Philippe Jaming
email: philippe.jaming@univ-orleans.fr

A CHARACTERIZATION OF FOURIER TRANSFORMS

Keywords: 1991 Mathematics Subject Classification. 42A, A3842A85, 42B10, 43A25 Fourier transform, convolution

The aim of this paper is to show that, in various situations, the only continuous linear map that transforms a convolution product into a pointwise product is a Fourier transform. We focus on the cyclic groups Z/nZ, the integers Z, the Torus T and the real line. We also ask a related question for the twisted convolution.

In memory of A. Hulanicki.

Introduction

The aim of this paper is to characterize the Fourier transform by some of its properties. Indeed, the Fourier transform is well known to change a translation into a modulation (multiplication by a character) and vice-versa and to change a convolution into a pointwise product. Moreover, these are some of its main features and are fundamental properties in many of its applications. The aim of this paper is to show that the Fourier transform is, to some extend, uniquely determined by some of these properties.

Before going on, let us introduce some notation. Let G be a locally compact Abelian group with Haar measure ν and let Ĝ be the dual group. Operations on G will be denoted additively. Let us recall that the convolution on G is defined for f, g ∈ L 1 (G) by

f * g(x) = G f (t)g(x -t) dν(t)
(and f * g ∈ L 1 (G)) while the Fourier transform is defined by

F (f )(γ) = f (γ) = G f (t)γ(t) dν(t).
We will here mainly focus on the four following cases, G = Ĝ = Z/nZ, G = Z and Ĝ = T and vice versa or G = Ĝ = R, (our results will then easily extend to products to such groups.)

We will here focus on two types of results. The first ones concerns the characterization of the Fourier transform as being essentially the only continuous linear transform that changes a convolution product into a pointwise product. To our knowledge the first results in that direction appear in the work of Lukacs [START_REF] Lukacs | An essential property of the Fourier transforms of distribution functions[END_REF][START_REF] Lukacs | A linear mapping of the space of distribution functions onto a set of bounded continuous functions[END_REF], pursued in [Em], and an essentially complete result appeared in [Fi] for all LCA groups, under the mild additional constraint that the transform has a reasonable kernel. We will show here that this hypothesis can be lifted. Further, a striking result, recently proved by Alekser, Artstein-Avidan, Milman [START_REF] Alesker | Milman A characterization of the Fourier transform and related topics[END_REF][START_REF] Alesker | A characterization of the Fourier transform and related topics[END_REF] is that, to some extend, continuity and linearity may be removed as well. More precisely, let us denote by S(R d ) the Schwartz functions on R d and by S ′ (R d ) the Schwartz (tempered) distributions.

Theorem (Alekser, Artstein-Avidan, Milman) Let T : S(R d ) → S(R d ) be a mapping that extends to a mapping T :

S ′ (R d ) → S ′ (R d ) that is bijective and such that (i) for every f ∈ S(R d ) and g ∈ S ′ (R d ), T (f * g) = T (f ).T (g); (ii) for every f ∈ S(R d ) and g ∈ S ′ (R d ), T (f.g) = T (f ) * T (g). Then there exists B ∈ M n (R) with det B = 1 such that T (f ) = F (f ) • B.
Note that T is neither assumed to be linear nor to be continuous. We will adapt the proof of this theorem to obtain an analogue result on the cyclic group. This has the advantage to highlight the main features which come into proof of this theorem. The main difference is that in this theorem, we assume that T sends smooth functions into smooth functions. In the case of the cyclic group, we do not have such functions at hand and are therefore lead to assume some mild continuity; see Theorem 2.2 for a precise statement.

A second set of results has its origin in the work of Cooper [START_REF]Cooper Functional equations for linear transformations[END_REF][START_REF]Cooper Linear transformations and group representations[END_REF].

Here one considers the Fourier transform as an intertwining operator between two groups of transforms acting on L p -spaces. In order to state the precise result, let us define, for α ∈ R and f a function on R,

τ α f (t) = f (t + α). Further, if ϕ : R → R, let M (ϕ) α f (t) = e iαϕ(t) f (t). It is easy to see that F τ α = M (t) α F and F M (-t) α = τ α F i.e.
the Fourier transform intertwines translations and modulations and vice versa. The converse is also true. More precisely:

Theorem (Cooper) Let T : L 2 (R) → L 2 (
R) be a continuous linear transformation such that there exists two measurable functions ϕ, ψ : R → R for which

T τ α = M (ϕ) α T and T M (ψ) α = τ α T.
Then ϕ(t) = bt + c, ψ(t) = bt + d with b, c, d ∈ R and T = F .

We will extend this theorem to Z/nZ and Z.

The remaining of the article is organized as follows. In the next section, we will prove the results for the groups G = Z and G = Z/nZ while Section 3 is devoted to the cases G = T d and G = R d . We conclude with some questions concerning the twisted convolution.

Before going on, let us introduce some more notation. If E ⊂ G, we will denote by χ E the function on G given by χ E (k) = 1 if k ∈ E and χ E (k) = 0 otherwise. The Kronecker symbol is denoted by δ j,k where δ j,k = 0 or 1 according to j = k or j = k.

The cyclic group and the integers

In this section, we consider G = Z/nZ or G = Z. We will write C( Ĝ) for the set of n-periodic sequences when G = Ĝ = Z/nZ or of continuous functions on Ĝ = T if G = Z. Our first result is the following:

Theorem 2.1. Let G = Z/nZ or G = Z. Let T be a linear continuous map T : L 1 (G) → C( Ĝ) such that T (f * g) = T (f ).T (g). Then there exists E ⊂ Ĝ and a map σ : Ĝ → Ĝ such that, for f ∈ L 1 (G) and almost every η ∈ Ĝ T (f )(η) = χ E (η) f σ(η) . *Moreover, σ is measurable if G = Z. Proof. Let δ k = (δ j,k ) j∈G ∈ L 1 (G). Then δ k * δ ℓ = δ k+ℓ , so that (2.1) T (δ k+ℓ ) = T (δ k * δ ℓ ) = T (δ k )T (δ ℓ ).
In particular, for each η ∈ Ĝ, the map k) so that if π η vanishes somewhere, it vanishes at 0. Conversely π η (k) = π η (k)π η (0) so that if π η vanishes at 0, it vanishes everywhere. Further π η (0) = π η (0) 2 so that π η (0) = 0 or 1.

π η : k → T (δ k )(η) is a group homomorphism from G to C. First note π η (0) = π η (k)π η (-
We will now assume that π η (0) = 1 and exploit π η (k + 1) = π η (k)π η (1) which implies that π η (k) = π η (1) k . We now need to distinguish the two cases:

-if G = Z/nZ, then 1 = π η (0) = π η (n) = π η (1) n π η (1) is an n-th root of unity i.e. T (δ 1 )(η) = e 2iπσ(η)/n for some σ(η) ∈ {0, 1 . . . , n -1} = Z/nZ. In follows that T (δ k )(η) = e 2iπkσ(η)/n . -if G = Z, as T was assumed to be continuous L 1 (G) → G( Ĝ), there is a constant C > 0 such that, for every f ∈ L 1 (G), T f ∞ ≤ C f 1 . In particular, for every k ∈ Z and every m ∈ Ĝ = T |π η (1) k | = |[T (δ 1 )(η)] k | = |T (δ k )(η)| ≤ C δ k 1 = C
thus, by letting k → ±∞, we obtain that π η (1) is a complex number of modulus 1 (as it is not 0 since π η (0) k = 0). We may thus write T (δ 1 )(η) = e 2iπσ(η) for some σ(η) ∈ [0, 1] ≃ T = Ĝ. Moreover, as η → T (δ 1 )(η) is measurable, we may assume that σ is measurable as well.

Let us now define

E = {η ∈ Ĝ : T (δ k )(η) = 0 ∀ k ∈ G}. Then, by linearity and continuity of T , for f ∈ L 1 (G), T f (η) = T k∈G f (k)δ k (η) = k∈G f (k)T (δ k )(η) =        k∈G f (k)χ E (η)e 2iπkσ(η)/n if G = Z/nZ k∈G f (k)χ E (η)e 2iπkσ(η) if G = Z = χ E (η) f σ(η) ,
which completes the proof.

