Thomas Batard
email: tbatar01@univ-lr.fr

CLIFFORD BUNDLES: A COMMON FRAMEWORK FOR IMAGES, VECTOR FIELDS AND ORTHONORMAL FRAME FIELDS REGULARIZATION *

Keywords: regularization, heat equations, Clifford algebras, vector bundles, differential geometry. AMS subject classifications. 68U10-53CXX-15A66-58J35

Clifford bundles: a unifying framework for images, vector fields and orthonormal frame fields regularization.

of initial condition I : (x 1 , x 2 ) -→ (I 1 (x 1 , x 2 ), • • • , I n (x 1 , x 2 )) a n-channels image, where f jk are real functions. We refer to [START_REF] Tschumperlé | Vector-valued Image Regularization with PDE's: A Common Framework for Different Applications[END_REF] for an overview on related works. From a geometric viewpoint, the set of right terms, for i = 1 • • • n, may be viewed as a second-order differential operator acting on sections of a vector bundle over a Riemannian manifold called a generalized Laplacian H [START_REF] Berline | Heat Kernels and Dirac Operators[END_REF]. As a consequence, it ensures existence and uniqueness of a kernel K t (x, y, H), called the heat kernel of H, generating the solution of the heat equation

∂I t ∂t + HI t = 0 (1.1)
from a 'convolution' with the initial condition I. A generalized Laplacian H on a vector bundle E over a Riemannian manifold X is determined by three pieces of data: the metric g of the base manifold X that determines the second order part, a connection on E that determines the first order part, and a zero-order operator F that determines the zero order part. For instance, the so-called oriented Laplacians can be viewed as generalized Laplacians [START_REF] Batard | Heat kernels of generalized Laplacians[END_REF]. Similarly, on the vector bundle C ∞ (X, g) of smooth functions on a Riemannian manifold (X, g), there is a canonical generalized Laplacian, the scalar Laplacian, which corresponds to the Laplace-Beltrami operator up to a sign. Considering each component I k of a multichannel image as a function over a well-chosen Riemannian manifold, the heat equation associated to the scalar Laplacian leads to the Beltrami flow of Sochen et al. in the context of image regularization [START_REF] Sochen | A General Framework for Low Level Vision[END_REF], [START_REF] Spira | A Short-time Beltrami Kernel for Smoothing Images and Manifolds[END_REF]. The aim of this paper is to extend the Beltrami flow on C ∞ (X, g) to vector fields and oriented orthonormal frame fields (i.e. SO(n)-valued fields) on (X, g).

Regularization of non-scalar fields such as vector fields and manifold-valued fields has been widely investigated in the last few years. Concerning vector fields, we may refer to [START_REF] Diewald | Anisotropic Diffusion in Vector Field Visualization on Euclidean Domains and Surfaces[END_REF], [START_REF] Tschumperlé | Vector-valued Image Regularization with PDE's: A Common Framework for Different Applications[END_REF], where the smoothing is made through the smoothing of a corresponding multichannel image and where the anisotropy is controlled at each point by the orientation of the vector field. More precisely, an oriented Laplacian is applied on each component of the image. The regularization of SO(n)-valued fields has been treated in [START_REF] Gur | Regularizing Flows over Lie Groups[END_REF] in the context of principal bundles using a geometric flow on the SO(n) group, and in [START_REF] Tschumperlé | Orthonormal vector sets regularization with PDE's and applications[END_REF] using a minimization problem with orthogonal constraints. For n = 2, this is the problem of orientations regularization, treated in [START_REF] Kimmel | Orientation Diffusion or How to Comb a Porcupine?[END_REF], [START_REF] Perona | Orientation Diffusions[END_REF], [START_REF] Tang | Diffusion of General Data on Non-Flat Manifolds via Harmonic Maps Theory: The Direction Diffusion Case[END_REF]. More generally, regularization of symmetric definite positive matrice fields were proposed in [START_REF] Gur | Fast GL(n)-Invariant Framework for Tensors Regularization[END_REF], [START_REF] Pennec | A Riemannian Framework for Tensor Computing[END_REF] and of Stiefel manifolds in [START_REF] Chefd'hotel | Regularizing flows for Constrained Matrix-Valued Images[END_REF], [START_REF] Edelman | The Geometry of Algorithms with Orthogonality Constraints[END_REF], all of them using geometric flows on manifolds.

In [START_REF] Chefd'hotel | Regularizing flows for Constrained Matrix-Valued Images[END_REF], [START_REF] Diewald | Anisotropic Diffusion in Vector Field Visualization on Euclidean Domains and Surfaces[END_REF], [START_REF] Tschumperlé | Vector-valued Image Regularization with PDE's: A Common Framework for Different Applications[END_REF], [START_REF] Tschumperlé | Orthonormal vector sets regularization with PDE's and applications[END_REF], the behaviour of the flow is determined by the field itself. In [START_REF] Edelman | The Geometry of Algorithms with Orthogonality Constraints[END_REF], [START_REF] Gur | Regularizing Flows over Lie Groups[END_REF], [START_REF] Gur | Fast GL(n)-Invariant Framework for Tensors Regularization[END_REF], [START_REF] Pennec | A Riemannian Framework for Tensor Computing[END_REF], it is determined by the choice of a Riemannian metric on the manifold too. In this paper, we make use of an additional geometric structure, called a connexion. Roughly speaking, a connection is a way to differentiate sections of a vector bundle. An example of connexion is the Levi-Cevita connection on the tangent bundle of a Riemannian manifold. By the use of heat equations on vector bundles in this paper, the behaviour of the diffusion is completely determined by the geometry of the vector bundle where the fields are considered as sections. By geometry of vector bundle, we mean a metric on the base manifold and a connection on the vector bundle. More precisely, the metric of the base manifold determines the anisotropy of the diffusion, whereas the connection determines the data averaged by the diffusion by the use of the transport parallel map.

In this paper, we extend the Beltrami flow to vector fields and orthonormal frame fields by extending the Laplace-Beltrami operator to vector fields and generators of orthonormal frame fields, i.e. fields with values in the Lie algebra so(n) of SO(n).

For this, we consider Clifford bundles, that are vector bundles where the fibers are endowed with a Clifford algebra structure. Clifford algebras framework [START_REF] Chevalley | The Algebraic Theory of Spinors and Clifford Algebras[END_REF], [START_REF] Hestenes | Clifford Algebra to Geometric Calculus[END_REF] finds a wide range of applications in computer science [START_REF] Sommer | Geometric Computing with Clifford Algebras[END_REF]. Application of Clifford bundles to image processing was introduced in [START_REF] Batard | A Metric Approach to nD Images Edge Detection with Clifford Algebras[END_REF] where the Di Zenzo's gradient, devoted to multichannel image segmentation, is generalized using covariant derivatives instead of usual derivatives.

We show that Clifford bundles provide a common framework to treat functions, vector fields, orthonormal frame fields and their generators over manifolds. We also give a general method to construct generalized Laplacians on Clifford bundles such that the subsequent heat equation gives a tool to regularize functions, vector fields and orthonormal frame fields. Even though we only treat the case of SO(n)-valued fields in this paper, we expect that the method we propose allows to treat and regularize fields with values in connected components of much more Lie groups. Indeed, a lot of Lie groups and their Lie algebras have representations in Clifford algebras [START_REF] Doran | Lie Groups as Spin Groups[END_REF], [START_REF] Hestenes | Clifford Algebra to Geometric Calculus[END_REF].

This paper is organized as follows. In Section 2, we treat the problem of approximations of heat equations solutions. We discuss both continuous and discrete settings. We treat the particular case of the scalar Laplacian, and relate it with the Laplace-Beltrami operator. In Section 3, we first introduce Clifford bundles, and show that functions, vector fields and generators of orthonormal frame fields can be viewed as sections of a Clifford bundle. Moreover, we show that rotation fields can be lifted to sections of a Clifford bundle called spinor fields. Then, we show how to construct generalized Laplacians leading to functions, vector fields and orthonormal frame fields regularization from connections compatible with metrics of vector bundles. Dealing with particular Clifford bundles, we construct Clifford-Beltrami and Clifford-Hodge operators. In Section 4, we consider base manifolds of dimension 2. We compare diffusions provided by Clifford-Beltrami and Clifford-Hodge flows in the context of image processing with applications to image, 2D vector fields and orientation fields regularization. The first Appendix is devoted to differential geometry of vector bundles. We give definitions of notions used throughtout the paper. In the second one, we introduce heat equations on vector bundles. In the last Appendix, we detail the construction of Clifford algebras and give their main properties. We also explain how orthogonal transformations can be represented in the Clifford algebras context using the spinor group Spin(n).

2. Approximation of generalized heat equations solutions.

2.1. The continuous setting. Generally speaking, for arbitrary base manifold (X, g) and vector bundle E, there is no explicit formula for the heat kernel K t (x, y, H) of a generalized Laplacian H on E. As a consequence, there is no explicit formulae for solutions of the corresponding heat equation.

However, there exist kernels K N t (x, y, H), for N ∈ N, approximating the heat kernel of H on normal neighborhoods of the base manifold, for small t. Based on these results, solutions of generalized heat equations may be approximated for small t by the convolution of the initial condition with such kernels.

In this paper, we approximate the heat kernel of H by the kernel K 0 t (x, y, H) defined by

Å 1 4πt ã m/2 e -d(x,y) 2 /4t Ψ(d(x, y) 2 ) τ (x, y) J(x, y) -1/2
where m is the dimension of the base manifold X, and d stands for the geodesic distance on X related to the Levi-Cevita connection on (T X, g). Ψ is a function such that the term Ψ(d(x, y) 2 ) equals 1 if y is inside a normal neighborhood of x and 0 otherwise. Hence we may assume that y is inside a normal neighborhood of x. The map τ (x, y) is the parallel transport map on E related to the connection ∇ E such that H = ∆ E + F (see Appendix B) along the unique geodesic joining y and x. Indeed, on normal neighborhoods, there is a unique geodesic joining any point to the origin. At last, J are the Jacobians of the coordinates changes from usual coordinates systems

T. BATARD

to normal coordinates systems.

Hence, we approximate the solution e -tH u of the heat equation ∂u t /∂t + Hu t = 0, of initial condition

u 0 = u, at x ∈ X by k 0 t u(x) = (1/4πt) m/2 × X e -d(x,y) 2 /4t Ψ(d(x, y) 2 ) τ (x, y) u(y) J(x, y) -1/2 dy
Remark: Whereas the heat kernel K t (x, y, H) of a generalized Laplacian H is unique, there is no one-one correspondance between H and the kernel K 0 t (x, y, H) approximating its heat kernel. Indeed, K 0 t (x, y, H) only depends of the Riemannian metric g on X and the connection ∇ E on E such that H = ∆ E + F . Then given F 1 a zero-order operator on E and the generalized Laplacian

H 1 = ∆ E + F 1 , we have K 0 t (x, y, H) = K 0 t (x, y, H 1 )
2.2. The discrete setting. For the purpose of applications to image processing, we discretize the computation of k 0 t u. We proceed by discrete convolutions with masks.

