3), it originates in the genus Emericiceras (Emericiceratidae Vermeulen, 2004), and T. vandenheckei [START_REF] Astier | Catalogue déscriptif des Ancyloceras appartenant à l'étage Néocomien d'Escragnolles et des Basses-Alpes[END_REF] is an intermediate species between Emericiceras emerici (Léveillé, 1837) and Gassendiceras alpinum (d 'Orbigny 1850) (Hemihoplitidae Spath, 1924). Vermeulen (2005, p. 159-160) considered Honnoratia Busnardo et al., 2003 (Emericiceratidae) as a direct ancestor of Toxancyloceras.

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4 [START_REF] Bert | Origin of the Tethyan Hemihoplitidae tested with cladistics (Ancyloceratina, Ammonoidea, Early Cretaceous): an immigration event?[END_REF] performed a phylogenetic analysis (Fig. 1) dealing with several taxa (Toxancyloceras, Honnoratia, Emericiceras, Moutoniceras Sarkar, 1955 and Gassendiceras).

The results showed that:

-(1) neither Emericiceras nor Honnoratia can be interpreted as potential ancestors of Toxancyloceras (Bert and Bersac, 2014, fig. 3 and 4, p. 262). This position is reinforced by the stratigraphic hiatus of nearly two ammonite zones between the last Emericiceratidae and the first Toxancyloceras; -(2) Gassendiceras (tested by the older forms G. essaouirae [START_REF] Bert | Origin of the Tethyan Hemihoplitidae tested with cladistics (Ancyloceratina, Ammonoidea, Early Cretaceous): an immigration event?[END_REF] and G. multicostatum [START_REF] Sarkar | Révision des Ammonites déroulées du Crétacé inférieur du Sud-Est de la France[END_REF]] in Bert and Bersac, 2014, fig. 3, p. 261) cannot be interpreted as a potential descendant of Toxancyloceras. Thus Hemihoplitidae and Ancyloceratidae are two independent families; -(3) the early Barremian genus Moutoniceras is a sister taxon of Toxancyloceras. This latter result, as well as its strong stratigraphic and morphological consistency, led these authors to consider Moutoniceras as an authentic representative of the Ancyloceratidae, rather than as a representative of the Heteroceratidae as it was admitted by a majority of experts until recently [START_REF] Klein | Fossilium catalogues, I: Animalia -Lower Cretaceous Ammonites III : Bochianitoidea, Protancyloceratoidea, Ancyloceratoidea, Ptychoceratoidea[END_REF]; see Bert and Bersac, 2014, p 264, for a historical account). Thus, to date, Moutoniceras is the oldest known Ancyloceratidae. A possible origin of the Moutoniceras within the Hauterivian Pseudomoutoniceras [START_REF] Autran | Discussion critique des genres d'ammonites déroulées: Toxoceras d'Orbigny, 1840, Hemibaculites Hyatt, 1900 et Moutoniceras Sarkar[END_REF] was proposed by Vermeulen (2006), but this hypothesis was challenged more recently by the same author (Vermeulen et al., 2010, p. 95) because of the very large stratigraphic gap that exists between these two genera. The review of the literature data (op. cit.) shows that there is a real gap in knowledge regarding the oldest Ancyloceratidae, and especially in their evolutionary modalities, in terms of patterns and processes. [START_REF] Reboulet | Report on the 5th International Meeting of the IUGS Lower Cretaceous Ammonite Working Group, the Kilian Group[END_REF]. The index species T. vandenheckei is thus revised, taking into account the review of its holotype (Astier collection).

Geological setting and sections studied

The fossil material studied in the present work is the result of the systematic bed-by-bed sampling of several stratigraphic sections near the Barremian stratotype (southeastern France). Other ex-situ specimens of Toxancyloceras are from the Lloseta and Biniamar area (Mallorca, Balearic Islands, Spain). These areas both belong to the Cretaceous of the northwestern Tethyan margin (Fig. 2).

The biostratigraphic framework used for the Barremian in this work is the one proposed by the I.U.G.S Lower Cretaceous ammonite working group; the Kilian Group [START_REF] Reboulet | Report on the 5th International Meeting of the IUGS Lower Cretaceous Ammonite Working Group, the Kilian Group[END_REF]-see Bert et al., 2008 for historical account). In the present work we also take into account the biostratigraphic proposals of [START_REF] Bert | Nouveaux biohorizons et propositions pour le découpage biozonal ammonitique du Barrémien supérieur du Sud-Est de la France[END_REF], [START_REF] Bert | Pseudoshasticrioceras bersaci nov. sp. (Ammonoidea, Gassendiceratinae), and new ammonite biohorizon for the Upper Barremian of southeastern France[END_REF]), and Bert et al. (2010, 2011), which allows a high stratigraphic precision for the area studied (Fig. 3).

The Vocontian Basin

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The Southeast France Basin (Fig. 4) is a large subsident intracratonic area, located in the southeast part of the country, between the Massif Central, which borders the West, the Alps that limit the East, the Jura in the North and the Mediterranean Sea in the South. Its Mesozoic sedimentation phase is between the Triassic (late Hercynian orogeny) and the Cretaceous (start of the Alpine orogeny), the Pyrenean-Provencal movements of the Late Cretaceous mark its demise as a basin.

Known as the Vocontian Basin [START_REF] Paquier | Recherches géologiques dans le Diois et les Baronnies orientales[END_REF] the Southeast basin began its reduction during the Early Cretaceous. Its southern part, in the Barremian stratotype area (near Angles, Alpes de Haute-Provence, southeastern France -red star in Fig. 4), is characterised by pelagic sedimentation with an alternation of marlstones and limestones in decimetric to metric beds. This particular area is very conducive to studies thanks to the abundance and the quality of its outcrops. The continuous sedimentation and good paleontological record reveal the ammonite succession in considerable detail. The deep marine conditions of the Vocontian Basin, largely open to the Alpine Tethys, are reflected in the dominance of ammonites.