Remark : Using tensorization, we may extend the result with no difficulty to

G = i∈I Z/n i Z × Z d .
We will now adapt the proof of [START_REF] Alesker | Milman A characterization of the Fourier transform and related topics[END_REF][START_REF] Alesker | A characterization of the Fourier transform and related topics[END_REF] to show that on Z n , a bijective transform that maps a product into a convolution is essentially a Fourier transform. We will need some notation:

Notation : We will consider the following particular elements of L 1 (Z/nZ) : 0 = (0, . . . , 0) and 1 = (1, . . . , 1). Further, if a ∈ L 1 (Z/nZ) we will write

E[a] = n-1 j=0 a j .
We can now state the main theorem: Theorem 2.2. Let T : L 1 (Z/nZ) → L 1 (Z/nZ) be a bijective transformation (not necessarily linear) such that the map

C → L 1 (Z/nZ), c → T (c1) is continuous. Assume that (i) for every a, b ∈ L 1 (Z/nZ), T (a.b) = T (a).T (b); (ii) for every a, b ∈ L 1 (Z/nZ), T (a * b) = T (a) * T (b).
Then there exists η ∈ {1, . . . , n -1} that has no common divisor with n such that -either, for every j ∈ Z/nZ and every a ∈ L 1 (Z/nZ), T (a)(ηj) = a(j); -or, for every j ∈ Z/nZ and every a ∈ L 1 (Z/nZ), T (a)(ηj) = a(j).

Remark : The fact that η has no common divisor with n implies that the map j → jη is a permutation of {0, . . . , n-1} so that the map T is actually fully determined.

Corollary 2.3. Let T : L 1 (Z/nZ) → L 1 (Z/nZ) be a bijective transformation (not necessarily linear) such that the map

C → L 1 (Z/nZ), c → T (c1) is continuous. Assume that (i) for every a, b ∈ L 1 (Z/nZ), T (a.b) = T (a) * T (b); (ii) for every a, b ∈ L 1 (Z/nZ), T (a * b) = T (a).T (b).
Then there exists η ∈ {1, . . . , n-1} that has no common divisor with n such that either, for every j ∈ Z/nZ and every a ∈ L 1 (Z/nZ), T (a)(ηj) = â(j) or, for every j ∈ Z/nZ and every a ∈ L 1 (Z/nZ), T (a)(ηj) = â(j).

Proof of Corollary 2.3. It is enough to apply Theorem 2.2 to T = F -1 T .

Corollary 2.4. k) and that one of the following two identities holds:

Let T : L 1 (Z/nZ) → L 1 (Z/nZ) be a bijective transforma- tion (not necessarily linear) such that the map C → L 1 (Z/nZ), c → T (c1) is continuous. Assume that for every a ∈ L 1 (Z/nZ), T 2 a(k) = a(-
(i) for every a, b ∈ L 1 (Z/nZ), T (a.b) = T (a) * T (b); (ii) for every a, b ∈ L 1 (Z/nZ), T (a * b) = T (a).T (b).
Then there exists η ∈ {1, . . . , n-1} that has no common divisor with n such that either, for every j ∈ Z/nZ and every a ∈ L 1 (Z/nZ), T (a)(ηj) = â(j) or, for every j ∈ Z/nZ and every a ∈ L 1 (Z/nZ), T (a)(ηj) = â(j).

Proof of Corollary 2.4. If T 2 a(k) = a(-k) then if one of the identities holds, so does the second one, so that Corollary 2.3 gives the result.

Proof of Theorem 2.2. The proof goes in several steps that are similar to those in [START_REF] Alesker | A characterization of the Fourier transform and related topics[END_REF]. The first one consists in identifying the image by T of some particular elements of L 1 (Z/nZ):

Step 1. We have T (δ 0 ) = δ 0 , T (0) = 0 and T (1) = T (1). Moreover, there is a k ∈ {-1, 1} and an α ∈ C with Re α > 0 such that, if we

define β : C → C by β(0) = 0 and β(c) = c |c| k |c| α for c = 0, then T (c1) = β(c)1.
Indeed, as T (a.b) = T (a).T (b), we immediately get the following: n 1], we may define β(c) = E T c n 1 so that T (c1) = β(c)1. Note that β is continuous since we have assumed that T acts continuously on constants and as T is one-to-one, so is β. Moreover, β is multiplicative :

T (c 1 c 2 1) = T (c 1 1).T (c 2
β(c 1 c 2 )1 = T (c 1 c 2 1) = T (c 1 1).T (c 2 1) = β(c 1 )β(c 2 )1.
It is then easy to check that there is a k ∈ {-1, 1} and an α ∈ C with Re α > 0 such that β(0) = 0 and

β(c) = c |c| k |c| α .
We will now take care of the image of δ j , j = 0, . . . , n -1.

Step 2. There is an η ∈ {1, . . . , n -1} with no common divisor with n such that T (δ j ) = δ ηj .

Assume that k = ℓ ∈ supp T (δ j ) thus δ k .T (δ j ) = 0 and δ ℓ .T (δ j ) = 0. Let a = T -1 (δ k ), b = T -1 (δ ℓ ), then a.δ j = T -1 (δ k ).T -1 T (δ j ) = T -1 δ k .T (δ j )) = T -1 (0)
since T is one-to-one. From Step 1, we know that T -1 (0) = 0, therefore a.δ j = 0. For the same reason, b.δ j = 0. In particular, a.b = 0, thus T (a.b) = 0. But this contradicts T (a.b) = δ k .δ ℓ with k = ℓ.

It follows that, for each j ∈ {1, . . . , n -1}, there exists c j ∈ C \ {0} and σ(j) ∈ {0, . . . , n -1} such that T (δ j ) = c j δ σ(j) . But then

1 = T (1) = T (1 * δ j ) = T (1) * T δ j = c j 1 * δ j = c j 1 thus c j = 1.
As T is one-to-one, it follows that σ(j) ∈ {1, . . . , n -1} and that σ is a permutation.

Next

δ σ(j+k) = T (δ j+k ) = T (δ j * δ k ) = T (δ j ) * T (δ k ) = δ σ(j) * δ σ(k) = δ σ(j)+σ(k) .
Thus σ(j + k) = σ(j) + σ(k) and therefore σ(j) = jσ(1). Further, the fact that σ is a permutation implies that σ(1) has no common divisor with n (Bezout's Theorem).

Step 3. Conclusion.

We can now prove that T is of the expected form: fix j ∈ {0, . . . , n -1} and a ∈ ℓ 2 n . Let k = σ -1 (j) so that T (δ k ) = δ j . Then

T (a)(j)δ j = T (a).δ j = T (a).T (δ k ) = T (a.δ k ) = T a(k)1.δ k = β a(k) 1.T (δ k ) = β a(k) δ j . It follows that T (a)(j) = β a • σ -1 (j) = a • σ -1 (j) |a • σ -1 (j)| k |a • σ -1 (j)| α . We want to prove that α = 1. But E[T (a)]1 = T (a) * 1 = T (a) * T (1) = T (a * 1) = T (E[a]1) = E[a] E[a] k |E[a]| α 1 so that E[T (a)] = E[a] |E[a]| k |E[a]| α or, in other words, n-1 ℓ=0 a(ℓ) |a(ℓ)| k |a(ℓ)| α = n-1 j=0 a σ -1 (j) a σ -1 (j) k a σ -1 (j) α = n-1 j=0 a j n-1 j=0 |a j | k n-1 j=0 a j α .
Taking a(0) = 1, a(1) = t > 0 and a(j) = 0 for j = 2, . . . , n -1, this reduces to 1 + t α = (1 + t) α . This implies that α = 1 (which is easiest seen by differentiating and letting t → 0). It follows that β(c) = c or c according to k = 1 or -1.

Remark : The proof adapts with no difficulty to any finite Abelian group.