Considering the standard Laplacian ∆ on R 2 , the gaussian kernel G t (x, y) is the heat kernel K t (x, y, ∆) of ∆. This means that the solution u t of the heat equation ∂u t /∂t = ∆u t of initial condition u 0 = u is given by the convolution of u with the Gaussian kernel

e -t∆ u(x) = Å 1 4πt ã R 2
e -x-y 2 /4t u(y) dy

The Gaussian kernel has an infinite support and satisfies the property

Å 1 4πt ã R 2 e -x-y 2 /4t dy = 1
In practice, the discrete Gaussian kernel is truncated such that G t ((i, j), (k, l)) = 0 if the pixel (k, l) is outside a given neighborhood N (i,j) of the pixel (i, j) (e.g. 5 × 5 neighborhood), and normalized inside N (i,j) , i.e.

G t ((i, j), (k, l)) = (1/4πt) e -(i,j)-(k,l) 2 /4t N (i,j) (1/4πt) e -(i,j)-(m,n) 2 /4t
Hence, the discrete approximation of e -t∆ u at a pixel (i, j) is given by the discrete convolution of u with such a mask. More precisely, we have (e -t∆ u)(i, j)

(k,l)∈N (i,j) G t ((i, j),(k, l)) u(k, l)
More generally, the kernel

Q t (x, y) = Å 1 4πt ã e -d(x,y) 2 /4t Ψ(d(x, y) 2 ) J(x, y) -1/2
on (X, g) may be viewed as the Riemannian counterpart of the Gaussian kernel on R 2 . Indeed, dealing with normal coordinates around a point x, they take both the form Å 1

4πt

ã e -y 2 /4t
where y are the normal coordinates of y.

Based on this observation, we propose to adapt the above method, devoted to compute a discrete approximation of e -t∆ u on R 2 , in order to compute more generally a discrete approximation of k 0 t u on (X, g) of dimension 2. This generalization requires the construction of discrete geodesic curves and normal neighborhoods.

For this, we use the following definitions and results on differential geometry of manifolds (see Appendix A and [START_REF] Helgason | Differential geometry, Lie groups, and symmetric spaces[END_REF] for more details).

1. Let X be a manifold equipped with a connection ∇ on its tangent bundle T X. The curve γ on X is said to be autoparallel if it satisfies ∇ γ γ = 0.

2. Normal neighborhoods may be characterized as neighborhoods where each point is joined to the origin by a unique autoparallel curve.

3. Dealing with a Riemannian manifold X equipped with the Levi-Cevita connection on its tangent bundle, an autoparallel curve on X is called a geodesic.

Then we construct geodesic curves from (i, j) and a normal neighborhood V (i,j) of (i, j) as follows.

Let us first fix a neighborhood N (i,j) of the pixel (i, j) (e.g. the 5 × 5 neighborhood of (i, j)). For each p, q ∈ {-1, 0, 1} such that (p, q) = (0, 0), we construct the geodesic γ such that γ(0) = (i, j) and γ(0) = (p, q) by solving a discrete counterpart of the parallel transport of γ along γ related to the Levi-Cevita connection on T X. Then, we state that a point (k, l) ∈ N (i,j) belongs to V (i,j) if it belongs to a unique geodesic from (i, j) and if it does not follow a point on this geodesic that belongs to several geodesics from (i, j).

Hence, we truncate Q t ((i, j), (k, l)) in such a way that Q t ((i, j), (k, l)) = 0 if (k, l) is outside V (i,j) , and we normalize it inside V (i,j) , i.e.

Q t ((i, j), (k, l)) = (1/4πt) e -d((i,j)-(k,l)) 2 /4t V (i,j) (1/4πt) e -d((i,j)-(m,n)) 2 /4t
Therefore, the discrete approximation of k 0 t u at a pixel (i, j) is given by the discrete convolution of the map (k, l) -→ τ ((i, j), (k, l)) u(k, l) with a such a mask. More precisely, we have

k 0 t u(i, j) (k,l)∈V (i,j)
Q t ((i, j), (k, l)) τ ((i, j),(k, l))u(k, l)

Remark: Let us consider the operator k 1 t u associated to the kernel K 1 t (x, y, H) approximating the heat kernel of H. In the continuous setting, we have

(k 1 t u)(x) = X K 1 t (x, y, H)u(y)dy = 1 4πt X e -d(x,y) 2 /4t Ψ(d(x, y) 2 ) (τ (x, y) + tΦ 1 (x, y, H)) u(y) J(x, y) -1 2 dy
In the discrete setting, we obtain

k 1 t u(i, j) (k,l)∈V (i,j) Q t ((i, j), (k, l)) [τ ((i, j),(k, l)) + t Φ 1 ((i, j), (k, l), H)] u(k, l)
Considering the neighborhood V i,j := {(i, j)} of the pixel (i, j), we have

k 1 t u(i, j) u(i, j) + t Φ 1 ((i, j), (i, j),H) u(i, j) Moreover, we have Φ 1 ((i, j), (i, j), H) = -J 1/2 • H • J -1/2 ((i, j), (i, j)).
From the property J(x, y) = 1 + O( y 2 ) where y are the normal coordinates of y around x,

(see [START_REF] Berline | Heat Kernels and Dirac Operators[END_REF] Chapter 1), we can assume in the discrete setting that J = 1 near the diagonal. Therefore, it gives Φ 1 ((i, j), (i, j), H) = -H, and

k 1 t u(i, j) u(i, j) -t Hu(i, j) (2.1)
Finally, we obtain the Euler scheme of the PDE ∂u t ∂t + Hu t = 0, u 0 = u 2.3. Example: the scalar/Beltrami Laplacian. The Laplace-Beltrami operator can be viewed as a generalized Laplacian called the scalar Laplacian. Definition 2.1. Let E be a vector bundle of rank 1 over a Riemannian manifold (X, g), endowed with the connection ∇ E = d + ω with ω ≡ 0. The scalar Laplacian is the connection Laplacian on E associated to the connection ∇ E .

In local frames ∂

i := ∂/∂x i of T X and e 1 of E, it is defined by ∆(f e 1 ) = - ij g ij ∂ i ∂ j - k Γ k ij ∂ k f e 1 (2.2)
where f ∈ C ∞ (X), and Γ k ij are the symbols of the Levi-Cevita connection of (T X, g) with respect to the frame ∂ i .

In the following Proposition, we determine the parallel transport map on E related to the connection ∇ E . Proposition 2.2. Let (X, g), E and ∇ E of Definition 2.1, γ be a C 1 curve in X such that γ(0) = y. The parallel transport Y along γ of the vector

Y 0 = Y 1 0 e 1 (y) is Y (t) = Y 1 0 e 1 (γ(t)).
Proof. The parallel transport of Y 0 along γ is the solution

Y (t) = Y 1 (t)e 1 (γ(t)) of the differential equation    ∇ E γ Y (t) = 0 Y (0) = Y 0 (2.3) ∇ E γ Y (t) = ∇ E γ (Y 1 e 1 )(t) = ∂Y 1 ∂t (t) e 1 (γ(t))
Finally, we obtain an ordinary differential equation

     ∂Y 1 ∂t (t) = 0 Y 1 (0) = Y 1 0
from which we deduce the solution of (2.3).

In other words, the parallel transport has no effect on coordinates. As a consequence, the kernel K 0 t (x, y, ∆) equals Å 1 4πt ã m/2 e -d(x,y) 2 /4t Id(x, y) (2.4) on normal neighborhoods of x, where Id(x, y) : E y -→ E x acts as the Identity map on coordinates.

The Laplace-Beltrami operator ∆ g is the differential operator of order 2 acting on smooth functions on a Riemannian manifold (X, g), and defined by

∆ g (f ) = 1 √ g jk ∂ j ( √ gg jk ∂ k f )
Considering the set C ∞ (X) of smooth functions on X as a trivial vector bundle of rank 1 over (X, g) endowed with the connection ∇ E determined by the 1-form ω = 0, we have

∆ g = -∆
This makes the PDE's ∂f /∂t = ∆ g f and ∂f e 1 /∂t + ∆ f e 1 = 0 be equivalent.

The PDE ∂f /∂t = ∆ g f appears in the Beltrami framework in the context of image regularization [START_REF] Sochen | A General Framework for Low Level Vision[END_REF], [START_REF] Spira | A Short-time Beltrami Kernel for Smoothing Images and Manifolds[END_REF]. In [START_REF] Sochen | A General Framework for Low Level Vision[END_REF], the solution is approximated by an Euler scheme of the form (2.1), whereas in [START_REF] Spira | A Short-time Beltrami Kernel for Smoothing Images and Manifolds[END_REF] the solution is approximated by the convolution of the initial condition with a kernel of the form (2.4).

3. Clifford bundles over Riemannian manifolds: a common framework to regularize functions, vector fields, and oriented orthonormal frame fields.

3.1. Clifford bundles: a common framework to treat functions, vector fields and oriented orthonormal frame fields on manifolds. In this section, we show that we can consider functions, vector fields and generators of rotation fields on a manifold X as sections of a trivial Clifford bundle Cl(E, h) over X. Moreover, we can also lift rotation fields to sections of Cl(E, h) called spinor fields. Definition 3.1 (Clifford bundle). Let E be a vector bundle of rank n over a manifold X, and equipped with a metric h. The Clifford bundle Cl(E, h) of (E, h) is the quotient bundle

Cl(E, h) = T (E) I(E, h)
where T (E) is the bundle whose fiber at x ∈ X is the tensor algebra of E x and I(E, h) is the bundle whose fiber at x ∈ X is the two-sided ideal

I(E x , h x ) in T (E x ), generated by elements v ⊗ v + h x (v, v) for v ∈ E x .
We obtain a bundle of rank 2 n over X whose fibers are endowed with a Clifford algebra structure, and the fiberwise multiplication in Cl(E, h) gives an algebra structure to the space Γ(Cl(E, h)) of sections of Cl(E, h).

More precisely, let (e 1 , • • • , e n ) be a local oriented orthonormal frame field of (E, h) on Ω ⊂ X. Then, any section s of Cl(E, h) takes the form

s = s 0 1 + k=1•••n i1<•••<i k s i1•••i k e i1 • • • e i k on Ω, for some functions s 0 , s i1•••i k ∈ C ∞ (Ω).
Example: Let X be a manifold of dimension m endowed with a metric g on its tangent bundle T X. The couple (T X, g) induces a Clifford bundle Cl(T X, g) over X of rank 2 m . The graduation on Clifford algebras carry over Clifford bundles. We denote by Γ

(Cl(E, h) k ) the set of sections s of Cl(E, h) such that s(x) is a k-vector ∀x ∈ X.
Let x ∈ X. From the embeddings of R and E x into the Clifford algebra Cl(E x , h(x)), both functions on X and sections of E may be viewed as sections of the Clifford bundle Cl(E, h). More precisely, we have the identifications

C ∞ (X) Γ(Cl(E, h) 0 ) Γ(E) Γ(Cl(E, h) 1 )
Let us denote by Γ(P SO(E, h)) the set of smooth oriented orthonormal frame fields of (E, h). On a local trivialization U of (E, h), we have the identifications

Γ(P SO(E |U , h |U )) C ∞ (U, SO(n)) Γ(Cl(E |U , h |U ) 2 ) C ∞ (U, so(n))
the latter arising from the Lie algebra isomorphism spin(n) so(n). We call such fields generators of oriented orthonormal frame fields.