The Barremian stratotype area belongs to the protected perimeter of the Geological National Nature Reserve of Haute-Provence, managed by the Departmental Council of the Alpes de Haute-Provence. Many field sections were surveyed and studied there for almost two decades, as part of a larger work done by one of us (DB). Some of them have already been published [START_REF] Bert | Pseudoshasticrioceras bersaci nov. sp. (Ammonoidea, Gassendiceratinae), and new ammonite biohorizon for the Upper Barremian of southeastern France[END_REF][START_REF] Bert | Nouveaux biohorizons et propositions pour le découpage biozonal ammonitique du Barrémien supérieur du Sud-Est de la France[END_REF][START_REF] Bert | The Dichotomus Horizon: a new biochronologic unit of the Giraudi Zone of the Upper Barremian of southeastern France, and considerations regarding the genus Imerites Rouchadze (Ammonoidea, Gassendiceratinae)[END_REF][START_REF] Bert | Paleontology, taxonomic revision and variability of some species of the genus Gassendiceras Bert et al., 2006 (Ammonitina, Upper Barremian) from southeastern France[END_REF]. The sections that have provided the majority of the Toxancyloceras and Moutoniceras specimens studied here are A* and G5 (see below).

Field section A* (Fig. 5)

Previous work: the Barremian historical stratotype (section A) has been designated by Busnardo in 1963 at the Symposium on the Lower Cretaceous (Busnardo, 1965b). It is located This area has been previously studied by ammonitologists, particularly by Busnardo (1965a), [START_REF] Delanoy | Biostratigraphie des faunes d'Ammonites à la limite Barrémien-Aptien dans la région d'Angles-Barrême-Castellane. Etude particulière de la Famille des Heteroceratidae (Ancyloceratina, Ammonoidea)[END_REF] and Vermeulen (2005). The complementary section A* was described for the first time by [START_REF] Bert | The Upper Barremian pro parte of the Angles stratotype (SE France)[END_REF].

Description: the trench of the road (section A) allows the entire stratigraphic succession to be seen, from bed No. 1 (upper Hauterivian) to bed No. 232 (lower Aptian), which underlies the Blue Marls Formation. The beds are almost tangential to the road, which allows them to be tracked in good conditions most of the time. Unfortunately, this exposure makes their study quite difficult. In this section, the lower part of the upper Barremian (T. vandenheckei and Gerhardtia sartousiana zones) is poorly exposed. Firstly because of access difficulties to the section that is along a cliff slope. Secondly because the growth faults disrupt the bed successions and make them impossible to study (see more explanations in Bert, 2012, p. 4). Given these difficulties in section A, a new reference section (denoted A*) was surveyed in the immediate lateral continuity of the stratotype (above the road), where there is no growth fault. For the T. vandenheckei Subzone, the bed numbering is the same as for stratotype A.

Field section G5 (Fig. 6)

Previous work: this outcrop corresponds to the site of "Les Lumières" mentioned by Delanoy (2003, p. 3), near Angles (Alpes de Haute-Provence, southeastern France). Although it allowed the study of several complete specimens of Toxancyloceras of the L. Ebbo collection by the latter author, this section was never studied by means of bed-by-bed sampling. Thus, these latter Toxancyloceras have been assigned to the T. vandenheckei Zone by Delanoy (2003, p. 3, 5) Description: the field section was surveyed in a small ravine, where the succession is clearly visible from the top of the upper Barremian (M. moutonianum Zone) to the Gassendiceras alpinum Subzone. The section borders the gigantic excavation dig that provided most of the Toxancyloceras collected in this section and figured by [START_REF] Delanoy | Toxancyloceras gen. nov. (Ammonoidea, Ancyloceratina) un nouveau genre du Barrémien supérieur[END_REF]. This intensive sampling almost completely destroyed the structural surface of bed No. G5/85, making it particularly complicated to study. Thus, given the presence of this gigantic excavation, it is very likely that most of the specimens from "Les Lumières" figured by [START_REF] Delanoy | Toxancyloceras gen. nov. (Ammonoidea, Ancyloceratina) un nouveau genre du Barrémien supérieur[END_REF] are from bed No. G5/85. The close proximity (a few hundred meters) of the field section G5 from the Angles stratotype (section A*) allows a bed-by-bed correlation between these two sections, which are almost identical.

Mallorca

Mallorca is the largest of the Balearic Islands (East of Spain). Geologically, it is closely related to the Betic System in Southeast Iberian Peninsula. In Mallorca, the Cretaceous is mostly known in the West of the Sierra de Tramuntana but also in the centre of the island (called Es Pla) and in Sierra de Llevant. These Sierras originated in the Alpine orogeny [START_REF] Colom | Geología de Mallorca[END_REF]. The Lower Cretaceous is formed by ammonitico rosso facies, but from the upper Berriasian to upper Barremian the marlstones and limestones of Maiolica facies are dominant. The lower Aptian is absent from the stratigraphical record and the upper Aptian is only present in the Southwest area. After this stratigraphical gap, the Lower Cretaceous is represented by marcasite-rich marls and marly limestones up to the upper Albian. The studied area (see below) is located in the middle part of Tramuntana (red star in Fig. 7).

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As they are collected ex-situ (see below), the specimens from Mallorca were not taken into account when considering the biostratigraphic part, but they were used (1) to enlarge the understanding of the morphology of the species studied (the fossils studied are often fragmentary due to the heteromorphic state of the shells, thus there is a real need to increase the number of specimens), and (2) for palaeobiogeographical purpose.

Can Negret Quarry

Previous work: the quarry was studied only recently in a work conducted by one of us (Juárez-Ruiz and Matamales-Andreu, unpublished), who described the Cretaceous ammonite biozonation of the area and part of the representative taxa.

Description: this active quarry, which represents the largest mining enterprise in the Balearic Islands, is located near the town of Lloseta, in the middle of Tramuntana. Can Negret is the richest Cretaceous fossil locality in Mallorca and one of the most abundant in Spain, with more than 220 recognised ammonite species. Unfortunately, the activity of the quarry and the extreme deformation of the series (common in most of the Mallorcan Cretaceous sites) make it almost impossible to carry out accurate bed-by-bed sampling. However many ammonite zones from the upper Valanginian to upper Albian are recognised by their index species and faunal association, including the M. moutonianum and T. vandenheckei zones.