To prove the same result on Z, the best is to first compose T with a Fourier transform and then to adapt the proof in [START_REF] Alesker | A characterization of the Fourier transform and related topics[END_REF] from the real line to the torus. We refrain from giving the details here. The lines of proof given here follows those given in [START_REF] Alesker | A characterization of the Fourier transform and related topics[END_REF] (up to the ordering and the removal of technicalities that are useless in the finite group setting). The main difference is that we need to assume that T acts continuously on constants. In [START_REF] Alesker | A characterization of the Fourier transform and related topics[END_REF] this hypothesis is replaced by the fact that T sends smooth functions into smooth functions.

Finally, it should also be noted that Hypothesis (i) and (ii) are only used when either a or b is either a constant c1 or a Dirac δ j .

We will conclude this section with a Cooper like theorem. Let us first introduce some notation: Notation : Let ϕ : Z/nZ → C. For k ∈ Z/nZ, we define the two following linear operators L 1 (Z/nZ) → L 1 (Z/nZ) :

τ k a(j) = a(j + k) and M (ϕ)
k a(j) = e kϕ(j) a(j).

Note that actually

ϕ : Z/nZ → C/2iπZ. As is well known, if ϕ(j) = 2iπj/n for some k ∈ Z/nZ, F τ -k = M (ϕ) k F and F M (ϕ) k = τ k F .
We can now state the following: Theorem 2.5. Let T : L 1 (Z/nZ) → L 1 (Z/nZ) be continuous linear operator such that there exist two maps ϕ, ψ : Z/nZ → C for which

T τ k = M (ϕ) k T and T M (ψ) k = τ k T . Then there exist k 0 , m 0 , m 1 ∈ Z/nZ, c ∈ C such that ϕ(j) = 2iπ n (k 0 j + m 0 ), ψ(j) = 2iπ n (-k 0 j + m 1 ) and T (a)(ℓ) = ce 2iπℓm 1 /n â(k 0 ℓ + m 0 ).
Proof. Without loss of generality, we may assume that T = 0. First note that the conditions are equivalent to

(2.2) a) T (δ k * a)(ℓ) = e -kϕ(ℓ) T (a)(ℓ) and b) T (e -kψ(•) a) = δ k * T (a).
Note that these two expressions are n-periodic in k so that ϕ and ψ take their values in {0, 2iπ n , . . . , 2iπ(n-1) n }. First a) implies

T (δ j )(ℓ) = T (δ j * δ 0 )(ℓ) = e -jϕ(ℓ) T (δ 0 )(ℓ). Next (2.2) b) implies that e -kψ(j) T (δ j )(ℓ) = T (e -kψ(j) δ j )(ℓ) = T (e -kψ(•) δ j )(ℓ) = δ k * T (δ j )(ℓ) = T (δ j )(ℓ -k).
In particular, T (δ j )(ℓ) = e ℓψ(j) T (δ j )(0), thus T (δ j )(ℓ) = e ℓψ(j)-jϕ(0) T (δ 0 )(0).

From linearity, we thus get that for a ∈ ℓ

2 n T (a)(ℓ) = n-1 j=0 a(j)T (δ j )(ℓ) = n-1 j=0 a(j)e ℓψ(j)-jϕ(0) T (δ 0 )(0).
As we assumed that T = 0, we thus have

T (δ 0 )(0) = 0. Then (2.2) reads n-1 j=0 a(j)e ℓψ(j+k)-(j+k)ϕ(0) = n-1 j=0 a(j)e ℓψ(j)-jϕ(0)-kϕ(ℓ)
thus ℓψ(j + k) -(j + k)ϕ(0) = ℓψ(j) -jϕ(0) -kϕ(ℓ) for all j, k, ℓ ∈ Z/nZ (modulo 2iπ/n). Taking k = 1, we get ϕ(ℓ) -ϕ(0) = ψ(j) -ψ(j + 1) ℓ so that ϕ and ψ are "affine". More precisely, ϕ(ℓ) = ψ(0) -ψ(1) ℓ + ϕ(0) modulo 2iπ/n and, as ϕ takes its values in 2iπ n Z/nZ, ϕ(ℓ) = 2iπ n (k 0 ℓ + m 0 ) (modulo 2iπ/n) with k 0 , m 0 ∈ {0, . . . , n -1} and b ∈ C. Further ψ(j + 1) = ψ(j) + ϕ(0) -ϕ(1) thus ψ(j) = ψ(0) + j ϕ(0) -ϕ(1) = 2iπ n (-k 0 j + m 1 ) (again modulo 2iπ/n).

We thus conclude that

T (a)(ℓ) = e 2iπ ℓm 1 n n-1 j=0 a(j)e -2iπ k 0 ℓ+m 0 n j .
as expected.

The real line and the Torus

We now consider the case

G = R d resp. G = T d so that Ĝ = R d resp. Ĝ = Z d . To simplify notation, we write C(Z d ) = L ∞ (Z d ). Theorem 3.1. Let d ≥ 1 be an integer and G = R d or G = T d . Let T be a continuous linear operator L 1 (G) → C( Ĝ) such that T (f * g) = T (f )T (g).
Then there exists a set E ⊂ G and a function ϕ

: Ĝ → Ĝ such that T (f )(ξ) = χ E (ξ) f ϕ(ξ) .
Proof. Let us fix ξ ∈ Ĝ and consider the continuous linear functional T ξ on L 1 (G) given by T ξ (f ) = T (f )(ξ). Then there exists a bounded function h ξ on G such that T ξ (f ) = G f (t)h ξ (t) dt. There is no loss of generality in assuming that h ξ = 0.

Let us now take A, B to sets of finite measure. Then Fubini's Theorem implies that

A×B h ξ (s + t) ds dt = R d χ A * χ B (t)h ξ (t) dt = T (χ A * χ B )(ξ) = T (χ A )(ξ)T (χ B )(ξ) = A h ξ (t) dt B h ξ (t) dt. (3.3) Now let ϕ n be defined on G 2 by ϕ n (x, y) = h ξ (x + y) -h ξ (x)h ξ (y) χ [-n,n] (x)χ [-n,n] (y) if G = R h ξ (x + y) -h ξ (x)h ξ (y) if G = T .
As ϕ n is bounded (since h ξ is) and has compact support, ϕ n ∈ L 1 (G 2 ) and (3.3) implies that A×B ϕ n (x, y) dx dy = 0 for every sets A, B of finite measure, so that ϕ n = 0 for every n. That is

(3.4) h ξ (x + y) = h ξ (x)h ξ (y)
for almost every x, y ∈ G.

If h ξ were continuous, this would imply that h ξ (x) = e i a ξ ,x and, by boundedness of h ξ , that a ξ ∈ R d . We will now overcome this difficulty by introducing

H ξ,j (x) = x 0 h ξ (te j ) dt
where j = 1, . . . , d and e j = (δ j,k ) k=1,...,d is the j-th vector in the standard basis. Clearly H ξ,j is continuous and satisfies

H ξ,j (x)H ξ,j (y) = x 0 H ξ,j (y + t) -H ξ,j (t) dt.
From this, we immediately deduce that H ξ,j is smooth, that H ′ ξ,j (t) = h ξ (te j ) almost everywhere and that H ′ ξ,j (x + y) = H ′ ξ,j (x)H ′ ξ,j (y) everywhere. Thus, for almost every x ∈ R or T, h ξ (xe j ) = e ia ξ,j x with a ξ,j real. Finally for x ∈ G,

h ξ (x) = h ξ (x 1 e 1 + • • • + x d e d ) = h ξ (x 1 e 1 ) • • • h ξ (x d e d ) = e i a ξ ,x
where a ξ = (a ξ,1 , . . . , a ξ,d ).

We have thus proved that there exists a map from ϕ : G → G and a set E such that

(3.5) T f (ξ) = χ E (ξ) f ϕ(ξ)
which completes the proof.

Remark : If T extends to a unitary operator from L

2 (R d ) onto L 2 (R d ) then E = R d and ϕ : G → G is bijection and is measure preserving i.e.
|ϕ -1 (E)| = |E| for every set E ⊂ G of finite measure. This last fact is a corollary of [Si] (see also [No]).