From these identifications, functions, vector fields and generators of rotation fields on X may be viewed as sections of a trivial Clifford bundle Cl(E, h) X × R n,0 , respectively of degree 0,1,2.

Indeed, a vector-valued function

v : X -→ R n of components (v 1 , • • • , v n ) may be viewed as a section V of the trivial vector bundle (E, 2 ) = X × R n of the form V (x) = (x, (v 1 (x), • • • , v n (x)))
and consequently as a section of the trivial Clifford bundle Cl(E, 2 ) = X × R n,0 of the form

V (x) = (x, (0, v 1 (x), • • • , v n (x), 0, • • • , 0))
This construction may be generalized endowing the vector bundle E of a definite metric h, in such a way that v may be viewed as a section V of a trivial vector bundle (E, h) X × R n . Let (e 1 , • • • , e n ) be an oriented orthonormal frame field of E with respect to h, and φ be the global trivialization of (E, h) such that

V (x) = V 1 (x)e 1 (x) + • • • + V n (x)e n (x) = φ(x, (v 1 (x), • • • , v n (x))) where (V 1 (x), • • • , V n (x)) are the components of the vector (v 1 (x), • • • , v n (x)) in the basis (e 1 (x), • • • , e n (x)
). As a consequence, v may be viewed as a section of the trivial Clifford bundle Cl(E, h) X × R n,0 of global trivialization Φ, of the form

V (x) = V 1 (x)e 1 (x) + • • • + V n (x)e n (x) = Φ(x, (0, v 1 (x), • • • , v n (x), 0, • • • , 0))
Then, we treat the case of rotation fields and their generators. Let r : X -→ SO(n) be a rotation field on X, considered as a section of the principal bundle X× SO(n) over X. From the covering ξ : Spin(n) -→ SO(n), r may be lifted to a spinor field s, i.e. a section of the principal bundle X× Spin(n). By the embedding of Spin(n) into R n,0 , we have X× Spin(n) ⊂ X × R n,0 . Hence we can consider s as a section of the trivial Clifford bundle X × R n,0 . Moreover, from the Lie algebra isomorphism so(n) R 2 n,0 , we can consider generators of rotation fields on X as sections s of X × R n,0 of the form

s(x) = (x, i<j a ij (x) e i e j )
for some functions a ij , where e i e j ∈ R 2 n,0 are induced by an orthonormal basis

(e 1 , • • • , e n ) of (R n , 2 ).
If we denote by exp the exponential map of Spin(n), the section s given by

s(x) = (x, exp( i<j a ij (x) e i e j ))
is a lifting of the rotation field r, i.e. we have

r(x) = (x, ξ • exp( i<j a ij (x) e i e j ))
Let us detail the construction of the section s. From the matrix representation of SO(n), we can construct generators of rotations r(x), x ∈ X, by the computation of log • r(x) where log : SO(n) -→ so(n) is the logarithm map of the Lie group SO(n). It gives a field of antisymmetric matrices on X. Then the Lie algebra isomorphism so

(n) -→ R 2 n,0 maps log • r(x) = i<j 2a ij (x)E ij to i<j a ij (x) e i e j
We identify the last expression with the section

s(x) = (x, i<j a ij (x) e i e j ))
of X × R n,0 .

Remark: For n ≥ 3, the lifting of the rotation field r to a spinor field s is not unique, due to the 2-sheeted covering Spin(n) -→ SO(n). Moreover, as the exponential map of the group SO(n) is not injective for any n, the field s generating the field r is not unique.

Example:

The case n = 2.
The lifting of a rotation field r to a spinor field s is unique from the isomorphism SO(2) Spin(2). Indeed, the field θ lifts to cos(θ) + sin(θ)e 1 e 2 . For any k ∈ Z, the field

s = (θ + 2kπ)e 1 e 2
is a generator of the field θ.

At last, any scalar field f : X -→ R may be viewed as a section F of a trivial vector bundle Cl(E, h) X × R n,0 of the form

F (x) = f (x)1(x)
In this paper, we are interested in the regularization of functions, vector fields and rotation fields over manifolds. By the above constructions, we can apply the context of heat equations on Clifford bundles to regularize these fields. By this approach, we can control the anisotropy of the regularization by the choice of a Riemannian metric of the base manifold X and the data we regularize by the choice of a connection on the Clifford bundle Cl(E, h), for a well-chosen metric h.

Through the regularization of sections of degree 0, resp. 1, we can regularize functions and vector fields over X. We propose to regularize rotation fields through the regularization of their generators, i.e. sections of degree 2 of Cl(E, 2 ) = X × R n,0 . Indeed, as mentioned above, we can construct a generator s ∈ Γ(Cl(E, 2 ) 2 ) of a rotation field r on X. The regularization of s provides sections st,t≥0 of the form

st (x) = (x, i<j a ij (t, x)e i e j )
Then computing

r t (x) = (x, ξ • exp( i<j a ij (t, x)e i e j ))
we obtain rotation fields r t,t≥0 on X regularizing the initial field r.

Finally, this general method only requires that the regularization process preserves the structure of function, vector field and generator of rotation field. Hence we are looking for heat equations preserving the graduation of Clifford bundles. More precisely, we are looking for PDEs of the form

∂s t ∂t + Hs t = 0
where H is a generalized Laplacian on Cl(E, h) preserving the graduation, i.e.

H : Γ(Cl(E, h) k ) -→ Γ(Cl(E, h) k ).
In the sequel, we show that such differential operators can be constructed from connections preserving the graduation. These connections can be constructed from connections on E compatible the metric h.

3.2.

Connections preserving the graduation. Let E be a vector bundle of rank n equipped with a metric h over a manifold X. Proposition 3.2. A connection ∇ E on E compatible with the metric h induces a connection ∇ C on the Clifford bundle Cl(E, h) preserving the graduation, i.e. a connection satisfying ∀k ∈ {0 • • • n}

∇ E ϕ ∈ Γ(T * X ⊗ Cl(E, h) k ) if ϕ ∈ Γ(Cl(E, h) k )
Proof. Any connection ∇ E on E may be extended in a unique way to a connection ∇ on T (E) by linearity and postulation of the Leibniz rule. For U ∈ Γ(T X) and V ∈ Γ(E), we have For the right term we obtain

∇ U (V ⊗ V -h(V, V )) = 1 2 (∇ U V + V ) ⊗ (∇ U V + V ) -h(∇ U V + V, ∇ U V + V ) - 1 2 (∇ U V -V ) ⊗ (∇ U V -V ) -h(∇ U V -V, ∇ U V -V ) (3.
∇ U V ⊗ V + V ⊗ ∇ U V -2h(∇ U V, V )
Developping the left term we have

∇ U (V ⊗ V -h(V, V )) = ∇ U V ⊗ V + V ⊗ ∇ U V -d U h(V, V )
However, by definition of a connexion compatible with the metric of a vector bundle, we have for any U ∈ Γ(T X) and W, Z ∈ Γ(E)

d U h(W, Z) = h(∇ U W, Z) + h(W, ∇ U Z) Then, taking W = Z = V , it gives d U h(V, V ) = 2h(∇ U V, V )
and the equality (3.1) is proved.

As the right term of (3.1) belongs to Γ(I(E, h)), it proves that ∇ preserves the Γ(I(E, h)).

We deduce that ∇ induces a connection ∇ C on Cl(E, h). Indeed, for a, b ∈ Γ(T (E)) in the same equivalence class (denoted by ȧ = ḃ), we have

∇ U b = ∇ U (a + I 1 ) for some I 1 ∈ Γ(I(E, h)). Hence ∇ U b = ∇ U a + I 2 for some I 2 ∈ Γ(I(E, h)). Therefore ∇ (a) = ∇ (b)
, and ∇ C is well-defined.

Let ϕ, ψ ∈ Γ(Cl(E, h)). By definition, there exist a, b ∈ Γ(T (E)) such that ϕ = ȧ and ψ = ḃ. Then,

∇ C (ϕ ψ) = ˙ ∇(a ⊗ b) = ˙ ∇(a) ⊗ b + a ⊗ ∇(b) = ˙ ∇(a) ⊗ b + ˙ a ⊗ ∇(b) = ∇ (a) ḃ + ȧ ∇ (b) = ∇ C (ϕ)ψ + ϕ∇ C (ψ)
from which follows that ∇ C preserves the graduation.

Remark: Dealing with the tangent bundle (T X, g) of a Riemannian manifold, the Levi-Cevita connection ∇ on (T X, g) is compatible with g. It follows that ∇ induces a connection ∇ C on Cl(T X, g) preserving the graduation [START_REF] Blau | Connections on Clifford bundles and the Dirac operator[END_REF].

As a consequence, from a connection compatible with the metric h on E and a zero-order operator F on Cl(E, h) preserving the graduation, we can construct a generalized Laplacian H on Cl(E, h) preserving the graduation, by the formula (see Appendix B)

H = ∆ C + F
where ∆ C is the connection Laplacian related to the connection ∇ C on Cl(E, h) induced by the connection ∇ E .

By this method, we construct in the sequel two generalized Laplacians on Clifford bundles preserving the graduation, and restricting to the Laplace-Beltrami on functions (i.e. sections of degree 0), called Clifford-Beltrami and Clifford-Hodge operators.

3.3. The Clifford-Beltrami operator. Let X be a manifold of dimension m, endowed with a Riemannian metric g on its tangent bundle T X. Over the Riemannian manifold (X, g), we consider the vector bundle T X endowed with the metric g given by the identity matrix I m in local frames (∂/∂x 1 , • • • , ∂/∂x m ) induced by local coordinates system (x 1 , • • • , x m ) on X. The vector bundle (T X, g) over X induces a Clifford bundle Cl(T X, g) over X. A local section s of Cl(T X, g) is of the form

s = s 0 1 + k=1•••m i1<•••<i k s i1•••i k ∂/∂x 1 • • • ∂/∂x k for some functions s 0 , s i1•••i k ∈ C ∞ (X).
The Levi-Cevita connection ∇ on (T X, g) is determined by ω ≡ 0 by construction of g. From Section 3.2, it induces an algebra connection ∇ C on Cl(T X, g) satisfying

∇ C s = ds 0 1 + k=1•••m i1<•••<i k ds i1•••i k ∂/∂x 1 • • • ∂/∂x k
In fact, ∇ C corresponds in this particular case to the differentation of components on Γ(Cl(T X, g)).

Then we construct the connection Laplacian ∆ C on Cl(T X, g) related to ∇ C . It takes the form

∆ C (s) = -∆ g s 0 1 - k=1•••m i1<•••<i k ∆ g s i1•••i k ∂/∂x 1 • • • ∂/∂x k
It consists in the action of minus the Laplace-Beltrami operator related to g on each component of a section. In this paper, we call this operator the Clifford-Beltrami operator.