Lloseta-Biniamar area

Previous work: several small and dispersed Cretaceous outcrops in the area between Lloseta and Biniamar were studied or mentioned from the end of the 19th Century [START_REF] Hermite | Études géologiques sur les îles Baléares (Majorque et Minorque)[END_REF][START_REF] Nolan | Structure géologique d'ensemble de l'archipel Baléare[END_REF]. The area near Lloseta is especially famous for having delivered the type specimen of Kotestishvilia sauvageaui [START_REF] Hermite | Études géologiques sur les îles Baléares (Majorque et Minorque)[END_REF]. [START_REF] Fallot | Sur quelques fossiles pyriteux du Gault des Baléares[END_REF][START_REF] Fallot | Etude géologique de la Sierra de Majorque[END_REF] and [START_REF] Fallot | Ammonites nouvelles des Iles Baléares[END_REF] also mentioned some species near this town. Wiedmann (1962aWiedmann ( , 1962b[START_REF] Wiedmann | Unterkreide-Ammoniten von Mallorca 2. Lieferung : Phylloceratina[END_REF] Recently, Juárez-Ruiz and Matamales-Andreu (unpublished) recognised most of the ammonite zones from the upper Valanginian to the upper Barremian, and they found taxa previously unknown in Mallorca, including taxa also unreported in Spain.

Description: this area shows some dispersed outcrops, which are hard to study because they are usually covered by cultivated land. The stratigraphy can be studied only with faunal association in ex situ samples in most of the cases. However, some small sections are located from the Kotetishvilia compressissima to T. vandenheckei zones, which are especially rich in ammonites in most of their beds.

Material

The explanations about the material studied, and an overview of the Astier collection, which contains the holotype of the index species Toxancyloceras vandenheckei, are in the additional online material of the present work.

Systematic paleontology

The detailed systematic part (including the description of Toxancyloceras canuti sp. nov. and the revisions of T. vandenheckei and Moutoniceras eigenheeri) and the photographic figures of the studied material are in the additional online material of the present work.

Results: phylogenetic and evolutionary perspective under patterns and processes

The new data exposed in the present study strengthens the phyletic link [START_REF] Bert | Origin of the Tethyan Hemihoplitidae tested with cladistics (Ancyloceratina, Ammonoidea, Early Cretaceous): an immigration event?[END_REF] proposed, based on a cladistic analysis (Fig. 1) between the genera Moutoniceras and Toxancyloceras. It appears that this link is consistent both stratigraphically and morphologically. The stratigraphic data obtained in the Vocontian Basin (southeastern

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France) shows that the genus Moutoniceras disappeared with the tuberculate species M. eigenheeri at the very end of the early Barremian, stratigraphically just below the First Occurence (FO) of the genus Toxancyloceras (beginning in the upper Barremian). In the Barremian stratotype area of Angles the latter genus appears only one bed higher from the former (Figs. 5,6), so a real continuity exists from a stratigraphic point of view between these two genera. On the other hand, the general morphology of the large tripartite ancyloceratic shell is very close to Moutoniceras and Toxancyloceras. Actually, in the beds around the boundary between the lower and upper Barremian of this area, there are no other large ancyloceratic ammonites aside from these two genera. Under these conditions, the following phylogenetic succession (from the oldest to the youngest) can be established:

M. eigenheeri -> T. canuti sp. nov. -> T. vandenheckei.
Paleontological and stratigraphic data, resulting from analytical systematics, enable us to highlight ontogenetic and evolutionary patterns in this phyletic lineage. This succession is completed for the lower Barremian by older Moutoniceras species. More data about the development of the Moutoniceras species are required to make an accurate cladistic analysis, which requires further research into the Nicklesia pulchella, K. compressissima and M. moutonianum zones (lower Barremian). In any case, the species of the genus Moutoniceras need a revision under intraspecific variability and evolutionary perspectives. In summary, the evolution of Moutoniceras / Toxancyloceras goes through a decrease in adult size from the upper M. moutonianum Zone (Fig. 8), and also by heterochronies in the duration of the different ontogenetic stages (Fig. 9), and finally by modification of the tubercles, which oscillate between appearance and disappearance (Fig. 9):

-The adult size seems relatively stable between M. nodosum (lower part of K. However, in the upper part of the M. moutonianum Zone the adult size of M. marii is reduced significantly (pseudo-dwarfism / progenetic process, in the sense of [START_REF] Dommergues | Les relations ontogenèse-phylogenèse : applications paléontologiques[END_REF]. Although no sufficiently complete specimen of this latter species is known to date, the height of around 600-700 mm is assumed by data collected by one of us [DB]. The tuberculation is present on the outer whorl of the coil (bituberculate pattern) and at the top of the shaft in M. nodosum (peri-ventral tubercles reduced to the state of bulges on the rest of the shaft). It then regresses gradually from M. berti to M. marii. In M. berti, the tuberculation is represented by peri-ventral clavi individualized on the outer whorl of the coil (the only currently known part of the shell in this species) and in very discrete lateral tubercles. In M. moutonianum, there are only mere peri-ventral bulges at the turn of the outer coil and the shaft. Finally, the tuberculation seems to disappear completely on the coil and the shaft in M. -(1) An ornamental oscillation of the appearance (M. nodosum), regression (M. berti, more in M. moutonianum), disappearance (M. marii), then reappearance (M. eigenheeri) and finally increasing amplification (T. canuti sp. nov., more in T. vandenheckei) of the tuberculate pattern over time. This oscillation is adjusted by: (1) the retardation of the ornamentation (the progressive disappearance of the tuberculation by the extension of the non-tuberculate young stage during ontogenesis -stage 1 in Fig. 9), which is a paedomorphosic process; and (2) by the acceleration of the ornamentation (invasion of the tuberculate pattern increasingly early through ontogenesis), which is a peramorphic process.

-(2) A differentiation of the ornamentation in main and intercalary ribs, increasingly clear in M. eigenheeri and the subsequent species. The tuberculation becomes more present and powerful on the main ribs over time, to become a true trituberculate pattern in Toxancyloceras (by the acquisition of the lower tubercles and the generalisation of the median tubercles on the main ribs).

-(3) A mitigation and disappearance of the marginal strengthening of the intermediate ribs. This evolutionary trend (as for the previous pattern) likely corresponds to graduaptation. This term was introduced by Chaline (1999) to designate the adaptations of a characteristic that take place gradually under the pressure of natural selection. These iterative and reversible changes (as this is the case here for the Moutoniceras / Toxancyloceras lineage) are understood under the model of the punctuated equilibrium/disequilibrium [START_REF] Chaline | Le concept d'évolution polyphasée et ses implications[END_REF][START_REF] Chaline | Arvicolid Data and Evolutionary Concepts[END_REF] in the sense of a continuous and flexible adaptation to environmental changes, possibly causing minimal genome changes.