Note that in this theorem, we have only used the L 1 -L ∞ duality to show that the operator is a kernel operator. A slightly more evolved theorem allows to obtain this directly. More precisely, this is a consequence of the following theorem that dates back at least to Gelfand [Ge] and Kantorovich-Vullich [KV] (see also [START_REF] Dunford | Pettis Linear operations on summable functions[END_REF]Theorem 2.2.5] or [AT,Theorem 1.3]): Theorem 3.2. Let (Ω 1 , µ 1 ) and (Ω 2 , µ 2 ) be two σ-finite measure spaces. There is a one-to-one correspondence between bounded linear operators T : L 1 (Ω 1 ) → L ∞ (Ω 2 ) and kernels k ∈ L ∞ (Ω 1 × Ω 2 ). This correspondence is given by T = T k where T k is defined by

T k f (ω) = Ω 1 k(ζ, ω)f (ζ) dµ 1 (ζ), f ∈ L 1 (Ω 1 ).
It follows that this proposition than essentially reduces to the results in [START_REF] Lukacs | An essential property of the Fourier transforms of distribution functions[END_REF][START_REF] Lukacs | A linear mapping of the space of distribution functions onto a set of bounded continuous functions[END_REF], However, a non explicit condition in those papers is that k be defined everywhere as it is applied to Dirac masses.

The twisted convolution

In this section, we consider the case of the twisted convolution (for background on this transform we refer to [Fo]). Recall that this is defined for f, g ∈ L 1 (R 2d ) by f ♮ g(x, y) = R d R d f (x -s, y -t)g(s, t)e iπ( x,t -y,s ) ds dt. This defines a new L 1 (R 2d ) function. Note also that this operation is non commutative.

Next, for p, q ∈ R d , let us define the following operator that acts on functions on R d : ρ(p, q)ϕ(x) = e 2iπ q,x +iπ p,q ϕ(x + p).

For f ∈ L 1 (R d × R d ) we define the (bounded linear) operator on

L 1 (R d ) → L 1 (R d ) ρ(f )ϕ(x) = R d R d
f (p, q)ρ(p, q)ϕ(x) dp dq = R d K f (x, y)ϕ(y) dy where K f (x, y) = R d f (y -x, q)e iπ q,x+y dq = F -1 2 [f ] y -x,

x + y 2 and F 2 stands for the Fourier transform in the second variable.

One then checks through a cumbersome computation that ρ(f ♮ g) = ρ(f )ρ(g) (here the product stands for composition of operators) or, for the kernels

K f ♮ g (x, y) = R d K f (x, z)K g (z, y) dz.
Question. To what extend does this characterize the transform f → ρ(f ).

  1) and T (c 1 δ j ) = T (c 1 1).T (δ j ) while from T (a * b) = T (a) * T (b) we deduce that T (δ j+k ) = T (δ j )T (δ k ) and T (a) = T (δ 0 * a) = T (δ 0 ) * T (a). Applying this last identity to a = T -1 (δ 0 ) we get δ 0 = T (δ 0 ) * δ 0 = T (δ 0 ). Further a = a.1 thus T (a) = T (a).T (1) and, applying this again to a = T -1 (b), we have b = b.T (1) for all b ∈ ℓ 2 n , thus T (1) = 1. Similarly, 0 = a.0 thus T (0) = T (a).T (0) and, applying this to a = T -1 (0) we get T (0) = 0.T (0) = 0. Finally, E[a]1 = a * 1 thus T (E[a]1) = T (a * 1) = T (a) * 1 = E[T (a)]1. As every c ∈ C may be written c = E[ c

The author wishes to thank S. Alesker for sending him the preprint [AAAV2] and O. Guedon for pointing to that paper. This work was partially financed by the French ANR project AHPI (ANR-07-BLAN-0247-01).


Karin Baur

CLUSTER CATEGORIES, m-CLUSTER CATEGORIES AND DIAGONALS IN POLYGONS

Keywords: 2000 Mathematics Subject Classification. Primary: Secondary: Key words and phrases. Cluster category, m-cluster category, diagonals, translation quiver. 1

Cluster categories, m-cluster categories and diagonals in polygons

Introduction

This expository article is the expanded version of a talk given at the conference at the Grenoble summer school "Geometric methods in representation theory" in July 2008. The goal of this talk was to explain how cluster categories and m-cluster categories can be described via diagonals and so-called m-diagonals in a polygon. And then how the latter can actually be described using the power of a translation quiver. The first section gives a very brief introduction to the theory of cluster algebras and cluster categories. It also introduces the notations used in the article. In Section 2, we explain the notions of a quiver given by the diagonals in a polygon and of the one given by m-diagonals. The results in this section are mainly due to Caldero-Chapoton-Schiffler ([CCS06]), to Schiffler ([S06]) and to Baur-Marsh ( [START_REF] Baur | A geometric description of m-cluster categories[END_REF], [START_REF] Baur | A geometric description of the m-cluster categories of type Dn[END_REF]). In the last section, we introduce the concept of the power of a translation quiver. Here, the results are from [START_REF] Baur | A geometric description of m-cluster categories[END_REF], [START_REF] Baur | A geometric description of the m-cluster categories of type Dn[END_REF] and from the masters thesis of C. Ducrest ([D08]).

Cluster algebras and cluster categories

Cluster algebras were introduced by Fomin and Zelevinsky ( [START_REF] Fomin | Cluster algebras I: Foundations[END_REF]) in order to provide an algebraic framework for the phenomena of total positivity and for the canonical bases of the quantized universal enveloping algebras.

We briefly illustrate the notion of total positivity: An n×n matrix is called totally positive if all its minors are positive. Originally, the term was used to describe matrices with non-negative minors: these matrices are nowadays called totally nonnegative. In the 1930s, Gantmacher-Krein and I. Schoenberg have independently started investigating such matrices. One of the motivations was to estimate the number of real zeroes of a polynomial.

Gantmacher showed that totally non-negative matrices have different real eigenvalues. The interest in total positivity was renewed in the 1990s when G. Lusztig extended the notion to reductive algebraic groups, cf. [START_REF] Lusztig | Total positivity in reductive groups[END_REF].

Example 1.1. To illustrate the notion on a (non-reductive) example, let us consider the group of 3 by 3-matrices with 1's on the diagonal and zeroes below. If

U = 1 a b 0 1 c 0 0 1 is such a matrix then U is totally positive if a, b, c > 0 and
acb > 0. One can check that it is actually enough to require a > 0, acb > 0 and b > 0: the condition c > 0 will follow automatically. Equivalently, the condition a > 0 can be dropped and is automatically satisfied by the remaining conditions. So there is only a certain number of minors that need to be checked.

Furthermore, if from the set {a > 0, acb > 0, b > 0} of conditions the first is omitted then we may replace it exactly with one other condition, namely with c > 0, to obtain total positivity of the matrix.

More generally, one shows that the minimal sets of minors to check all have the same cardinality. And it is often the case that if you remove one minor from such a minimal set, there exists exactly one other minor to replace it with.

1.1. Cluster algebras. A cluster algebra A ⊂ Q(u 1 , . . . , u n ) of rank n is an algebra with possibly infinitely many generators. These generators are called cluster variables; they are arranged in overlaping sets of the same cardinality n, the clusters. There are relations between the cluster variables, encoded in an n × n matrix, the mutation matrix. Through mutation, one element of a cluster is exchanged by exactly one other element and this exchange process is prescribed by the exchange matrix.

If there are only finitely many generators, the cluster algebra is of finite type. Finite type cluster algebras have been classified by Fomin and Zelevinsky ([FZ03]). Their classification describes the finite type cluster algebras in terms of Dynkin diagrams.

More concretely: a seed is a pair (x, M ) where x = {x 1 , . . . , x n } is a basis of Q(u 1 , . . . , u n ) and M = (M ij ) ij is a sign skew symmetric n × n-matrix with integer entries, called the exchange matrix. That means that the sign of M ij is the opposite of the sign of M ji .