As the Clifford-Beltrami operator is a generalized Laplacian, we may consider the corresponding heat equation, whose solution are provided by the convolution of the initial conditions with the heat kernel K t (x, y, ∆ C ) of ∆ C . Definition 3.3. Let s ∈ Γ(Cl(T X, g)). The Clifford-Beltrami flow of s is the solution s t, t≥0 of the heat equation

∂s t ∂t + ∆ C s t = 0, s 0 = s
Then, as the Clifford-Beltrami operator preserves the graduation on Cl(T X, g), it preserves the structures of function, vector field and generator of orthonormal frame field. As a consequence, the Clifford-Beltrami flow provides a common tool to regularize functions, vector fields and oriented orthonormal frame fields.

3.4. The Clifford-Hodge operator. Let (X, g) be a Riemannian manifold. We denote by T * X the vector bundle of differential forms on (X, g). The vector space isomorphism (C.2) between V * and Cl(V, Q) carries over vector bundles. It follows a canonical vector bundle isomorphism between T * X and Cl(T X, g) that maps differential forms of degree k to sections of degree k of Cl(T X, g).

The Hodge Laplacian ∆ is a generalized Laplacian acting on differential forms on a Riemannian manifold (X, g). It is defined by

∆ = dδ + δd
where d is the exterior derivative operator and δ its formal adjoint [START_REF] Rham | Variétés différentiables: Formes, Courants, Formes harmoniques[END_REF]. In particular, when applied to 0-forms, i.e. functions, ∆ corresponds to minus the Laplace-Beltrami operator.

Under the identification between T * X and Cl(T X, g), the Hodge Laplacian can also be applied to Γ(Cl(T X, g)), and consequently to vector fields and generators of orthonormal frame fields. In the context of Clifford bundles, we call this operator the Clifford-Hodge operator. It is defined as the square of a first order differential operator called the Dirac operator. Definition 3.4. Let ∇ C be the connection on Cl(T X, g) induced by the Levi-Cevita connection on (T X, g). Let (e 1 , • • • , e m ) be a local oriented orthormal frame field of T X with respect to g. The Dirac operator is the first-order differential operator D : Γ(Cl(T X, g)) -→ Γ(Cl(T X, g)) defined locally by

Dσ = m i=1 e i ∇ C e i σ (3.2)
This definition is independant of the choice of the local oriented orthonormal frame field.

By the Bochner identity, the Clifford-Hodge operator D 2 is a generalized Laplacian on Cl(T X, g) preserving the graduation. Indeed, we have

D 2 = ∆ C + i<j e i e j R ei,ej
where i<j e i e j R ei,ej is a zero-order operator preserving the graduation. The term R ei,ej is called the curvature transformation of ∇ C associated to e i and e j [START_REF] Lawson | Spin Geometry[END_REF].

As the operator D 2 is a generalized Laplacian, we may consider the corresponding heat equation, whose solution are provided by the convolution of the initial conditions with the heat kernel K t (x, y, D 2 ) of D 2 .

Definition 3.5. Let s ∈ Γ(Cl(X, g)). The Clifford-Hodge flow of s is the solution s t,t≥0 of the heat equation

∂s t ∂t + D 2 s t = 0, s 0 = s
Then, as the Clifford-Hodge operator preserves the graduation on Cl(T X, g), it preserves the structures of function, vector field and generator of orthonormal frame field. As a consequence, the Clifford-Hodge flow provides a common tool to regularize functions, vector fields and oriented orthonormal frame fields.

Applications.

In this paper, we approximate the solution of the heat equation associated to a generalized Laplacian H by the convolution of the initial condition with the kernel K 0 t (x, y, H) approximating the heat kernel of H near the diagonal. This kernel is determined by the tranport parallel map associated to the connection ∇ E such that H = ∆ E + F (see Appendix B), and by geodesic distances on the base manifold. In this Section we compute parallel transport maps on Cl(T X, g) and Cl(T X, g) related to the Clifford-Beltrami and Clifford-Hodge operators for base manifolds of dimension 2. Then we present applications in the context of image processing.

Clifford-Beltrami operator and subsequent parallel transport map.

We denote by (∂ x1 , ∂ x2 ) the frame field (∂/∂x 1 , ∂/∂x 2 ). From Section 3.3, we have

∇ C ∂x 1 1 = 0 ∇ C ∂x 2 1 = 0 ∇ C ∂x 1 ∂ x1 = 0 ∇ C ∂x 2 ∂ x1 = 0 ∇ C ∂x 1 ∂ x2 = 0 ∇ C ∂x 2 ∂ x2 = 0 ∇ C ∂x 1 ∂ x1 ∂ x2 = 0 ∇ C ∂x 2 ∂ x1 ∂ x2 = 0 Then, for ϕ = ϕ 1 1 + ϕ 2 ∂ x1 + ϕ 3 ∂ x2 + ϕ 4 ∂ x1 ∂ x2 we have ∆ C ϕ = -(∆ g ϕ 1 )1 -(∆ g ϕ 2 )∂ x1 -(∆ g ϕ 3 )∂ x2 -(∆ g ϕ 4 )∂ x1 ∂ x2 Proposition 4.1 (Parallel transport on Cl(T X, g)).
Let X be a manifold of dimension 2. Let ∇ C be the connection on Cl(T X, g) induced by the Levi-Cevita connection on (T X, g).

Let Y 0 = Y 1 0 1(y)+Y 2 0 ∂ x1 (y)+Y 3 0 ∂ x2 (y)+Y 4 0 ∂ x1 ∂ x2 (y) ∈ Cl(T X, g) y , and γ be a C 1 curve in X such that γ(0) = y. The parallel transport Y of Y 0 along γ is Y (t) = Y 1 0 1(γ(t)) + Y 2 0 ∂ x1 (γ(t)) + Y 3 0 ∂ x2 (γ(t)) + Y 4 0 ∂ x1 ∂ x2 (γ(t))
Proof. The parallel transport of Y 0 along γ is the solution

Y (t) = Y 1 (t) 1(γ(t)) + Y 2 (t) ∂ x1 (γ(t)) + Y 3 (t) ∂ x2 (γ(t)) + Y 4 (t) ∂ x1 ∂ x2 (γ(t)) of the differential equation    ∇ C γ Y (t) = 0 Y (0) = Y 0 (4.1) ∇ C γ Y (t) = ∇ C γ Y 1 1 + Y 2 ∂ x1 + Y 3 ∂ x2 + Y 4 ∂ x1 ∂ x2 (t) = ∂Y 1 ∂t (t) 1(t) + Y 1 (t)( γ1 (t)∇ C ∂x 1 1 (t) + γ2 (t)∇ C ∂x 2 1 (t)) + ∂Y 2 ∂t (t) ∂ x1 (t) + Y 2 (t)( γ1 (t)∇ C ∂x 1 ∂ x1 (t) + γ2 (t)∇ C ∂x 2 ∂ x1 (t)) + ∂Y 3 ∂t (t) ∂ x2 (t) + Y 3 (t)( γ1 (t)∇ C ∂x 1 ∂ x2 (t) + γ2 (t)∇ C ∂x 2 ∂ x2 (t)) + ∂Y 4 ∂t (t) ∂ x1 ∂ x2 (t) + Y 4 (t)( γ1 (t)∇ C ∂x 1 ∂ x1 ∂ x2 (t) + γ2 (t)∇ C ∂x 2 ∂ x1 ∂ x2 (t)) = ∂Y 1 ∂t (t) 1(t) + ∂Y 2 ∂t (t) ∂ x1 (t) + ∂Y 3 ∂t (t) ∂ x2 (t) + ∂Y 4 ∂t (t) ∂ x1 ∂ x2 (t)
Finally, we obtain four ordinary differential equations on R of the form ∂Y i /∂t = 0 for i = 1 • • • 4, from which we deduce the parallel transport of Y 0 along γ.

4.2.

Clifford-Hodge operator and subsequent parallel transport map. Let us denote by Γ k ij the Levi-Cevita connection's symbols of (T X, g) with respect to a local orthonormal frame field (e 1 , e 2 ). By Section 3.2, we have

∇ C e1 1 = 0 ∇ C e2 1 = 0 ∇ C e1 e 1 = Γ 2 11 e 2 ∇ C e1 e 2 = -Γ 2 11 e 1 ∇ C e2 e 1 = Γ 2 21 e 2 ∇ C e2 e 2 = -Γ 2 21 e 1 ∇ C e1 e 1 e 2 = 0 ∇ C e2 e 1 e 2 = 0
Proposition 4.2. Let (X, g) be a Riemannian manifold of dimension 2. Let (e 1 , e 2 ) be a local oriented orthonormal frame field of (T X, g) and ϕ = ϕ 1 1 + ϕ 2 e 1 + ϕ 3 e 2 + ϕ 4 e 1 e 2 ∈ Γ(Cl(T X, g)). Then Proof. We obtain D 2 (ϕ) from (3.2) and the relations above defining ∇ C in the frame (e 1 , e 2 ). Then we simplify the expression using some properties of the Levi-Cevita connection: (i) In an orthonormal frame, its symbols satisfy

D 2 (ϕ 1 1) = -d 2 e 1 ,e 1 ϕ 1 -d 2 e 2 ,e 2 ϕ 1 -Γ 2 21 d e 1 ϕ 1 + Γ 2 11 d e 2 ϕ 1 1 = -∆ g (ϕ 1 )1 D 2 (ϕ 2 e 1 + ϕ 3 e 2 ) = -d 2 e 1 ,e 1 ϕ 2 -d 2 e 2 ,
Γ k ij = -Γ j ik . (ii) [e 1 , e 2 ] = -Γ 2 11 e 1 -Γ 2 21 e 2 .
Proposition 4.3 (Parallel transport on Cl(T X, g)). Let (X, g) be a Riemannian manifold of dimension 2. Let (e 1 , e 2 ) be a local oriented orthonormal frame of T X, and

Y 0 = Y 1 0 1(y) + Y 2 0 e 1 (y) + Y 3 0 e 2 (y) + Y 4 0 e 1 e 2 (y) ∈ Cl(T X, g) y . Let γ be a C 1 curve in X such that γ(0) = y. The parallel transport Y of Y 0 along γ is Y (t) = Y 1 0 1(γ(t)) + Re exp i t 0 γ1 (s) Γ 2 11 (s) + γ2 (s) Γ 2 21 (s) ds (Y 2 0 + i Y 3 0 ) e 1 (γ(t)) +Im exp i t 0 γ1 (s) Γ 2 11 (s)+ γ2 (s) Γ 2 21 (s) ds (Y 2 0 +i Y 3 0 ) e 2 (γ(t))+Y 4 0 e 1 e 2 (γ(t))
Proof. The parallel transport of Y 0 along γ is the solution

Y (t) = Y 1 (t) 1(γ(t)) + Y 2 (t) e 1 (γ(t)) + Y 3 (t) e 2 (γ(t)) + Y 4 (t) e 1 e 2 (γ(t)) of the differential equation    ∇ C γ Y (t) = 0 Y (0) = Y 0 (4.2) ∇ C γ Y (t) = ∇ C γ Y 1 1 + Y 2 e 1 + Y 3 e 2 + Y
          dY 1 /dt dY 2 /dt dY 3 /dt dY 4 /dt           =           0 0 0 0 0 0 γ1 Γ 2 11 + γ2 Γ 2 21 0 0 -γ1 Γ 2 11 -γ2 Γ 2 21 0 0 0 0 0 0                     Y 1 Y 2 Y 3 Y 4           of initial condition Y 1 (0) = Y 1 0 , Y 2 (0) = Y 2 0 , Y 3 (0) = Y 3 0 and Y 4 (0) = Y 4 0 . It leads to Y 1 (t) = Y 1 0 , Y 4 (t) = Y 4 0 and a differential equation on R 2 Ñ dY 2 /dt dY 3 /dt é = Ñ 0 γ1 Γ 2 11 + γ2 Γ 2 21 -γ1 Γ 2 11 -γ2 Γ 2 21 0 é Ñ Y 2 Y 3 é (4.3) of initial condition Y 2 (0) = Y 2 0 and Y 3 (0) = Y 3 0 .
Under the identification between R 2 and C, (4.3) becomes

   ∂(Y 2 + i Y 3 )/∂t = i ( γ1 Γ 2 11 + γ2 Γ 2 21 )(Y 2 + i Y 3 ) Y 2 (0) + i Y 3 (0) = Y 2 0 + i Y 3 0 (4.4) T. BATARD
The solution of (4.4) is

Y 2 (t) + i Y 3 (t) = exp i t 0 γ1 (s) Γ 2 11 (s) + γ2 (s) Γ 2 21 (s) ds (Y 2 0 + i Y 3 0 ) (4.5)
from which follows the parallel transport of Y 0 along γ.