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The second major evolutionary trend of this group regards the reduction in adult size throughout the transition from Moutoniceras to Toxancyloceras (Fig. 8), which seems to be expressed by a decrease in the duration of the different stages in a rather balanced way. This could be interpreted in terms of heterochronies (accelerated progenesis = pseudo-dwarfism in the sense of [START_REF] Dommergues | Les relations ontogenèse-phylogenèse : applications paléontologiques[END_REF]. It is important to note that, as this is usually the case for numerous heteromorph ammonites species (see [START_REF] Bert | L'influence de la variabilité intraspécifique sur la taxinomie, la biostratigraphie et l'évolution des ammonites : une approche paléobiologique[END_REF], the adult size is subject to variation from one specimen to another (see the systematic part in the online additional material), thus, the evolutionary trend recognised here concerns only the maximal possible reached size.

Discussion

Moutoniceras marii and M. eigenheeri are both at the limit of the two phases recognised in the evolution of the lineage formed by the Moutoniceras and the Toxancyloceras species.

Obviously, the Toxancyloceras morphology is already predefined in Moutoniceras where there already exists a tendency to form tubercles. In addition, the peri-ventral bulges on smooth ribs, which persist in earlier Toxancyloceras, are an inherited trait of the

Moutoniceras.

Thus, from a merely taxonomic point of view, the choice could have been made to separate the small Moutoniceras species (M. marii and M. eigenheeri) from their giant ancestors. This proposition would have been coherent in evolutionary terms, since the use of evolutionary progenesis events are usually used to separate taxa at the genus-group level [START_REF] Dommergues | Les relations ontogenèse-phylogenèse : applications paléontologiques[END_REF]. There were two taxonomic options: either (1) we group M. marii and M. eigenheeri to the Toxancyloceras to match the generic cut-off with the drastic reduction in the size of M. Unfortunately, neither of these solutions is really satisfactory. In the former, morphological coherence would be lost since M. eigenheeri and M. marii have an ornamentation strongly rooted in the Moutoniceras morphology (this is especially the case for M. marii). They do not have the apomorphies known in the more recent Toxancyloceras, and especially the trituberculate pattern, which is a diagnostic criterion for defining the genus (discriminative character). The second solution has the disadvantage of pulverizing the group by introducing a third artificial taxon to create the link which doesn't make much sense in terms of palaeobiology and clearly poses a problem in overestimating biodiversity. These risks have been evaluated and criticised by several authors from other ammonites groups (see a review in [START_REF] Bert | L'influence de la variabilité intraspécifique sur la taxinomie, la biostratigraphie et l'évolution des ammonites : une approche paléobiologique[END_REF].

Ideally, it is better to foster generic cuts to major changes in the evolutionary rhythms. So the choice was made in the present work to maintain M. marii and M. eigenheeri in the genus Moutoniceras, with the interest of maintaining a certain nomenclatural stability, and also to reserve this taxonomic hyphenation of the genus-group level for another major morphological aspect in the evolution of the group: the appearance of the trituberculate pattern. All this shows that even if taxa of the genus-group level are essential to separate formally phylogenetic lineages, they are also very conventional. Thus, they should not be used to measure biodiversity without some care to ensure they were introduced on a solid phylogenetic basis, taking into account the variability of species and evolutionary data [START_REF] Bert | L'influence de la variabilité intraspécifique sur la taxinomie, la biostratigraphie et l'évolution des ammonites : une approche paléobiologique[END_REF].

When interpreting (see above, chapter 5) the two major evolutionary patterns of the Moutoniceras / Toxancyloceras lineage in terms of heterochronies, it is understood that two

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17 evolutionary processes are underlying these patterns in the history of this group. The first is paedomorphic and concerns the large sized Moutoniceras, in which it causes the progressive reduction in tuberculation by retardation of the ontogeny with the disappearance of ornamental stages (Fig. 9). However, during this process the morphology and the adult size remain remarkably stable (Fig. 8). The second process is peramorphic and, determines the second phase of the evolutionary history of the group (small Moutoniceras species and transition to the genus Toxancyloceras -Fig. 9): an increasingly early appearance of the ontogenetic stages (acceleration). Between these two phases, the start of the size reduction is effective from M. marii and is quite sudden, usually corresponding to a progenesis (interpreted here as pseudo-dwarfism in the sense of [START_REF] Dommergues | Les relations ontogenèse-phylogenèse : applications paléontologiques[END_REF]. This change in size also corresponds to the minimum expression of the tubercles, which reappear later (oscillation) and are strengthened to give the Toxancyloceras morphology. It is tempting to interpret this progenetic phenomenon as a result of environmental stress, e.g. due to environmental changes. At the same time, the maximum regressive (named SbB3 in [START_REF] Arnaud | The South-East France Basin (SFB) and its Mesozoic Evolution[END_REF] occurs, which corresponds to the top-early Barremian peak of δ 18 O (Baudin et al., 2009), corresponding with a "drastic" decrease in temperature.

It is very interesting to consider the ornamental oscillation, with the gradual disappearance and then reappearance of the tuberculate pattern along the evolutionary trend of the group.

Ornamental oscillation implies that some evolutionary patterns are ultimately not completely irreversible and that they are certainly subject to external variations [START_REF] Guex | Retrograde Evolution During Major Extinction Crises[END_REF]. This is the very idea of the graduaptation (see above chapter 5), which designates the adaptations of a characteristic that take place gradually under the pressure of natural selection. If these conditions are reversed, some evolutionary trends seem able to reverse also, even to oscillate, but not without resulting in some internal upheavals requiring morphological adaptations.

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT

18 And that's what seems to happen here with the progenetic event of M. marii, which causes the reduction in size and the heterochrony inversion (peramorphosis versus paedomorphosis previously).