Then one defines an involutive map µ k (for k ∈ {1, . . . , n}) on the seeds, called the mutation in direction of k, through µ k (x) = (x 1 , . . . , x k , x ′ k , . . . , x n ) where x ′ k is given by the relation

x k • x ′ k = x i ∈ x M ik > 0 x M ik i + x i ∈ x M ik < 0 x -M ik i
In a similar way, one defines M ′ by

M ′ ij := -M ij if i = k or j = k M ij + 1 2 (|M ik |M kj + M ik |M kj |)
otherwise. and thus obtains (x ′ , M ′ ) as µ k ((x, M )) (the matrix M ′ is also a sign skew symmetric n × n-matrix over Z). For more details we refer to Section 1 of the survey article [START_REF] Buan | Cluster-tilting theory[END_REF] of Buan-Marsh. The x i obtained through successive mutations are the so-called cluster variables. The cluster algebra A = A(x, M ) is then defined as the algebra generated by the cluster variables. There can be infinitely many of them. Fomin-Zelevinsky have shown in [START_REF] Fomin | Cluster algebras IV: Coefficients[END_REF] that A lies in Z[x ±1 1 , . . . , x ±1 n ] (Laurent-phenomenon). First examples of cluster algebras are coordinate rings of SL 2 , SL 3 .

The field of cluster algebras is a young and very dynamic field. Since its first introduction, there have been many different directions in its development. We only mention a few connections to other areas (in parentheses: the objects corresponding to the cluster variables): the theory of Teichmüller spaces (Penner coordinates), see work of Fock-Goncharov, [START_REF] Fock | Moduli spaces of local systems and higher Teichmüller theory[END_REF] and [START_REF] Fock | Dual Teichmüller and lamination spaces[END_REF]; the representation theory of finite dimensional algebras (tilting modules), cf. [START_REF] Buan | Tilting theory and cluster combinatorics[END_REF]), triangulations of surfaces (diagonals), see the work [START_REF] Fomin | Cluster algebras and triangulated discs. Part I: Cluster complexes[END_REF] of Fomin-Shapiro-Thurston.

1.2. Cluster categories. Cluster categories were introduced independently in the work [START_REF] Buan | Tilting theory and cluster combinatorics[END_REF] of Buan-Marsh-Reineke-Reiten-Todorov, and by Caldero-Chapoton-Schiffler, [START_REF] Caldero | Quivers with relations arising from clusters (An case)[END_REF] to provide a categorification of the theory of cluster algebras.

We will use the approach of [START_REF] Buan | Tilting theory and cluster combinatorics[END_REF] to describe cluster categories and will consider the approach of [CCS06]later, cf. Section 2.

Let Q be a simply-laced Dynkin quiver (i.e. a quiver whose underlying graph is of type A, D or E). Let k be an algebraically closed field and kQ the path algebra of Q (for more details, we refer to the lecture notes of M. Brion, [B08] in the same volume). Take the bounded derived category D b (kQ) of finitely generated kQ-modules (for details on D b (kQ) we refer to [H88]) with shift denoted by [1] and Auslander-Reiten translate denoted by τ . By Happel ([H88]), the category D b (kQ) is triangulated, Krull-Schmidt, and has almost split sequences. To understand the category D b (kQ) it is helpful to study its Auslander-Reiten quiver: The Auslander-Reiten quiver of a category is a combinatorial tool which helps understanding the category. Its vertices are by definition the indecomposable modules up to isomorphism and the number of arrows between two points are given by the dimension of the space of irreducible maps between two representatives of the corresponding modules.

We now associate a quiver ZQ to Q. Its vertices are (n, i) for n ∈ Z, and where i a vertex of Q. For every arrow i → j in Q there are arrows (n, i) → (n, j) and (n, j) → (n + 1, i) in ZQ. So ZQ has the shape of a Z-strip of copies of Q. Together with the map τ : (n, i) → (n -1, i) (n ∈ Z, i a vertex of Q), ZQ is a stable translation quiver as defined by Riedtmann (see [START_REF] Riedtmann | Algebren, Darstellungsköcher, Überlagerungen und zurück[END_REF]). For a precise definition of stable translation quivers we refer the reader to Section 3 below. We illustrate ZQ in Example 1.2. Happel has shown in [H88], that the Auslander-Reiten quiver AR

(D b (kQ)) of D b (kQ) is just ZQ. In particular, the category D b (kQ) is independent of the orientation of Q. Example 1.2. Let Q be a quiver of type A 3 , Q : 3 1 2
Then, ZQ has the shape with one copy of the quiver Q highlighted to show how it appears inside ZQ. The dotted arrows indicate the Auslander-Reiten translate τ which sends each vertex to its leftmost neighbor. It is an auto-equivalence of the Auslander-Reiten quiver.

On the other hand, the Auslander-Reiten quiver of the module category kQ-mod of finitely generated kQ-modules looks like a triangle: AR(kQ-mod):

Observe that the infinite quiver ZQ can be viewed as being covered by copies of the Auslander-Reiten quiver of the module category kQ-mod, with additional arrows and dotted arrows introduced to connect the copies of the triangle of kQmod. With this picture in mind, we can describe the shift [1] on AR(D b (kQ)): it sends each vertex to the "same" vertex in the next copy of the triangle AR(kQ-mod) to the right.

Back to the general situation, where Q is of simply-laced Dynkin type. The shift [1] then is the auto-equivalence of AR(D b (kQ)) which sends a vertex to the corresponding vertex in the next copy of the Auslander-Reiten quiver of the module category kQ-mod and the translation τ sends a vertex to its leftmost neighbor. As an abbreviation, we write F for the auto-equivalence τ -1 • [1] of D b (kQ). Now we are ready to define the cluster category associated to Q.

Definition. The cluster category

C := C Q := D b (kQ)/F of
type Q is the orbit category whose objects are the F -orbits of objects of D b (kQ) and whose morphisms are given as follows:

Hom C ( X, Y ) = i∈Z Hom D b (kQ) (F i X, Y )
where X and Y are representatives of the F -orbits through X and Y respectively.

Note that for any pair of objects X, Y of D b (kQ) there are only finitely many i such that Hom D b (kQ) (F i X, Y ) is non-zero. The cluster category is Krull-Schmidt ([BMRRT05]), triangulated and Calabi-Yau of dimension 2 ( [START_REF] Keller | On triangulated orbit categories[END_REF]).

The connection between C Q and the cluster algebra of the same type is given by the following result.

Theorem 1.3 ([BMRRT05]

). There is a bijection between the cluster variables of the cluster algebra of type A n (resp. D n , E n ) and indecomposable objects of

C Q where Q is of type A n (resp. D n , E n ).
To understand the cluster categories better, we consider its Auslander-Reiten quiver. By definition, it has the form of one copy of the module category, together with a copy of the quiver Q (with additional arrows, dotted arrows), as illustrated in types A and D below (Figures 1 and2). In particular, it is a finite quiver. In the pictures of Figures 1 and2 we have repeated one copy of the quiver Q to indicate how the quivers are glued together: both quivers wrap around. The Auslander-Reiten quiver of type A n can be viewed as lying on a Möbius strip and the one of type D n as lying on a cylinder.

Let G be the underlying graph of Q, G of Dynkin type A, D or E. We recall that a famous result of P. Gabriel establishes a bijection between the indecomposable objects of kQ-mod (up to isomorphism) and the positive roots of the Lie algebra of type G. For a recent description of this result we refer to Section 5 of the lecture notes [START_REF] Krause | Representations of quivers via reflection functors[END_REF] There exists an analogous result for cluster categories. To state it we need to enlarge the set of roots considered: we add the negatives of the simple roots. An almost positive root of a Lie algebra of type G is a positive root or the negative of a simple root.