4.3.

The particular context of images. Let us consider a n-channels image

I : (x 1 , x 2 ) -→ (I 1 (x 1 , x 2 ), • • • , I n (x 1 , x 2 )) defined on a domain Ω ⊂ R 2 . I determines a surface S embedded in R n+2 parametrized by ϕ : (x 1 , x 2 ) -→ (x 1 , x 2 , I 1 (x 1 , x 2 ), • • • , I n (x 1 , x 2 )) Then we endow R n+2 of a metric h of matrix representation diag(1, 1, h 1 , • • • , h n )
where h 1 , • • • , h n are positive functions. We denote by g the metric on S induced by h. This construction makes the couple (S, g) be a Riemannian manifold of dimension 2 of global chart (Ω, ϕ).

On the trivial vector bundle T S, the natural global frame is (∂/∂x 1 , ∂/∂x 2 ) induced by the cartesian coordinates system (x 1 , x 2 ) on Ω. However, the Clifford-Hodge operator D 2 on Cl(T S, g) is defined with respect to an oriented orthonormal frame field of (T S, g) (see Section 3.4). In what follows, we construct an oriented orthonormal frame field (e 1 , e 2 ) of (T S, g) and compute the transformation of Levi-Cevita connection's symbols with respect to the frame change from (∂/∂x 1 , ∂/∂x 2 ) to (e 1 , e 2 ). Let λ + and λ -(λ + ≥ λ -) be the two eigenvalues of the induced endomorphism. Then a positively oriented orthonormal basis (e 1 , e 2 ) of T p S may be constructed from eigenvectors, distinguishing four cases:

(i) if F = 0, take (e 1 , e 2 ) = àà λ + -G √ λ + F 2 + (λ + -G) 2 F √ λ + F 2 + (λ + -G) 2 í , sign(F ) à λ --G √ λ -F 2 + (λ --G) 2 F √ λ -F 2 + (λ --G) 2 íí in the basis (∂/∂x 1 , ∂/∂x 2 )(p). (ii) if F = 0 and E > G, take (e 1 , e 2 ) = ∂/∂x 1 (p) √ E , ∂/∂x 2 (p) √ G (iii) if F = 0 and E < G, take (e 1 , e 2 ) = ∂/∂x 2 (p) √ G , - ∂/∂x 1 (p) √ E
(iv) if F = 0 and E = G, the whole space T p S is eigenspace. Then for any θ, the two vectors àà cos(θ)

√ E sin(θ) √ E í , à -sin(θ) √ E cos(θ) √ E íí in the basis (∂/∂x 1 , ∂/∂x 2 )(p)
form a positively oriented orthonormal basis of T p S.

Proof. As unit eigenvectors, they form an orthonormal basis of T p S. The orientation is clearly positive in the cases (ii), (iii) and (iv). For the case (i), one just need to compute the 2-form

ω = λ + -G √ λ + F 2 + (λ + -G) 2 dx 1 + F √ λ + F 2 + (λ + -G) 2 dx 2 ∧ λ --G √ λ -F 2 + (λ --G) 2 dx 1 + F √ λ -F 2 + (λ --G) 2 dx 2 Then ω = F (λ + -λ -) √ λ + λ -F 2 + (λ + -G) 2 F 2 + (λ --G) 2 dx 1 ∧ dx 2
and the sign of the scalar term is given by the sign of F .

Following this construction for each p ∈ S, we obtain a positively oriented orthonormal frame field (e 1 , e 2 ) of (T S, g), where e 1 is the unit vector field of highest variations (eigenvectors associated to the eigenvalues λ + ) and e 2 the unit vector field of lowest variations (eigenvectors associated to the eigenvalues λ -).

By the antisymmetry property of its symbols Γ k ij in an orthonormal frame field, the Levi-Cevita connection is entirely determined by the symbols Γ 2 11 and Γ 2 21 in such frames. In the next proposition, we determine the expressions of these two symbols in a frame field (v 1 , v 2 ) in function of the symbols Γ k ij of the connection in the frame field (∂/∂x 1 , ∂/∂x 2 ).

Proposition 4.5. Let (v 1 , v 2 ) be a frame field such that v 1 = a ∂/∂x 1 + b ∂/∂x 2 and v 2 = c ∂/∂x 1 + d ∂/∂x 2 . Then Γ 2 11 = 1/(ad -bc)× -ab ∂a ∂x 1 -a 2 b Γ 1 11 -2ab 2 Γ 1 12 -b 2 ∂a ∂x 2 -b 3 Γ 1 22 +a 3 Γ 2 11 +a 2 ∂b ∂x 1 +2a 2 b Γ 2 12 +ab ∂b ∂x 2 +ab 2 Γ 2 22 Γ 2 21 = 1/(ad -bc)× -bc ∂a ∂x 1 -acb Γ 1 11 -(bc + ad)b Γ 1 12 -bd ∂a ∂x 2 -b 2 d Γ 1 22 + a 2 c Γ 2 11 + ac ∂b ∂x 1 + (bc + ad)a Γ 2 T. BATARD +ad ∂b ∂x 2 + abd Γ 2 22
Proof. By definition, we have

∇ v 1 v 1 = Γ 1 11 v 1 + Γ 2 11 v 2 ∇ v 2 v 1 = Γ 1 21 v 1 + Γ 2 21 v 2
With respect to the frame field (∂ x1 , ∂ x2 ) := (∂/∂x 1 , ∂/∂x 2 ), we obtain

∇ v 1 v 1 = ∇ a∂x 1 +b∂x 2 a ∂ x1 + b ∂ x2 = a∇ ∂x 1 a ∂ x1 + a∇ ∂x 1 b ∂ x2 + b∇ ∂x 2 a ∂ x1 + b∇ ∂x 2 b ∂ x2 = a ∂a ∂x 1 ∂ x1 + a Γ 1 11 ∂ x1 + Γ 2 11 ∂ x2 + a ∂b ∂x 1 ∂ x2 + b Γ 1 12 ∂ x1 + Γ 2 12 ∂ x2 +b ∂a ∂x 2 ∂ x1 + a Γ 1 21 ∂ x1 + Γ 2 21 ∂ x2 + b ∂b ∂x 2 ∂ x2 + b Γ 1 22 ∂ x1 + Γ 2 22 ∂ x2 = a ∂a ∂x 1 +a 2 Γ 1 11 +2ab Γ 1 12 +b ∂a ∂x 2 +b 2 Γ 1 22 ∂ x1 + a ∂b ∂x 1 +a 2 Γ 2 11 +2ab Γ 2 12 +b ∂b ∂x 2 +b 2 Γ 2 22 ∂ x2
Then, since

∂ x1 = 1 ad -bc (d v 1 -b v 2 ) ∂ x2 = 1 ad -bc (-c v 1 + a v 2 )
we obtain

∇ v 1 v 1 = 1 ad -bc ad ∂a ∂x 1 + a 2 d Γ 1 11 + 2abd Γ 1 12 + bd ∂a ∂x 2 + b 2 d Γ 1 22 -a 2 c Γ 2 11 -ac ∂b ∂x 1 -2abc Γ 2 12 -bd ∂b ∂x 2 -b 2 c Γ 2 22 v 1 + 1 ad -bc -ab ∂a ∂x 1 -a 2 b Γ 1 11 -2ab 2 Γ 1 12 -b 2 ∂a ∂x 2 -b 3 Γ 1 22 + a 3 Γ 2 11 + a 2 ∂b ∂x 1 +2a 2 b Γ 2 12 + ab ∂b ∂x 2 + ab 2 Γ 2 22 v 2
from which we deduce Γ 2 11 , and

∇ v 2 v 1 = 1 ad -bc cd ∂a ∂x 1 + acd Γ 1 11 + (bc + ad)d Γ 1 12 + d 2 ∂a ∂x 2 + bd 2 Γ 1 22 -ac 2 Γ 2 11 -c 2 ∂b ∂x 1 -(bc + ad)c Γ 2 12 -cd ∂b ∂x 2 -cbd Γ 2 22 v 1 + 1 ad -bc -bc ∂a ∂x 1 -acb Γ 1 11 -(bc + ad)b Γ 1 12 -bd ∂a ∂x 2 -b 2 d Γ 1 22 + a 2 c Γ 2 11 +ac ∂b ∂x 1 + (bc + ad)a Γ 2 12 + ad ∂b ∂x 2 + abd Γ 2 22 v 2
from which we deduce Γ 2 21 .

From this proposition, we deduce the expression of the Levi-Cevita's connection symbols in the frame field (e 1 , e 2 ).

Experiments.

We apply the Clifford-Beltrami and Clifford-Hodge flows in the context of image processing. The base manifold X we consider is the surface S embedded in R 5 parametrized by the graph of a color image I = (I 1 , I 2 , I 3 ) given with its RGB components. We endow S with the Riemannian metric g induced by the metric h on R 5 , for some functions h i (see Section 4.3). Then we construct the trivial Clifford bundles Cl(T S, g) = S × R 2,0 and Cl(T S, g) S × R 2,0 over (S, g).

Clifford-Beltrami flow in Γ(Cl(T S, g) 0 ) and Clifford-Hodge flow in Γ(Cl(T S, g) 0 ) can be applied to regularize the color image I. Indeed we can consider each component I i of I as a section of degree 0 of the Clifford bundles Cl(T S, g) and Cl(T S, g), of the form I i 1. Then the heat equations associated to Clifford-Beltrami ∆ C and Clifford-Hodge D 2 operators lead in both cases to the following 3 PDEs

∂I i t ∂t = ∆ g I i t , I i 0 = I i
Finally, we obtain the 3 PDEs of Beltrami framework of Sochen et al. in the context of color image regularization (see e.g. [START_REF] Sochen | A General Framework for Low Level Vision[END_REF], [START_REF] Spira | A Short-time Beltrami Kernel for Smoothing Images and Manifolds[END_REF]). Clifford-Beltrami flow in Γ(Cl(T S, g) 1 ) and Clifford-Hodge flow in Γ(Cl(T S, g) 1 ) can be applied to regularize vector fields related to I. Indeed, let v = (v 1 , v 2 ) be a vector field on Ω. By the global chart ϕ of S, it can be considered as a tangent vector field on S. Then, v is the section v 1 ∂/∂x 1 +v 2 ∂/∂x 2 of Cl(T S, g) and the section ṽ1 e 1 +ṽ 2 e 2 of Cl(T S, g) where (e 1 , e 2 ) is the oriented orthonormal frame field of (T S, g) constructed in Section 4.3. Then we consider the heat equations related to ∆ C on Cl(T S, g) and Results are given by the computations of k 0 t (x, y, ∆ C )v and k 0 t (x, y, D 2 )v, where convolutions are done on 5x5 neighborhoods. Fig. 4.3 shows results for t = 0.3 after 200 iterations where g is induced by h i = 0.1. Fig. 4.4 shows results for t = 0.3 after 200 iterations where g is induced by h i = 0.01.