Evidence appears to suggest that this oscillation occurs as a possible case of evolutionary reversibility, which at first glance seems to contradict the Dollo law on the irreversibility of evolution [START_REF] Dollo | Les lois de l'évolution[END_REF][START_REF] Gould | Dollo on Dollo's Law: Irreversibility and the Status of Evolutionary Laws[END_REF]. Although genetic [START_REF] Marshall | Dollo's Law and the death and resurrection of genes[END_REF] and molecular experimentations [START_REF] Bridgham | An epistatic ratchet constrains the direction of glucocorticoid receptor evolution[END_REF] seem to show the veracity of this law, exceptions have already been reported in literature, often based on heterochrony (eg. Collin and Cipriani, 2003;[START_REF] Diogo | Violation of Dollo's Law: evidence of muscle reversions in primate phylogeny and their implication for the understanding of the ontogeny, evolution, and anatomical variations of modern humans[END_REF][START_REF] Kerney | Do larval traits re-evolve? Evidence from the embryogenesis of a direct-developing salamander, Plethodon cinireus[END_REF]Whiting et al., 2003;see review in Wiens, 2011), to the extent that its relevance could be questioned [START_REF] Colin | Reversing opinions on Dollo's Law[END_REF]. In any case, and as Gould stated (1970), Dollo Law strictly concerns the impossibility for the organisms (or a sufficiently complex biological structure) to return to their ancestral state using the same complete but reversed path (to a question of probabilities based on complexity). However, Dollo Law doesn't objectively exclude the possibility of certain characteristics returning to an ancestral state, regardless of the function they occupied before.

Without being a real violation of Dollo Law in the strict sense [START_REF] Gould | Dollo on Dollo's Law: Irreversibility and the Status of Evolutionary Laws[END_REF], the example of Moutoniceras / Toxancyloceras shows that there may be a form of repeatability in the evolutionary patterns in ammonites. It is here favored by heterochrony (due to the progenesis impact), as it was demonstrated with the re-evolution of coiling in some gastropods of the family Calyptraeidae (Collin and Cipriani, 2003). Besides the developmental process highlighted by heterochrony, it is also tempting to think that some of these iterative phenomena may cause minimal modifications of the genome and may occur through epigenetic processes, whose influence seems to be confirmed as might suggest some recent studies [START_REF] Goudarzi | Dynamic Competing Histone H4 K5K8 Acetylation and Butyrylation Are Hallmarks of Highly Active Gene Promoters[END_REF].

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Remarks about the dimorphism in the Moutoniceras / Toxancyloceras lineage: in the literature data, the dimorphism has never been demonstrated for the genera Moutoniceras and Toxancyloceras [START_REF] Delanoy | Sur le dimorphisme des Ancyloceratina (Ammonoidea) du Crétacé inférieur[END_REF]. The specimens studied in the present work do not document it either, even if size variation is observed between some specimens. There is no evidence that this size variation could be due to dimorphism, especially because the extreme variants seem to be linked by intermediates. This size variation very likely corresponds to the instraspecific "normal" variation, as it is known for other genera of heteromorph ammonites (see for example [START_REF] Delanoy | Biostratigraphie des faunes d'Ammonites à la limite Barrémien-Aptien dans la région d'Angles-Barrême-Castellane. Etude particulière de la Famille des Heteroceratidae (Ancyloceratina, Ammonoidea)[END_REF] for the genus Heteroceras d 'Orbigny, 1849). Of course, this non-documentation does not mean that the dimorphism is absent from these genera, but that further research is needed to highlight it.

Biostratigraphic implications

Some biostratigraphic implications can be drawn from the understanding of the relationship between the genera Toxancyloceras and Moutoniceras at the turn of the early and late Barremian in the northwestern Tethyan margin. On the one hand, the revision of the classic species T. vandenheckei reinforces its status as index for the T. vandenheckei zone and subzone (lower upper Barremian). On the other hand, the review of M. eigenheeri and the discovery of T. canuti sp. nov., which was confused in the past with T. vandenheckei for various reasons (see above, chapter 4), allow us to propose three new biostratigraphic horizons for the Vocontian Basin (Fig. 3): the M. eigenheeri Horizon at the top of the lower Barremian (M. moutonianum Zone), and the T. canuti and T. vandenheckei horizons at the lower upper Barremian (T. vandenheckei Zone and Subzone). The stratigraphic distribution of all the index species proposed in the present work (Fig. 3) is very restricted and now precisely known. The importance of their being part of an evolutionary context is now well established 20 (this work,Figs. 8 and 9). The relative frequency of these species, in respect to the other contemporary ammonite faunas, makes them serious and legitimate candidates to the status of index species. Finally, the presence of representatives of the Moutoniceras and Toxancyloceras genera over a large geographical area, coupled with the discovery of T. canuti sp. nov. from Spain (Mallorca) to Tyrol (= T. vandenheckei, fig. 8a-b in Lukeneder, 2012) and Hungary (Fözy and Janssen, 2009, fig. 5b, 5f-g), suggests a high potential of correlation for these horizons on the whole northwestern Tethyan margin.

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Moutoniceras eigenheeri Horizon (new)

Index species: Moutoniceras eigenheeri (Vermeulen, 2003), which is revised in the present work. This species is currently known in the Vocontian Basin only, but as it is the motherspecies of T. canuti sp. nov., it could potentially extend to the same area.

Reference section: the section G5 of the Angles area (southeastern France, Vocontian Basin) is chosen as the reference section of the M. eigenheeri Horizon. The index species is present in the beds 78, 80 and 82, which are equivalent to the stratotype beds 141, 143-1a and 143-1c respectively (sections A, A' and A*, see Figs 5 and 6 and Bert, 2012).

Status: this horizon is defined by the First Occurrence (FO) of its index-species. In fact, the latter is present in the uppermost thick bed of the lower Barremian in the stratotype area, and the horizon corresponds to the total distribution of the index species. In 2009, Reboulet et al. restored the division of the M. moutonianum Zone into two horizons: a Coronites darsi Horizon at the base, and a Heinzia caicedi Horizon at the top. The latter horizon encompasses the upper half of the M. moutonianum Zone; however, Heinzia caicedi (Karsten, 1856) is virtually absent from the basin, where the horizon is rendered of difficult use. The latter species is almost exclusively present in the platform borders where the levels are often quite condensed. Compared to the reference zonal scheme [START_REF] Reboulet | Report on the 5th International Meeting of the IUGS Lower Cretaceous Ammonite Working Group, the Kilian Group[END_REF], the M. eigenheeri

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 21
Horizon is more recent than the H. caicedi Horizon, as the disappearance of M. eigenheeri coincides with the end of the lower Barremian. It immediately precedes the appearance of T. canuti sp. nov., which serves as index species of the overlying horizon: the first one of the upper Barremian.