Buan et al. have shown in [START_REF] Buan | Tilting theory and cluster combinatorics[END_REF] that there is a bijection between the indecomposable objects of the cluster category C Q and the almost positive roots of the Lie algebra of type Q. Note that the Auslander-Reiten quiver of C m thus consists of m copies of the Auslander-Reiten quiver of the module category kQ-mod and additionally, of a copy of the quiver Q, connected with additional (dotted) arrows. In the case of A n , we observe that if m is odd, the Auslander-Reiten quiver of C m Q lies on a Möbius strip, whereas if it m is even, it lies on a cylinder. As an example, Figure 3 shows the Auslander-Reiten quiver of the 2-cluster category of type A 2 . Again, we have repeated a slice of the quiver to indicate how it wraps around.

Polygons and diagonals

In the first part of this section we present the approach of Caldero, Chapoton and Schiffler, [START_REF] Caldero | Quivers with relations arising from clusters (An case)[END_REF], who described the cluster category of type A n in terms of the diagonal of a polygon. This was later adapted to type D n by Schiffler in [S06], using a punctured n-gon. In the second part, we explain how to describe the m-cluster category of type A n in terms of so-called m-diagonals in a polygon.

2.1. Quiver of diagonals. Let Π be a polygon with n + 2 vertices, labeled clockwise by {1, 2, . . . , n + 2}. To it, we associate a quiver Γ(n, 1) as follows:

The vertices of Γ(n, 1) are the diagonals

{(i, j) | 1 ≤ i, j ≤ n + 2, |i -j| > 1} of Π.
The arrows are (i, j) → (i, j + 1) and (i, j) → (i + 1, j), where i + 1 and j + 1 are taken modulo n + 2, provided the image is also a diagonal of Π. Furthermore, we define a bijection τ on the vertices of Γ(n, 1) as follows: τ : (i, j) → (i -1, j -1) (again, taking i -1 and j -1 modulo n + 2). The quiver Γ(n, 1) together with this map τ is a stable translation quiver (cf. definition in Section 3 below). The translation quiver Γ(4, 1) obtained from the hexagon is:

(1, 5) o o A A A A A
(2, 6)

A A A A A o o A A A A A (1, 3) A A A A A (1, 4) o o A A A A A
Remark. Every maximal collection of non-crossing diagonals of a polygon Π (punctured or not) is a triangulation of Π. All maximal collections have the same number of elements, this number is an invariant of Π. It is called the rank of the polygon.

The rank of an n + 2-gon is n -1, the rank of a punctured n-gon is n. This leads us back to cluster algebras -for the connection between (punctured) polygons of rank n and cluster algebras of type A n (of type D n respectively) we refer the reader to [START_REF] Fomin | Cluster algebras and triangulated discs. Part I: Cluster complexes[END_REF].

2.2. Quiver of m-diagonals. In a similar way, the m-cluster categories can be modelled using a certain quiver Γ(n, m). We will now explain how this works. Let Π be an nm + 2-gon. The vertices of Γ(n, m) are the m-diagonals, i.e. the diagonals of the form (1, m + 2), (1, 2m + 2), etc., where vertices are taken modulo nm + 2. More precisely, an m-diagonal divides Π into an mj +2-gon and an m(n-j)+2-gon (for 1 ≤ j ≤ n-1 2 ). The arrows send (i, j) to (i, j +m) and to (i+m, j) whenever the image is also an m-diagonal. Furthermore, we define a translation τ m on Γ(n, m) which sends (i, j) to (im, jm) (taking vertices modulo nm + 2). Then Γ(n, m) is also a stable translation quiver. With m = 1 we just recover the case of the usual diagonals as described above. (

Then the quiver Γ(3, 2) is:

(1, 6) o o " " D D D (3, 8) o o " " D D D (2, 5) " " D D D o o (4, 7) o o " " D D D (1, 6) (1, 4) o o < < z z z (3, 6) o o < < z z z (5, 8) o o < < z z z (2, 7) o o < < z z z (1, 4) < < z z z
Here, we have also repeated the first slice (1, 4) → (3, 6) to indicate how the quiver wraps around.

Observe that the quiver Γ(3, 2) is just the Auslander-Reiten quiver of the 2cluster category of type A 2 . So the 2-diagonals in the octagon model the cluster category C 2 Q for Q of Dynkin type A 2 . This holds more generally by the following result.

Theorem 2.4 ([BaM08]

). Let Q be a quiver of Dynkin type A n-1 , let m ≥ 1. Then the Auslander-Reiten quiver of C m Q is isomorphic to the quiver Γ(n, m) of m-diagonals in an nm + 2-gon.

Note that we recover Theorem 2.2 in the case m = 1.

Remark. (1) Theorem 2.4 tells us that in order to understand the m-cluster category of type A n it is enough to study Γ(n, m).

(2) To model the m-cluster categories of type D n , one defines so-called m-arcs in a punctured nmm + 1-gon and obtains a quiver Γ ⊙ (n, m). One can show that BAUR this is the Auslander-Reiten quiver of C m Q where Q is of type D n , cf. Theorem 3.6 in [START_REF] Baur | A geometric description of the m-cluster categories of type Dn[END_REF].

The maximal collections of non-crossing m-diagonals in an nm + 2-gon (resp. in a punctured nmm + 1-gon) correspond to m + 2-angulations of the polygon. The number of elements in such a maximal collection is again an invariant of the polygon. It is equal to n -1 (resp. to n).

Powers of translation quivers

We now provide another way of obtaining m-cluster categories. It uses the concept of the power of a translation quiver which was introduced in [START_REF] Baur | A geometric description of m-cluster categories[END_REF]. In order to explain this, let us give the precise definition of a translation quiver.

Definition. A translation quiver is a pair (Γ, τ ) where Γ is a quiver, possibly with infinitely many vertices and arrows; τ is an injective map from a subset of the vertices of Γ to the vertices of Γ, such that the following holds: the number of arrows going from a vertex x to y equals the number of arrows from τ y to x for all vertices x, y of Γ. The map τ is called the translation of (Γ, τ ).

If τ is defined on all vertices (and thus bijective) then (Γ, τ ) is a stable translation quiver.

We remark that in all examples of stable translation quivers appearing in this article, the number of arrows between 2 vertices is always at most 1.

We recall that a composition

x 0 → x 1 → • • • → x m-1 → x m of m arrows x i → x i+1 (
where the x i are vertices of Γ) is a path of length m. Such a path is said to be sectional if τ x i+1 = x i-1 for i = 1, . . . , m -1 (for which τ x i+1 is defined), cf. [START_REF] Ringel | Tame Algebras and Integral Quadratic Forms[END_REF].

Definition. Let (Γ, τ ) be a translation quiver. The m-th power Γ m of Γ is the quiver whose vertices are the same as the vertices of Γ and whose arrows are the sectional paths of length m of Γ.

One can show that if (Γ, τ ) is a stable translation quiver, then the pair (Γ m , τ m ) is also a stable translation quiver ([BaM08], Section 6). Note however that (Γ m , τ m ) is not connected in general, even if (Γ, τ ) is so. The following example illustrates this.

Example 3.1. Let Γ be the quiver Γ(6, 1) of diagonals in an octagon, let m = 2. The quiver of the octagon has five rows, with first slice (1, 3) As before, we repeat the first slice at the end to indicate how the quiver wraps around.