D 2 on Cl(T S, g) of initial condition v 0 = v ∂v t ∂t + ∆ C v t = 0 ∂v t ∂t + D 2 v t = 0 T. BATARD (a) (b)
The role of the functions h i is to control the anisotropy of the diffusion. Indeed, we have explained throughtout the paper that the anisotropy of the diffusion is determined by the metric g of the base manifold. In this context, g is determined by the functions h i . Therefore, the more the functions h i are low, the more the diffusion is isotropic, as it can be seen comparing Clifford-Beltrami flow in Γ(Cl(T S, g) 2 ) and Clifford-Hodge flow in Γ(Cl(T S, g) 2 ) can be respectively applied to regularize rotation fields in (T S, g) and (T S, g). As we deal with vector bundle of rank 2, rotation fields can be given by unit vector fields. We propose to compare regularizations of unit vector fields depending on they are treated as vector fields or as rotation fields. By our choice of the unit vector field treated in this paper (the unit vector field of edge orientations), the Clifford-Hodge flow can not be considered in this application. Indeed, unit vector fields in the frame (∂/∂x 1 , ∂/∂x 2 ) are not unit in the frame (e 1 , e 2 ). Therefore, we are only concerned with the Clifford-Beltrami flow.

Let ψ be the rotation field given by v = (cos ψ, sin ψ) on Ω ⊂ R 2 , where v is the unit vector field of edge orientations (see Fig. 4.2(b)). Following the method of Section 3.1, we construct the section

‹ Ψ : x -→ (x, ψ(x) ∂/∂x 1 ∂/∂x 2 ) ∈ Γ(Cl(T S, g) 2 )
such that for each x ∈ S, ψ(x) is an infinitesimal generator of the rotation ψ(x). The 2π periodicity in the choice of ψ is discussed in the Remark below. From Section 4.1, the Clifford-Beltrami flow of ‹ Ψ leads to

∂ ψt ∂t = ∆ g ψt , ψ0 = ψ
Remark: The 2π periodicity in the choice of ψ may be removed by constructing ‹ Ψ not as a global section of Γ(Cl(T S, g)), but locally on small neighborhoods. Indeed, for x 0 ∈ S, we state ψ(x) = ψ(x) -2π if ψ(x) -ψ(x 0 ) ≥ π, ψ(x) = ψ(x) + 2π if ψ(x) -ψ(x 0 ) < -π and ψ(x) = ψ(x) otherwise. By this construction ψ takes values locally in [ψ(x 0 ) -π, ψ(x 0 ) + π[. Hence, it makes sense to the computation of (k 0 t ‹ Ψ)(x 0 ). Then we extend this construction for each point in S. This method can be viewed as the Lie group counterpart of the choice of maximal normal coordinates systems on manifolds. Indeed, we always work in the tangent space of the neutral element and consider the exponential map of the Lie group. , the regularization provides similar orientations but the unit norm of the initial unit vector field is preserved when it is treated as an oriented orthonormal frame field.

Conclusion.

In this paper, we have proposed a new framework to treat scalar, vector and oriented orthonormal frame fields on manifolds, by considering Clifford bundles. We have shown that scalar and vector fields can be viewed as sections of Clifford bundles, respectively of degree 0 and 1, and that oriented orthonormal frame fields can be lifted to sections of Clifford bundles called spinor fields. We have also shown that sections of degree 2 can be identified with generators of orthonormal frame fields. In this paper, we were particulary concerned with the problem of regularization of these fields. Using the framework of heat equations associated to generalized Laplacians on vector bundles over Riemannian manifolds, we have shown that the behaviour of the regularization of these fields is determined by the choice of a connection on a Clifford bundle and a Riemannian metric on the base manifold. We have considered the Clifford-Beltrami and Clifford-Hodge flows generalizing the Beltrami flow to sections of Clifford bundles. Dealing with base manifolds of dimension 2, we have shown applications in the context of image processing. By the choice of different metrics and connections, different regularizations can be performed. Moreover, by the choice of other base manifolds and vector bundles, regularization of fields in other contexts can be envisaged. For instance, dealing with base manifolds of dimension 3, we can extend the applications presented in this paper to the context of video processing.

Appendix A. Differential Geometry of Vector Bundles.

See [START_REF] Greub | Connections, Curvature and Cohomology[END_REF], [START_REF] Helgason | Differential geometry, Lie groups, and symmetric spaces[END_REF], [START_REF] Husemoller | Fibre Bundles[END_REF], [START_REF] Spivak | A Comprehensive Introduction to Differential Geometry[END_REF] for some references on differential geometry of manifolds and vector bundles.

Definition A.1. A smooth vector bundle of rank n is a triplet (E, π, X) where X and E are two C ∞ manifolds, and π : E -→ X is a surjective map such that the preimage π -1 (x) of x ∈ X is endowed with a structure of vector space of dimension n. X is called the base manifold and E the total space of the vector bundle. The set π -1 (x) is called the fiber over x, and is denoted by E x .

The vector bundle is said to be locally trivial if the following conditions hold: for each x ∈ X , there is a neighborhood U of x and a diffeomorphism φ : U × R n -→ π -1 (U ) satisfying π • φ(x, f ) = x, and such that the map φ x : R n -→ E x is a vector space isomorphism. The couple (U, φ) is called a local trivialization.

The vector bundle is said to be trivial if there exists a diffeomorphism Φ : X × R n -→ E such that π • Φ(x, f ) = x, and Φ x : R n -→ E x is a vector space isomorphism.

Example: Let X be a C ∞ manifold of dimension m. The disjoint union of tangent spaces T X := T x X for x ∈ X, is the total space of a vector bundle (T X, π, X) of rank m called the tangent bundle of X. Tangent space T x X is the fiber over x. Definition A.2. A metric h on a vector bundle is the assigment of a scalar product h x on each fiber π -1 (x).

Example: A Riemannian metric on a manifold is a definite positive metric on its tangent bundle.

Definition A.3. A section of a smooth vector bundle (E, π, X) is a differentiable map S :

X -→ E such that π • S = Id X . Let (f 1 , • • • , f n ) be a basis of R n .
In a local trivialization (U, φ) of (E, π, X), any section may be written

S(x) = n i=1 s i (x)φ(x, f i ) for some functions s i ∈ C ∞ (X). The set {φ(•, f 1 ), • • • , φ(•, f n )} is called a local frame of (E, π, X).
The set of sections of (E, π, X) is denoted by Γ(E).

Example: Tangent vector fields on X are the sections of the tangent bundle (T X, π, X).

Definition A.4. A connection on (E, π, X) is a map ∇ E : Γ(T X) × Γ(E) -→ Γ(E) satisfying the following axioms: -∇ E f u+gv Y = f ∇ E u Y + g∇ E v Y -∇ E u f Y = (d u f )Y + f ∇ E u Y for f, g ∈ C ∞ (X), u, v ∈ Γ(T X) and Y ∈ Γ(E).
Hence, a connection on (E, π, X) may be written as d+ω, where d is the differentation of components and ω ∈ Γ(T * X ⊗ End(E)).

In local frames

(e 1 , • • • , e n ) of E and (u 1 , • • • , u m ) of T X, a connection is determined by n 2 × m functions Υ k ij such that ∇ E ui e j = n k=1 Υ k ij e k
Example: The Levi-Cevita connection is the connection on the tangent bundle of a Riemannian manifold (X, g) determined by the m 3 functions

Γ k ij = 1 2 g kl (∂ j g li + ∂ i g lj -∂ l g ij ) with respect to the local frame (∂/∂x 1 , • • • , ∂/∂x m ) of T X given by a local coordi- nates system (x 1 , • • • , x m ) of X.
Definition A.5. Let ∇ E be a connection on a vector bundle (E, π, X), and γ a C 1 curve in X such that γ(0) = y. The parallel transport of Y 0 ∈ E y along the curve γ is the section Y (t) that is solution of the following differential equation

   ∇ E γ Y (t) = 0 Y (0) = Y 0 A section Y along a curve γ is parallel if ∇ E γ Y (t) = 0 for each t.
Example: Let (X, g) be a Riemannian manifold endowed with the Levi-Cevita connection ∇ on its tangent bundle. A Geodesic curve on X is a C 1 curve γ whose tangent vector field γ is parallel along γ, i.e. ∇ γ γ = 0.

Appendix B. Heat Kernels of Generalized Laplacians.

We refer to [START_REF] Berline | Heat Kernels and Dirac Operators[END_REF] for more details on this part.

Definition B.1. Let E be a vector bundle over a Riemannian manifold (X, g), endowed with a connection ∇ E . Let ∇ be the Levi-Cevita connection of (T X, g). To any pair of tangent vector fields V and W on X, we associate an invariant second derivative

∇ 2 V,W : Γ(E) -→ Γ(E) by setting ∇ 2 V,W ϕ ≡ ∇ E V ∇ E W ϕ -∇ E ∇ V W ϕ
Then the connection Laplacian ∆ E : Γ(E) -→ Γ(E) is defined by

∆ E ϕ = -trace(∇ 2 •,• ϕ)
where trace denotes the contraction with the metric g.

In particular, if e i is a local orthonormal frame of T X, the operator ∆ E is given by

∆ E = - i ∇ E ei ∇ E ei -∇ E ∇ e i e i
With respect to the local frame ∂ i := ∂/∂x i defined by a local coordinates system of X, we have

∆ E = - ij g ij ∇ E ∂ i ∇ E ∂ j - k Γ k ij ∇ E ∂ k
where the symbols Γ k ij are defined by

∇ ∂ i ∂ j = k Γ k ij ∂ k .
Definition B.2. Let E be a vector bundle over a Riemannian manifold (X, g). A Generalized Laplacian on E is a second-order differential operator H : Γ(E) -→ Γ(E) that may be written

H = ∆ E + F
for some connection ∇ E and a zero-order operator F .

In particular, any connection Laplacian is a generalized Laplacian.

To any generalized Laplacian H on a vector bundle E over a compact manifold X, one may associate an operator e -tH : Γ(E) -→ Γ(E), for t > 0, with the property that u t (x) = e -tH u(x) satisfies the heat equation

∂u t ∂t + Hu t = 0, u 0 = u
We shall define e -tH as an integral operator of the form (e -tH u)(x) = X K t (x, y, H)u(y)dy where K t (x, y, H) : E y -→ E x is a linear map depending smoothly on x, y and t. It is called the heat kernel of H.