Faunal assemblage: the ammonites are relatively scarce at the top of the lower Barremian. In spite of that, M. eigenheeri is relatively abundant (often in fragments) with respect to other ammonite faunas of the M. eigenheeri Horizon, where it is the most significant and easy-touse ammonite for non-specialists. The index species is associated with numerous Barremitidae; the holcodiscids are well represented with Holcodiscus uhligi (Karakasch, 1907) and Parasaynoceras sp.; Kotetishvilia sauvageaui (Hermite, 1890) and Silesites vulpes (Matheron, 1880) are also present.

Toxancyloceras canuti Horizon (new)

Index species: Toxancyloceras canuti sp. nov., which is introduced and described in an evolutionary context in the present work. It is known currently in southeastern France, in South Tyrol, in Hungary and in Spain (Mallorca), where the new horizon could be used.

Toxancyloceras vandenheckei Horizon (new)

Index species: Toxancyloceras vandenheckei [START_REF] Astier | Catalogue déscriptif des Ancyloceras appartenant à l'étage Néocomien d'Escragnolles et des Basses-Alpes[END_REF], which is revised in the present work. This species is currently known in southeastern France, Spain (incl. Mallorca), Slovakia, Italy, and maybe in South Tyrol and Japan.

Reference section: the field section A* is chosen as the reference for the T. vandenheckei Horizon. The index species appears there from bed No. 146. This level corresponds to bed No. 86 of the neighbouring section G5 (see Figs 5 and6).

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Status: this horizon is defined by the FO of its index species, and the horizon corresponds to the total distribution of its index species as understood in the present work. T. vandenheckei, now revised, is strengthened as index species for the T. vandenheckei Zone and Subzone. On the other hand, while it is now recognised that T. vandenheckei does not appear at the base of its zone, it is not necessary to rename it the "T. canuti Zone". Indeed, the lower limit of an interval zone, of which the use has been recommended by the IUGS Lower Cretaceous Ammonite Working Group (Reboulet et al., 2011, p. 790), is defined by a horizon [START_REF] Thierry | La biostratigraphie, du taxon à la biobiozone et aux échelles biozonales. in : Stratigraphie, terminologie française, Comité[END_REF]. In this case, it is not mandatory that a horizon starts with its index taxon (unlike a distribution zone, for example).

Faunal assemblage: T. vandenheckei is common in the horizon of which the ammonite fauna is very similar to the T. canuti Horizon. Heinzia sayni specimens however, become more frequent, while Holcodisciidae become scarcer. As in the previous horizon, Macroscaphites rakusi ([m&M]), Kotetishvilia sauvageaui, Silesites vulpes and numerous Barremitidae are present.

Conclusions

The new data from several stratigraphic sections near the Barremian stratotype (the Angles area in southeastern France) and from the Lloseta and Biniamar area (Mallorca, Balearic The second process is peramorphosis with the acceleration of the ornamentation in the small Moutoniceras species and the Toxancyloceras (Figs. 8,9). Between these two phases occurs a progenesis with the quite sudden reduction in adult size effective from M. marii. This size reduction is interpreted here as pseudo-dwarfism (in the sense of [START_REF] Dommergues | Les relations ontogenèse-phylogenèse : applications paléontologiques[END_REF] where the duration of the different ontogenic stages decreases. This change in size also corresponds to the pattern of minimum expression of the tubercles, which reappear later (oscillation). This progenesis is concomitant with some environmental changes (maximum regressive of the late Barremian, and "drastic" decrease in temperatures) and it would be tempting to correlate the two phenomenon.

The oscillation of the tuberculate pattern is adjusted by: (1) a process of paedomorphosis (retardation of the ornamentation with extension of the non-tuberculate young stage during growth that implies the progressive disappearance of the tuberculation); and (2) by a peramorphic process (acceleration of the tuberculate pattern, which appears increasingly early through ontogenesis). Finally, two other ornamental patterns are related to graduaptation processes. The first is the differentiation in main and intercalary ribs, with the increase of the tuberculation to a true trituberculate pattern (in Toxancyloceras). The second is the mitigation and disappearance of the marginal strengthening of the intermediate ribs.

The ornamental oscillation of the tuberculate pattern could act a possible case of evolutionary reversibility. Actually, the Dollo law is not violated as this latter does not objectively exclude the possibility for certain characteristics to return to an ancestral state, regardless of the function they occupied before. Here, the example of Moutoniceras / Toxancyloceras shows that there exist a form of repeatability in the ammonites evolution favoured by the heterochronies. [START_REF] Bert | Origin of the Tethyan Hemihoplitidae tested with cladistics (Ancyloceratina, Ammonoidea, Early Cretaceous): an immigration event?[END_REF](reconstructed from Barron et al., 1981[START_REF] Dercourt | Atlas Peri-Tethys, Palaeogeographical maps[END_REF], with the position of the Vocontian Basin (southeastern France), and the Balearic Islands (Spain). Fig. 3. Barremian biostratigraphy of the stratotype area (southeastern France), according to [START_REF] Reboulet | Report on the 5th International Meeting of the IUGS Lower Cretaceous Ammonite Working Group, the Kilian Group[END_REF] completed by the data of [START_REF] Bert | Nouveaux biohorizons et propositions pour le découpage biozonal ammonitique du Barrémien supérieur du Sud-Est de la France[END_REF][START_REF] Bert | Pseudoshasticrioceras bersaci nov. sp. (Ammonoidea, Gassendiceratinae), and new ammonite biohorizon for the Upper Barremian of southeastern France[END_REF][START_REF] Bert | Problems in the identity of "Crioceras" barremense Kilian, 1895, (Ancyloceratida, Upper Barremian), and their proposed resolution[END_REF][START_REF] Bert | The Dichotomus Horizon: a new biochronologic unit of the Giraudi Zone of the Upper Barremian of southeastern France, and considerations regarding the genus Imerites Rouchadze (Ammonoidea, Gassendiceratinae)[END_REF]. The three new horizons are in red (Moutoniceras eigenheeri, Toxancyloceras canuti and T. vandenheckei). Note that the Hemihoplites astarte Horizon replaces the H. casanovai Horizon of [START_REF] Bert | Nouveaux biohorizons et propositions pour le découpage biozonal ammonitique du Barrémien supérieur du Sud-Est de la France[END_REF], because of the objective synonymy of H. casanovai Delanoy, 1992 with H. astarte [START_REF] Fallot | Ammonites nouvelles des Iles Baléares[END_REF] (new unpublished data -forcoming work). Bituberculate pattern (shaft); 7, Irregular trituberculate pattern (shaft); 8, Bend and hook (this stage is considered here as an 'adult variation' without any evolutionary implications -see [START_REF] Delsol | Embryogenesis, Morphogenesis, Genetics and Evolution[END_REF]. See text, chapter 5, for explanations.