→ (1, 4) → (1, 5) → (1, 6) → (1, 7): (1, 7) o o $ $ (2, 8) $ $ o o (1, 3) $ $ (1, 6) o o $ $ : : (2, 7) o o $ $ : : (3, 8) $ $ o o : : (1, 4) $ $ (1, 5) o o $ $ : : (2, 6) o o $ $ : : (3, 7) $ $ o o : : (4, 8) o o $ $ : : (1, 5) $ $ (1, 4) o o $ $ : : (2, 5) o o $ $ : : (3, 6) $ $ o o : : (4, 7) o o : : $ $ (5, 8) o o $ $ : : (1, 6) $ $ (1, 3)
The second power of Γ(6, 1) has three components. One containing the vertex (1, 3), one containing the vertex (1, 4) and the third containing the vertex (2, 4):

(1, 6) o o " " D D D (3, 8) o o " " D D D (2, 5) " " D D D o o (4, 7) o o " " D D D (1, 6) (1, 4) o o < < z z z (3, 6) o o < < z z z (5, 8) o o < < z z z (2, 7) o o < < z z z (1, 4) < < z z z (1, 7) o o " " D D D (1, 3) " " D D D (1, 5) o o < < z z z " " D D D (3, 7) o o < < z z z " " D D D (1, 5) " " D D D (1, 3) o o < < z z z (3, 5) o o < < z z z (5, 7) o o < < z z z (1, 7) and (2, 8) o o " " D D D (2, 4) " " D D D (2, 6) o o < < z z z " " D D D (4, 8) o o < < z z z " " D D D (2, 6) " " D D D (2, 4) o o < < z z z (4, 6) o o < < z z z (6, 8) o o < < z z z (2, 8)
We observe that the component through the vertex (1, 4) is the same as the quiver Γ(3, 2) from Example 2.3. This is a property that holds in general, cf. Theorem 3.2. The other two components are isomorphic to the Auslander-Reiten quiver of the orbit category

D b (kQ)/[1] where Q is of Dynkin type A 3 . Theorem 3.2. The quiver Γ(n, m) is a connected component of Γ(nm, 1) m .
From Theorem 3.2 one obtains that the m-cluster category of type A n is a full subcategory of the additive category generated by the mesh category of Γ(nm, 1) (for a definition of the mesh category we refer the reader to [BaM08, Section 3]). In other words: in order to understand the m-cluster category, there is an alternative approach to the one presented in Subsection 2.2. Namely, we can consider the m-th power of the quiver given by the usual diagonals. The Auslander-Reiten quiver of the m-cluster category of type A n-1 appears as a connected component of Γ(nm, 1) m .

Remark. We can actually prove an analogous result as Theorem 3.2 for type D n , cf. [START_REF] Baur | A geometric description of the m-cluster categories of type Dn[END_REF]: Let Γ ⊙ (n, 1) denote the quiver obtained from the arcs of a punctured n-gon (note the difference: here, the polygon has n vertices instead of n + 2) and by Γ ⊙ (n, m) the quiver of m-arcs in a punctured nm-m+1-gon. Then the Auslander-Reiten quiver of the m-cluster category of type D n is Γ ⊙ (n, m) and it is a connected component of Γ ⊙ (nmm + 1, 1) m . For details we refer to Section 5 of [START_REF] Baur | A geometric description of the m-cluster categories of type Dn[END_REF] 1 .

A natural question to ask at this point is what the other connected components of the mth power of Γ(nm, 1) are. Remarkably, this question is much easier in type D. There, we have a complete answer:

The connected components of the (restricted) mth power of the Auslander-Reiten quiver of the cluster category of type D nm-m+1 are exactly the union of the Auslander-Reiten quiver of the m-cluster category of type D n with m -1 copies of the Auslander-Reiten quiver of D b (A n-1 )/τ nm-m+1 , cf. Theorem 5.2 of [START_REF] Baur | A geometric description of the m-cluster categories of type Dn[END_REF].

The difficulty arising in type A has to do with the additional symmetry of the Dynkin graph of type A, i.e. with the involution sending the first to the last node, the second to the second-to-the-last, etc.

Let now Q be of Dynkin type A n . For odd m, C. Ducrest([D08]) was recently able to answer the question of the components of the mth power. For even m, she provides a partial description. Her result is the following (see Section 4 of [D08]):

1 to be precise: the arrows in the m-power arise from restricted sectional paths of length m, and we are taking the restricted mth power.

Figure 1 .Figure 2 .

 12 Figure 1. The Auslander-Reiten quiver of C for A 3

1. 3 .

 3 The m-cluster category. In 2005, Keller ([Ke05]) has introduced the mcluster categories as a natural generalisation of the cluster categories. Again, let Q be a quiver whose underlying graph is of Dynkin type A, D or E. Let [1] be the shift and τ the Auslander-Reiten translate as before. Let F m be the auto-equivalence τ -1 • [m] of D b (kQ), for m ≥ 1. Definition. The m-cluster category C m := C m Q := D b (kQ)/F m (of type Q) is the orbit category with objects the F m -orbits of objects of D b (kQ) and with morphisms Hom C m ( X, Y ) = i∈Z Hom D b (kQ) (F i m X, Y ) where X and Y are representatives of the F m -orbits of X and Y respectively.

Figure 3 .

 3 Figure 3. The Auslander-Reiten quiver of C 2 for A 2

  Example 2.1 (Hexagon). Let us illustrate this in the case n = 4

  Example 2.3. Let m = 2 and n = 3; Π is an octagon. Its 2-diagonals are of the form (1, 4), (1, 6), (2, 5), etc.

We have repeated the first slice (1, 3) → (1, 4) → (1, 5) at the end to indicate how the quiver wraps around.

Observe that the quiver Γ(4, 1) is equal to the Auslander-Reiten quiver of the cluster category of type A 3 (Figure 1). More generally, one can show that the quiver of diagonals of an n + 2-gon encodes the cluster category of type A n-1 :

As a consequence of this, the cluster category C Q is equivalent to the additive category generated by the mesh category of the stable translation quiver Γ(n, 1) of diagonals of an n + 2-gon. For details, we refer the reader to [START_REF] Caldero | Quivers with relations arising from clusters (An case)[END_REF] and [START_REF] Baur | A geometric description of m-cluster categories[END_REF].

Remark. The cluster category of type D n can be modelled in a similar way if we use a punctured n-gon. This has been done by Schiffler in [S06] using arcs in the punctured polygon. BAUR Theorem 3.3.

where Γ i is the Auslander-Reiten quiver of an orbit category of D b (A n ) under an auto-equivalence of the form τ -s • [r] for some t and for some s, r where we can assume s < n.

Furthermore, if m is odd then r = (m -1)/2 and s = 1 2 ((m -1)(n -1)) + 1. The even case is more complicated. One can show that for even m, we have m/2 ≤ r ≤ m. Example 3.1 above shows that r = m does occur.

Remark. If m is odd, one can show that the mth power only has one connected component per row of the original quiver Γ(nm, 1). In the even case, there are examples where we obtain one component per row and examples where there are two components per row. For details we refer to [D08].

C. Ducrest has developed a programme to calculate all components of the mth power of Γ(m, n) for all n, m ≤ 20. This programme is available online at http://www.math.ethz.ch/∼baur/algo/ and the documentation explaining how it works is [D208]. We hope that this programme will help us finding the complete answer for m even.


Mathieu Lugiez

Michel Ménard

Abdallah El-Hamidi

Dynamic Texture Extraction And Video Denoising

 the decomposition models based on Total Variation (TV) appear as a very good way to extract texture from image sequences. Indeed, videos show up characteristic variations along the temporal dimension which can be catched in the decomposition framework. However, there are very few works in literature which deal with spatio-temporal decompositions. Thus, we devote this paper to spatio-temporal extension of the spatial color decomposition model. We provide a relevant method to accurately catch Dynamic Textures (DT) present in videos. Moreover, we obtain the spatio-temporal regularized part (the geometrical component), and we distinctly separate the highly oscillatory variations, (the noise). Furthermore, we present some elements of comparison between several models in denoising purpose.

Introduction

Decomposing an image into meaningful components appears as one of major aims in recent development in image processing. The first goal was image restoration and denoising; but following the ideas of Yes Meyer [START_REF] Meyer | Oscillating patterns in image processing and nonlinear evolution equations[END_REF], in the Total Variation minimization framework of L. Rudin, S. Osher and E. Fatemi [START_REF] Rudin | Nonlinear total variation based noise removal[END_REF], image decomposition into geometrical and oscillatory components grants access to the textured part of an image, rejected by the previous model. We only cite, among many others, most recent works which we appear like most relevant and useful papers. In this way, reader can refer to the work of Aujol et al. [START_REF] Aujol | Image decomposition into a bounded variation component and an oscillating component[END_REF] [4], Aujol and Chambolle [START_REF] Aujol | Dual norms and image decomposition models[END_REF], Aujol and Kang [START_REF] Aujol | Color image decomposition and restoration[END_REF], Vese and Osher [START_REF] Vese | Color texture modeling and color image decomposition in a variational-PDE approach[END_REF], Gilles [START_REF] Gilles | Noisy image decomposition: A new structure, texture and noise model based on local adaptivity[END_REF] and more recently Bresson and Chan [START_REF] Bresson | Fast minimization of the vectorial total variation norm and applications to color image processing[END_REF], Duval et al. [START_REF] Duval | A projected gradient algorithm for color image decomposition[END_REF] and Aubert et al. [START_REF] Aubert | On a class of ill-posed minimization problems in image processing[END_REF].