In the following theorem, we summarize some results on approximations of the heat kernel and solutions of the heat equation (see [START_REF] Berline | Heat Kernels and Dirac Operators[END_REF] p. 84).

Theorem B.3. Let x ∈ X. We denote by y i the normal coordinates of a point y in the injectivity radius of X at x, ∂ i the corresponding partial derivatives, and by g ij (y) the scalar product of ∂ i and ∂ j at y. Moreover, we define J(x, y) = det(g ij (y)) 1/2 for y = exp x (y)

Let chosen smaller than the injectivy radius of X. Let Ψ : R + -→ [0, 1] be a smooth function such that Ψ(s) = 1 if s < 2 /4 and Ψ(s) = 0 if s > 2 . Let τ (x, y) : E y -→ E x be the parallel transport along the unique geodesic curve joining x and y, and d(x, y) its length.

for all v in V (1 denotes the unit of A), then there exists a unique morphism

g : Cl(V, Q) -→ A of R-algebras such that f = g • i Q .
The solution is unique up to isomorphisms and is given as the (non commutative) quotient

T (V )/(v ⊗ v + Q(v).1)
of the tensor algebra of V by the two-sided ideal generated by v ⊗ v + Q(v).1, where v belongs to V (see [START_REF] Lawson | Spin Geometry[END_REF] for a proof ).

It is well known that there exists a unique anti-automorphism t on Cl(V, Q) such that

t(i Q (v)) = i Q (v)
for all v in V . It is called reversion and usually denoted by x -→ x † , x in Cl(V, Q).

In the same way there exists a unique automorphism α on Cl(V, Q) such that

α(i Q (v)) = -i Q (v)
for all v in V . In this paper we write v for i Q (v) (according to the fact that i Q embeds V in Cl(V, Q)).

When it is defined, we denote x = √ xx † and say that x is a unit if xx † = ±1. Let V be a vector space on R of dimension n, Q a quadratic form on V and (e 1 , • • • , e n ) an orthonormal basis of V with respect to Q. As a vector space Cl(V, Q) is of dimension 2 n on R and a basis is given by the set These products extend by linearity on Cl(V, Q). Clearly, if a and b are vectors of V , then the inner product of a and b coincides with the scalar product defined by Q.

T. BATARD

Remark: For Q ≡ 0, the Clifford algebra Cl(V, Q) corresponds to the exterior algebra V of V . Indeed, in this case the product in the Clifford algebra is the outer product (C.1), and we have an algebra isomorphism between Cl(V, Q) and V . For arbitrary quadratic form Q, there is a vector space isomorphism between V and Cl(V, Q), that maps the subspace k V to Cl(V, Q) k . It follows a vector space isomorphism between the space V * of linear forms on V and the Clifford algebra Cl(V, Q), that maps k-linear forms to k-vectors. More precisely, for (e 1 , • • • , e n ) a orthonormal basis of (V, Q), we have the identifications k-form :

i1<•••<i k 1≤i k ≤m ω i1•••i k e i1 ∧ • • • ∧ e i k ←→ i1<•••<i k 1≤i k ≤m ω i1•••i k e i1 • • • e i k 0-form : l ←→ l 1 (C.2)
In this paper, we deal in particular with the Clifford algebra of the Euclidean space (R n , 2 ) denoted by R n,0 . R k n,0 is the subspace of elements of degree k and R * n,0 is the group of elements that admit an inverse in R n,0 .

C.2. The Lie groups Spin(n) and SO(n). The group Spin(n) is defined by

Spin(n) = 2k i=1 a i , a i ∈ R 1 n,0 , a i = 1
It is well known that Spin(n) is a connected compact Lie group that universally covers SO(n) (n ≥ 3). Under the identification of (R n , 2 ) and its embedding R 1 n,0 into R n,0 , the covering group is given by the map ξ

Spin(n) -→ SO(n) R 1 n,0 -→ R 1 n,0 ξ : s -→ R s : x -→ sxs -1
For instance, one can verify that Spin(3) is the group The exponential map exp : spin(n) -→ Spin(n) is onto and corresponds to the usual matrix exponential map. As a consequence, every spinor can be written as

S = ∞ i=0 1 i! A i
for some 2-vector A.

From the covering group Spin(n) -→ SO(n), we have a Lie algebra isomorphism spin(n) so(n)

Dealing with the matrix representation of the group SO(n), we have so(n) {n × n skew symmetric matrices}

Denoting by E ij , i < j the elementary skew symmetric matrix such that E ij (i, j) = -1

and E ij (j, i) = 1, the Lie algebra isomorphism maps E ij to 1 2 e i e j .

In the rest of this part, we show how rotations in R n can be interpreted in the Clifford algebras formalism. From Hestenes and Sobczyk [START_REF] Hestenes | Clifford Algebra to Geometric Calculus[END_REF], we know that every A in R 2 n,0 can be written as

A = A 1 + A 2 + • • • + A m
where m ≤ n/2 and for all σ in the permutation group S(m). Actually, as A 2 k is negative we have e Ai = cos( A i ) + sin( A i )

A j = A j
A i A i
The corresponding rotation R i : x -→ e -Ai xe Ai acts in the oriented plane defined by A i as a plane rotation of angle 2 A i . The vectors orthogonal to A i are invariant under R i . It then appears that any element R of SO(n) is a composition of commuting simple rotations, in the sense that they have only one invariant plane. The vectors left invariant by R are those of the orthogonal subspace to A. If m = n/2 this latter is trivial. The previous decomposition is not unique if A k = A j for some j and k with j = k. In this case infinitely many planes are left invariant by R.

1 .

 1 Introduction. Most multichannel image regularization methods are based on PDEs of the form of order ≤ 1)

  1)We prove (3.1) by developping both right and left terms of the equality.

Proposition 4 . 4 (

 44 Positively oriented orthonormal basis of T p S). representation of g at p in the basis (∂/∂x 1 , ∂/∂x 2 )(p).

Fig. 4 .

 4 Fig. 4.1 is an illustration of color image regularization induced by Clifford-Beltrami and Clifford-Hodge flows. It is given by the computations of k 0 t (x, y, ∆ C )I i 1 and k 0 t (x, y, D 2 )I i 1, i = 1, 2, 3, where convolutions are done on 5x5 neighborhoods (see Section 2). Fig. 4.1(a) is taken from the Berkeley image segmentation database [20]. Fig. 4.1(b) is the result of the diffusions after 10 iterations for t = 0.2, and g induced by h 1 = h 2 = h 3 = 0.01. We obtain a smoothing of the initial image on regions of low color variations whereas high edges are preserved.

Fig. 4 . 1 .

 41 Fig. 4.1. Color image regularization

Fig. 4 . 2 .Fig. 4 . 3 .Fig. 4 . 4 .

 424344 Fig. 4.2. Unit vector field of edge orientations

Fig. 4 .

 4 3 and Fig. 4.4. By the base manifold we have chosen for the applications, g measures color variations. It explains why the anisotropy of the diffusions is related with color variations of the image on Fig. 4.2(a). By definition, the Clifford-Beltrami flow consists in a Beltrami flow of each component of v. We see on Fig. 4.3(a) and Fig. 4.4(a) that it preserves the vector field on high color variations and smoothes it on low color variations. The Clifford-Hodge flow also preserves the vector field on high color variations, but it vanishes it on regions of low color variations, as it can be seen on Fig. 4.3(b) and Fig. 4.4(b).

1 Fig. 4 .

 14 Fig. 4.5. Unit vector field of edge orientations regularization for h i = 0.1

  {e i1 e i2 • • • e i k , i 1 < i 2 < • • • < i k , k ∈ {1, . . . , n}}and the unit 1. An element of degreek i1<•••<i k α i1...i k e i1 e i2 • • • e i kis called a k-vector. A 0-vector is a scalar and e 1 e 2 • • • e n is called the pseudoscalar. We denote x k the component of degree k of an element x of Cl(V, Q). The inner product of x r of degree r and y s of degree s is defined by x r • y s = x r y s |r-s| if r and s are positive and by x r • y s = 0 otherwise. The outer product of x r of degree r and y s of degree s is defined by x r ∧ y s = x r y s r+s (C.1)

{a1 + be 1 e 2 +

 2 ce 2 e 3 + de 3 e 1 , a 2 + b 2 + c 2 + d 2 = 1} and is isomorphic to the group H 1 of unit quaternions. We have Spin(2)= {a1 + be 1 e 2 , a 2 + b 2 = 1}. It follows the identifications Spin(2) S 1 SO(2) The Lie algebra spin(n) of Spin(n) is R 2 n,0 with Lie bracket A × B = AB -BA.

  a j b j , j ∈ {1, . . . , m}, with {a 1 , . . . , a m , b 1 , . . . , b m } a set of orthonormal vectors. ThusA j A k = A k A j = A k ∧ A j whenever j = k and A 2 k = -A k 2 < 0This means that the planes encoded by A k and A j are orthogonal and implies that e A1+A2+•••+Am = e A σ(1) e A σ(2) . . . e A σ(m)

Acknowledgments. The author thanks Pr. Michel Berthier for helpful discussions. He also thank the anonymous reviewers for helpful remarks and suggestions.

* This work was partially supported by the ONR Grant N00014-09-1-0493, and by the "Communauté d'Agglomération de La Rochelle".

T. BATARD

Then the kernels K N t (x, y, H) defined by

m 2 e -d(x,y) 2 /4t Ψ(d(x, y) 2 )

where the sections Φ i are given by Φ 0 (x, y, H) = τ (x, y) and τ (x, y)

where H x is the operator H with respect to the first variable, satisfy 1. For every N > m/2, the kernel

where l is a norm on C l sections.

2. Denoting by k N t the operator defined by

we have lim t→0 k N t u -u l = 0 for every N .

Moreover, we have the following estimate:

which justifies the notation e -tH .

Appendix C. Clifford algebras and the Lie group Spin(n).

C.1. Clifford algebras. We refer to [START_REF] Chevalley | The Algebraic Theory of Spinors and Clifford Algebras[END_REF], [START_REF] Hestenes | Clifford Algebra to Geometric Calculus[END_REF], [START_REF] Lawson | Spin Geometry[END_REF] for more details on Clifford algebras.

Definition C.1 (Clifford algebra). Let V be a vector space of finite dimension n over R equipped with a quadratic form Q. Formally speaking, the Clifford algebra Cl(V, Q) of (V, Q) is the solution of the following universal problem. We search a couple

for all v in V (1 denotes the unit of Cl(V, Q)) such that for each R-algebra A and each R-linear map f : V -→ A with (f (v)) 2 = Q(v).1


V Bertin
email: bertin@cppm.in2p3.fr

First data from the ANTARES neutrino telescope

This contribution reviews the recent progress achieved towards building the ANTARES neutrino telescope. The first results obtained by the operation of a Mini Instrumentation Line with Optical Modules, "MILOM", and the first complete detector line are highlighted.