  the area of the Barremian stratotype of Angles (southeastern France), in levels dated at the top of the M. moutonianum Zone and the basal T. vandenheckei Zone, at the transition between the lower and upper Barremian, shed new light on the questions of the Toxancyloceras origin and evolution. Other data from the Mallorcan area (Balearic Islands) close gaps in the knowledge about the morphology of the studied species. Seven successive species of Moutoniceras and Toxancyloceras are identified in the present work, including a new one, and their phylogenetic relationships are considered, based on an evolutionary perspective. Biostratigraphic implications are potentially significant, given the importance of the genera Moutoniceras and Toxancyloceras in the standard Tethyan ammonite zonation

  of Angles (Alpes de Haute-Provence, France) where the section A extends approximately 660 m (500 m for the Barremian itself) along two East-West oriented hills.

  without further detail. The current and more detailed work of this section characterise the presence of the lower barremian that was never reported previously (Fig.6, M. moutonianum Zone).

  some Valanginian, Barremian and Albian ammonites from this area.

  compressissima Zone -H=1160 mm) and M. moutonianum (lower M. moutonianum Zone -M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 12 H=1280 mm), and also probably in M. berti (upper part of K. compressissima Zone).

  marii.-M. eigenheeri (top of the M. moutonianum Zone, in the new M. eigenheeri Horizon in this work): the adult size of M. eigenheeri continues its reduction compared to M. marii (H around 500-600 mm) with a large spiral part of about 200 mm in diameter. The tubercles, which were missing in M. marii, reappear on the early whorls of the coil (bituberculation with the fibula pattern) and the shaft (peri-ventral tubercles). The acquisition of the lateral tubercles on the shaft is very discreet, but this is new for the genus Moutoniceras on this part of the shell (lateral tubercles are known in M. nodosum but only on the outer coil). It is here that the trituberculation known in the immediately more recent Toxancyloceras begins to occur. This trituberculation appears however in a very fleeting manner at the very top of the shaft (the peri-dorsal bulges are very discreet). There are four ontogenetic stages in M. eigenheeri: (1) the inner part of the coil with differentiated ribs; (2) the outer part of the coil M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 13 with all identical ribs; (3) the shaft with ribs differentiated again; and (4) the hook with a more irregular ornamentation.-T. canuti sp. nov. (lower part of the T. vandenheckei Zone and Subzone, in the new T. canuti Horizon, in this work): the adult size is a little further reduced (360<H<550 mm), as the spiral part reduces (110-140 mm). This implies an increase of the shell curvature and an increase in the whorl height growth. The main ribs of the coil and the shaft become systematically trituberculate; however the tubercles never become strongly expressed. The four ontogenetic stages are still present, although there are some changes from M. eigenheeri:(1) the inner part of the coil has strong differentiated ribs and regular alternation between main and intercalary ribs. The new pattern with looped ribs is likely an evolution of the pattern with fibula ribs known in M. eigenheeri: the lower tubercle closes the fibula and forms the loop; (2) the alternation of main / intercalary ribs of the outer part of the coil becomes much less regular. As in M. eigenheeri, the intercalary ribs maintain their strengthened appearance on the venter and on the peri-ventral margin, where they are sometimes even slightly tuberculate; (3) the shaft has differentiated ribs, but the number of intercalary ribs increases; and (4) the hook is almost identical.-T. vandenheckei (lower part of the T. vandenheckei Zone and Subzone, in the new T.vandenheckei Horizon, in this work): the adult size is reduced and seems to be stabilised (H around 425 mm), as well as the spiral uncoiled part (95-160 mm). In terms of ornamentation, the intermediate bituberculate ribs, which were ornamental remains of Moutoniceras, disappear. At the same time the tubercles become stronger on the main ribs, while the periventral strengthening of the intercalary ribs becomes anecdotal. There are no more than three ontogenetic successive stages during growth since the stage with differentiated ribs invaded the entire coil.

  two imbricated evolutionary trends are highlighted in the Moutoniceras / Toxancyloceras lineage. The first major trend concerns the ornamental changes, itself determined by three imbricated patterns (Fig.9):

  marii;(2) either we group M. marii and M. eigenheeri into a separate genus (the genus Vermeulen in 2003 for the latter species) separate from Moutoniceras and Toxancyloceras.

  Reference section: the field section A* is chosen as reference for the T. canuti Horizon. The index species is present in beds No. 144 and 145. It may be present in the small beds No. 143-2 and 143-3 (where the change in lithology known at the lower/upper Barremian boundary occurs), but so far these beds have delivered no fossils. The T. canuti Horizon corresponds to the beds No. 84 and 85 of the neighbouring section G5 (see Figs5 and 6).

  Status: this horizon is defined by the FO of its index species immediately above the last thick beds of the lower Barremian in the stratotype area (Angles, southeastern France), and the horizon corresponds to the total distribution of the index species. The choice to place the T. canuti Horizon into the upper Barremian instead of the lower Barremian does not modify the the T. vandenheckei Zone. The first reason is that T. canuti sp. nov. has often been confused with T. vandenheckei (see synonymy, chapter 4), and therefore the T.vandenheckei Zone was already starting de facto with the appearance of T. canuti sp. nov. The second reason is that the ammonite fauna of the T. canuti Horizon is closer to that of the upper of Heinzia sayni Hyatt, 1903, which was used as index of the first zone of the upper Barremian in the past[START_REF] Vermeulen | Biohorizons ammonitiques dans le Barrémien du Sud-Est de la France (de la zone à Hugii à la zone à Sartousiana)[END_REF]. The acceptance of the T. canuti Horizon as the lowest horizon of the upper Barremian at the basis of the T. vandenheckei Zone requires only very slight modification of its definition. It no longer starts with its index species, but with the emergence of the Toxancyloceras genus (in the evolutionary context Moutoniceras -> Toxancyloceras), which ultimately facilitates its recognition by non-ammonite specialists. Faunal assemblage: the T. canuti Horizon undoubtedly marks the appearance of the Toxancyloceras. T. canuti sp. nov., which is relatively abundant, appears simultaneously with rare specimens of Heinzia sayni. Note the extreme scarcity of the Holcodisciidae (scarce Holcodiscus uhligi and Parasaynoceras sp.), and the total disappearance of the Moutoniceras. Present in this horizon are also: Macroscaphites rakusi (Uhlig, 1883) ([m&M]), Kotetishvilia sauvageaui, Silesites vulpes and numerous Barremitidae.

  Islands, Spain), allow us to demonstrate that the upper Barremian ammonite genusToxancyloceras originates in the lower Barremian Moutoniceras. The genus Moutoniceras disappeared with the tuberculate species M. eigenheeri at the very top of the lower Barremian, just below the FO of T. canuti sp. nov., which is the oldest species of the genus Toxancyloceras at the base of the upper Barremian. Thus, Moutoniceras is now the oldest currently known representative of the ammonite family Ancyloceratidae, and this genus as a Heteroceratidae. Of course, this classification does not exclude the hypothesis that the Heteroceratidae themselves probably derive from the Ancyloceratidae.This new data sheds new light on the comprehension of the Moutoniceras andToxancyloceras. Especially, the revision of the species M. eigenheeri and T. vandenhecki, and a new species (T. canuti sp. nov.) is described. T. vandenheckei is considered in the historical context of the acquisition of the Astier collection by the Natural History Museum of London, and its holotype is figured here for the first time other than by a drawing. The examination of this latter specimen, which is not totally identical with the original drawing of 1851, partly explains some misinterpretations of the species. Seven successive species of Moutoniceras and Toxancyloceras are considered and their phylogeny is established as follow: M. nodosum-> M. berti -> M. moutonianum -> M. marii -> M. eigenheeri -> T. canuti sp. nov. -> T.vandenheckei. However, more data about the development of the Moutoniceras species is required to make an accurate cladistic analysis, which requires further research into the Nicklesia pulchella, K. compressissima and M. moutonianum zones (lower Barremian). In any case, the species of the genus Moutoniceras need revision with consideration into the perspectives of intraspecific variability and evolutionary.Biostratigraphically, the status of T. vandenheckei as zonal and subzonal index at the base of the upper Barremian is reinforced, and three new ammonite horizons are defined: the Moutoniceras eigenheeri, Toxancyloceras canuti and T. vandenheckei horizons. The stratigraphic distribution of all their index species is very restricted, forming part of a well established evolutionary context.There are two phases in the evolutionary history of the Moutoniceras / Toxancyloceras lineage, underlined by two processes. The first is paedomorphosis and it concerns the large the progressive disappearance of the tubercle pattern over time through a retardation of the ontogeny and the disappearance of ornamental stages(Figs. 8, 9).

Fig. 2 .

 2 Fig. 2. Palaeogeographic map of the Western Tethyan Realm at the Barremian, modified after

Fig. 4 .

 4 Fig. 4. Palaeogeographic map of the Vocontian Basin (southeastern France -modified after

Fig. 5 .

 5 Fig. 5. Outcrop section A* near Angles (stratotype of the Barremian stage, southeastern

Fig. 6 .

 6 Fig. 6. Outcrop section G5 in the Angles area (southeastern France). ME: Moutoniceras

Fig. 7 .

 7 Fig. 7. Simplified geological map of Mallorca (Balearic Islands, Spain). The red star points

Fig. 8 .

 8 Fig. 8. Size and reconstructed morphology of the successive species of the lineage

Fig. 9 .

 9 Fig. 9. Ontogeny and heterochronies within the Moutoniceras / Toxancyloceras lineage. The

Acknowledgments

The authors wish to thank Peter Bengtson for his proposal to submit a paper in the special volume dedicated to the memory of Richard A. Reyment. Phil Hurst is acknowledged for the photos of the holotype of Toxancyloceras vandenheckei, and Pere Miralles for his help with Figure 7. This work was supported by the Departmental Council of the Alpes de haute-Provence (France), and by the Laboratory of the G.P.A. (Research group in ammonites palaeobiology and biostratigraphy). The Réserve naturelle nationale géologique de Haute-Provence is aknowledged for allowing the prospections on its territory. The curators of the French National Museum of Natural History (Sylvain Charbonnier) and of the Faculty of Science Claude Bernard (Abel Prieur), and the private collectors Léon Canut (GPA) and Roger Quelquejeu, are aknowledged for allowing the study of the specimens under their care.

Finally, the Editor-in-Chief Eduardo Koutsoukos, Josep A. Moreno Bedmar and two anonymous reviewers are acknowledged for their constructive remarks, which greatly helped to improve the manuscript.

Gassendiceras). The table of characters is given in

 Bert and Bersac (2014, appendix).

Standard Tethyan ammonite zonation of [START_REF] Reboulet | Report on the 5th International Meeting of the IUGS Lower Cretaceous Ammonite Working Group, the Kilian Group[END_REF] with the biostratigraphic proposals of [START_REF] Bert | Nouveaux biohorizons et propositions pour le découpage biozonal ammonitique du Barrémien supérieur du Sud-Est de la France[END_REF][START_REF] Bert | Pseudoshasticrioceras bersaci nov. sp. (Ammonoidea, Gassendiceratinae), and new ammonite biohorizon for the Upper Barremian of southeastern France[END_REF][START_REF] Bert | Problems in the identity of "Crioceras" barremense Kilian, 1895, (Ancyloceratida, Upper Barremian), and their proposed resolution[END_REF][START_REF] Bert | The Dichotomus Horizon: a new biochronologic unit of the Giraudi Zone of the Upper Barremian of southeastern France, and considerations regarding the genus Imerites Rouchadze (Ammonoidea, Gassendiceratinae)[END_REF] Distribution of the Moutoniceras / Toxancyloceras

Stages