The ANTARES detector

The European Collaboration ANTARES aims at building and operating a large undersea neutrino telescope located at a depth of 2500 m in the Mediterranean Sea, offshore from Toulon in France [1]. Neutrinos will be detected through their interaction in the matter surrounding the detector, producing muons radiating Cherenkov light while propagating in the sea water. Photons are recorded by a lattice of Optical Modules (OMs) [2], consisting of 10" hemispherical photomultiplier tubes (PMTs) [3] housed in pressure resistant glass spheres, installed on a set of mooring lines. The reconstruction of the muon track direction, pointing to a fraction of a degree towards the direction of the parent neutrino source for high energy neutrinos, is achieved from the measurement of the arrival times of the Cherenkov photons on the OMs, as well as their position in space.

The complete ANTARES detector will consist of 12 lines of 25 storeys, each storey being equipped with a triplet of Optical Modules looking at 45° downward and an electronics container mounted on a titanium frame, giving a grand-total of 900 OMs. On each storey, the local electronics container includes the front-end electronics of the PMTs, an Ethernet board for the data acquisition and the detector Slow Control, electronics boards for clock distribution and for Dense Wavelength Division Multiplexing of the Ethernet transmission, and a tiltmeter-compass board measuring the local tilt and orientation of the storey. Some storeys also support a hydrophone for acoustic positioning or an LED Optical Beacon used for inter-string time calibration.

The vertical distance between storeys is 14.5 m, the first one being placed at 100 m above the sea bed, leading to a total height of the detector strings of 480 m. Each string is anchored on the sea floor at a distance of 70 m from its neighbours. Every line is individually connected to a Junction Box by an interconnection cable a few hundred metres long. The Junction Box is itself linked to the shore by a 40 km long electro-optical cable equipped with 48 optical fibres.

The construction of the ANTARES detector started in October 2001 with the deployment of the main electro-optical cable from the ANTARES site, located in the Mediterranean Sea (42°48'N-6°10'E) offshore Toulon (France) at a depth of 2475 m, to the beach of La Seyne-sur-Mer where the shore station of the experiment is situated. In December 2002, the Junction Box was connected at the end of this cable and immersed on the site. In Spring 2003, two small test lines, the Prototype Sector Line (PSL) and the Mini Instrumentation Line (MIL), were installed, connected and operated for a few months. The genuine operation of the ANTARES neutrino telescope really started in April 2005 after the connection of the Mini Instrumentation Line with Optical Modules (MILOM) and more recently the installation of the first full complete detector line, Line 1, in March 2006. The second line, Line 2, was in integration phase in June 2006, at the time of the Neutrino'06 Conference, it has been successfully deployed on the ANTARES site at the end of July 2006 as scheduled. Thanks to two assembly sites running in parallel, the ANTARES detector is foreseen to be fully deployed and operational by the end of 2007 for several years of physics data taking.

The main performance parameters expected for the complete ANTARES neutrino telescope are summarized in figure 1. The left plot shows the effective area for neutrinos as a function of the neutrino energy for various incident angles. The effective area reaches 1 m² for E ν >100 TeV, the energy above which the Earth shadowing starts to be of some importance. The right plot shows the angular resolution for the reconstructed muon compared to the true muon direction and to the parent neutrino direction, as a function of the neutrino energy. While the angular resolution is dominated by the physical angle between the muon and the parent neutrino at low energy, it is dominated by the reconstruction for E ν ≥ 10 TeV and is expected to be as good as 0.3°. In this regime, the angular resolution is mainly dominated by two effects: the scattering and the chromatic dispersion of the Cherenkov light during its travel into the sea water, contributing for a time arrival spread of σ ~ 1 ns; the transit time spread (TTS) of the PMT signals being σ ~ 1.3 ns. To achieve this good angular resolution, the ANTARES detector is designed such as additional electronics contributions to the time calibration contribute for less than ~0.5 ns to the time-stamping of the detected photons. In addition, the relative position reconstruction of the OM has to be controlled with a precision of ~10 cm. The time coincidences between pairs of neighbouring Optical Modules have also been studied. These distributions exhibit a flat background of random coincidences and a Gaussian peak of few ns width due to genuine coincidences of 40 K radioactive decays producing two detected photons. The 40 K coincidence rate is measured to be 13.0±0.5 Hz and is in good agreement with a simulation the signals induced by 40 K decays which leads to a coincidence rate of 12 Hz with a 4 Hz systematic error due to uncertainties in the effective area and angular response of the OMs.

Time calibration with the LED Optical Beacon. The time calibration of the MILOM Optical

Modules has been checked by flashing the LED Optical Beacon located on the bottom storey. This device consists of a glass cylinder container containing 36 blue LEDs synchronised in time in order to produced intense light flashes with a time dispersion < 0.5 ns. The time calibration of the OMs can be checked either by looking at the arrival time of the signal on the PMT relative to the time of the flash, or by the time difference of the flash arrival time measured by two adjacent OMs. Due to the large intensity of the light flashes and the short 15 m distance of propagation of the light into the water, the time stamping of the OM signal is dominated by its electronics contribution and not by the TTS of the PMT in this case. The measured distributions confirm that the electronics contribution to the time calibration is ≤ 0.5 ns as expected.

The Line 1

The first complete line of the ANTARES neutrino telescope, Line 1, has been deployed on the site on February 14 th 2006 and connected two weeks later on March 2 nd by using the Remote Operated Vehicle Victor of IFREMER. This line is made of a BSS and of 25 storeys, holding a total of 75 OMs. It also includes 4 LED Optical Beacons and 5 acoustic positioning hydrophones spread along the line, as well as an acoustic transducer on its BSS.

Time calibration with the MILOM LED Optical Beacon. The time calibration of the Line 1

OMs can also be checked by flashing the LED Optical Beacon located on the MILOM bottom storey. Figure 3 (left) shows the detection time spread of the LED Optical Beacon flashes by an OM of Line 1 located on a storey at the same altitude than the MILOM LED Optical Beacon. In this case, the measured distribution width is σ = 0.7 ns for a "horizontal" travel path of ~80 m of the light in the sea water. Figure 3 (right) shows in comparison the detection time spread for an OM of Line 1 located at a higher altitude corresponding to a "diagonal" travel path of the flash light of ~150 m. A wider distribution is clearly observed due to the smaller intensity of the detected signal, as well as a tail of delayed photons coming from scattering light. All measured distributions have been found in good agreement with expectations. Figure 3. Detection time spread of the MILOM LED Optical Beacon flashes by an OM of Line 1 located at the same altitude as the LED Optical Beacon at an "horizontal" distance of ~80 m (left) and by an OM at a higher altitude at a "diagonal" distance of ~150 m (right).

Acoustic positioning measurements.

The reconstruction of the detector geometry in real time is primarily based on acoustic triangulation of a small number of hydrophones scattered along every line. The triangulation is performed from distance measurements of each hydrophone to several fixed acoustic emitters located either on every line anchor base (transducers) or on autonomous pyramidal structures anchored around the detector field (transponders). A relative positioning of the hydrophones in space with a precision of few cm is necessary in order to obtain a precision of ~10 cm on the OM positions as a result of the line shape reconstruction performed by the addition of the tilts and heading measurements of every storey. The concomitant operation of MILOM and Line 1 has allowed a check of the acoustic system performance by performing the Line 1 hydrophone triangulation based on the acoustic emission of the MILOM transducer and two autonomous transponders. The good resolution of the acoustic system, found to be well within the specification, can be appreciated on figure 4 which shows the radial displacement of the lowest hydrophone of Line 1, located on its bottom storey at 100 m above the sea bed, with respect to the line axis during a period of two weeks.

Reconstruction of atmospheric muons.

Although the OMs point at 45° downwards, the ANTARES detector has a none negligible efficiency for the detection and the reconstruction of downward-going atmospheric muons. The event selection is first performed by an online filter algorithm, running at the shore station, which looks for a set of ≥ 4 local coincidence hits, the triggered hits, causally compatible with a muon track passing through the detector during a 4 µs time window. The muon reconstruction is then performed with a χ² fit of the hit times as function of their altitudes, to a hyperbola corresponding to the intersection of the muon Cherenkov light front with the line plane, in order to determine the zenith angle of the muon track. An example of such a muon track fit is shown on figure 5. Several thousand of atmospheric muons have already been reconstructed after a few weeks of operation of the Line 1, the first being detected only two days after its connection. The study of the muon angular distribution is in progress. The preliminary results already show that the muon reconstruction is working well and that the hunt for the first undersea neutrino can be started.

Conclusion

The ANTARES Collaboration has made a major step forward during the last year by the operation of a Mini Instrumentation Line with Optical Modules, the MILOM, for more than a year, and the installation of the first complete line of the detector in Spring 2006. All studies performed with these two lines show that the detector behaves well within the design specification and that all technical problems are solved. The detector should be fully installed by the end of 2007 and in operation for science during at least five years. It is also considered as a milestone towards the building of a km 3 underwater detector for which a design study is under preparation.

Figure 1 .

 1 Figure 1. Expected performance for the complete 12 lines ANTARES neutrino telescope. The left figure shows the effective area for neutrinos as a function of the neutrino energy for several incident angles. The right figure shows the angular resolution of the reconstructed muon compared to the true muon direction and to the parent neutrino as a function of the neutrino energy.

2 .

 2 Results from the first ANTARES lines2.1. The MILOM lineThe current data taking of the ANTARES detector started in March 2005 with the operation of the MILOM[START_REF] Aguilar | First results of the Instrumentation Line for the deep-sea ANTARES neutrino telescope[END_REF]. This instrumentation line, partly devoted to multi-disciplinary and environmental studies, consists of an instrumented releasable anchor, the Bottom String Socket (BSS), and three storeys respectively located at 100 m, 117 m and 169 m above the sea bed. The middle storey is a standard ANTARES storey housing a triplet of Optical Modules. The main other devices are a water current profiler located on the top storey, an LED Optical Beacon held on the bottom storey, an acoustic positioning transducer attached to the BSS and a seismometer buried into the sea floor 50 m away from the MILOM.2.1.1. Optical background measurements. The operation of the MILOM allowed a continuous monitoring of the background rates of the Optical Modules. A typical OM counting rate display exhibits a baseline of ~60-80 kHz largely dominated by optical background due to 40 K decays and bioluminescence activities coming from bacteria, as well as bursts of a few seconds duration produced by bioluminescent emission of macro-organisms[START_REF] Amram | [END_REF]. Figure2(left) shows the baseline rates recorded by the three OMs of the MILOM during a period of three months in Autumn 2005. A seasonal variation of the bioluminescence component of the baseline is clearly observed. The 15% higher counting rate of OM1 during the full period is due to a lower threshold set on the readout of this Optical Module. Figure2(right) shows the burstfraction, defined as the fraction of the time when the counting rate is higher than the baseline by 20% during a 15 min interval, as a function of the water current intensity. A strong correlation of these two quantities is clearly observed.

Figure 2 .

 2 Figure 2. Baseline rates recorded by the three OMs of the MILOM during Autumn 2005 (left) and burstfraction as function of the water current intensity (right).

Figure 4 .

 4 Figure 4. Radial displacement of the Line 1 lowest hydrophone with respect to the line axis measured by the acoustic relative positioning system during a period of two weeks.

Figure 5 .

 5 Figure 5. Example of a downward-going atmospheric muon track reconstruction with the ANTARES Line 1.