We are aiming to decompose videos into three components, regular one, the geometry u, oscillating one v, representative of texture and highly oscillating one, the noise w (see Fig. 1 for a grayscale image decomposition example). So, we deal with color image sequences in extending to time existing reliable models. Moreover, through the texture component, we attempt to determinate the dynamics present in sequences, which will be suited for future work on Dynamic Texture (DT).

After have introduced different minimization functionals and a DT definition, we present the color spatio-temporal decomposition framework and subsequently their implementation. Then, we show some significant results, in particular the dynamicity of DT and comparing the denoising ability of the different models.

Decomposition models

In 2001, Y. Meyer in [START_REF] Meyer | Oscillating patterns in image processing and nonlinear evolution equations[END_REF] has proposed an extension of the famous image restoration model proposed by Rudin, Osher and Fatemi [START_REF] Rudin | Nonlinear total variation based noise removal[END_REF], based on the BV 3 space:

inf u∈BV F ROF (u) J(u) + 1 2λ f -u 2 L 2 (1)
where f is the original image and the parameter λ controls the L 2 -norm of the residual part f -

(u + v + w).
He has suggested an appropriate space (G 4 ) to model oscillating signals (i.e texture and noise), close to the dual space of BV . Aujol et al. in [START_REF] Aujol | Image decomposition into a bounded variation component and an oscillating component[END_REF] solve the Meyer algorithm, and propose the following minimization functional:

inf (u,v)∈BV ×G/f =u+v F A2BC (u, v) J(u) + J * (u) + 1 2λ f -u -v 2 L 2 (2)
where J * is the Legendre-Fenchel transform 5 of J, so J is the indicator function on

G η = v ∈ G/ v G ≤ η 6 .
To decompose image sequences in suitable components we propose to extend the Aujol-Chambolle [START_REF] Aujol | Dual norms and image decomposition models[END_REF] decomposition model. It rely on dual norms derived from BV , G and E 7 spaces. Authors propose to minimize the following discretized functional:

inf (u,v,w)∈X 3 /f =u+v+w F AC (u, v, w) = J(u) Regularization: TV + J * v η Texture extraction + B * w δ
Noise extraction by wavelet shrinkage

+ 1 2λ f -u -v -w 2 X Residual part (3)
where X is the Euclidian space R N ×N . J * and B * , are the Legendre-Fenchel transform of respectively J and B. See Fig. 1 for an image decomposition example through F AC .

3 BV (Ω) is the subspace function u ∈ L 1 (Ω) such that the following quantity, the total variation of u, is finite:

J(u) = sup Ω u(x)div(ξ(x))dx such that ξ ∈ C 1 c (Ω, R 2 ), ||ξ|| L ∞ (Ω) ≤ 1 4 G, is a Banach space, composed of all distributions v: v = ∂1g1 + ∂2g2 = div(g) with g1 and g2 ∈ L ∞ (R 2 ) endowed with the following norm v G = inf { g L ∞ (Ω,R 2 ) /v = div(g), g = (g1, g2), |g(x)| = (|g1| 2 + |g2| 2 )(x)} 5 F * (v) = supu( u, v L 2 -F (u))
, where ., . L 2 stands for the L 2 inner product. 6 i.e: J * ( v η ) = 0 if v ∈ Gη, +∞ elsewhere 7 E is a Banach space to model very oscillating patterns: E =

Repetitivity in video and Dynamic Texture (DT)

A new issue in texture analysis is its extension to the temporal dimension, a field known as Dynamic Texture Analysis. In DTA, the notion of self-similarity central to conventional image textures is extended to the spatio-temporal domain. DT typically results from processes such as water flows, smoke, fire, flag blowing in the wind or moving escalator. Important tasks are thus detection [START_REF] Dedeoglu | Real-time fire and flame detection in video[END_REF], segmentation [START_REF] Chetverikov | A brief survey of dynamic texture description and recognition[END_REF], synthesization and perceptual characterization [START_REF] Péteri | Dynamic texture recognition using normal flow and texture regularity[END_REF] of DT.

Image sequences present variations along time dimension. These variations could be, purely temporal (led blinking), spatial in movement (a leaf waving into wind like as shown in Fig. 2:(1)), periodic or pseudo-periodic (like an escalator or the low frequencies dues to quasi periodic anemone contractions as shown in Fig. 2:(3)) or spatio-temporal (flowing water and wavelets as shown in Fig. 2:(2)).

To properly catch all this spatio-temporal variation (i.e DT) and effectively separate them from spatio-temporal structures we consider a video as an pseudo 3-D image, i.e a volume, so that we can apply 2-D image algorithms correctly extended to the 2-D+t case. Indeed, straight spatial (i.e frame by frame) point of view is not sufficient to rightly discern patterns presenting dynamic process or movement. Moreover since static informations are redundant in a sequence, and since we take into account temporal variations, we grant to decomposition process to be more accurate and efficient for discerning the geometry and texture.

2 Spatio-temporal extension

Spatio-temporal structure and discretization

We assume that we have a given image sequence f ∈ L 2 (Ω), where Ω is an open and bounded domain on R 3 , with Lipschitz boundary. In order to recover u, v and w from f, we propose: -an extended discrete version of gradient vector: |∇u| xyt given by:

(∇u) i,j,k = (∇u) x i,j,k , (∇u) y i,j,k , (∇u) t i,j,k (4) where,
(∇u

) t i,j,k = u i,j,k+1 -u i,j,k-1 if k < N 0 if k = 1 or N
-an extended discrete total variation definition:

J(u) = Ω c=R,G,B |((∇u) x i,j,k ) c | 2 + |((∇u) y i,j,k ) c | 2 + α|((∇u) t i,j,k ) c | 2 dxdydt (5)
We introduce the constant α to maintain homogeneity between space and time components. The quantization steps can be different along space and time dimensions. In practice, we often set it to one, but user can tune it.

-an adapted definition of G: inspired by [START_REF] Vese | Color texture modeling and color image decomposition in a variational-PDE approach[END_REF] for the vectorial RGB case, extended to the third dimension:

Definition 1. Let G denote the Banach space consisting of all generalized vectorvalued functions v(x, y, t) = (v R (x, y, t), v G (x, y, t), v B (x, y, t)) which can be written as:

v(x, y, t) = (div g R , div g G , div g B ) (6)
g x,c , g y,c , g t,c ∈ L ∞ (R 3 ), c = R, G, B ,
induced by the norm v * defined as the lower bound of all L ∞ norms of functions | g|, where

| g| = | g R | 2 + | g G | 2 + | g B | 2 = c=R,G,B ((g x,c ) 2 + (g y,c ) 2 + (g t,c ) 2 ),
and where the infinitum is computed over all decompositions (6) of v.

-an adapted shrinkage scheme: In [START_REF] Dubois | Adding a noise component to a color decomposition model for improving color texture extraction[END_REF], authors use a connection given by Weickert in [START_REF] Weickert | Diffusion filters and wavelets: can they learn from each other?[END_REF] between wavelet shrinkage and non linear diffusion filtering to introduce the noise component in color image decomposition model. Indeed, by considering an explicit discretization and relating it to wavelet shrinkage, Weickert gives shrinkage rules where all channels are coupled. The formula states a general correspondence between a shrinkage function W ST (S θ (w i j , δ)) and the total variation diffusivity of an explicit nonlinear diffusion scheme. So, in order to steer the evolution of all three channels in spatio-temporal domain, the following shrinkage function W ST , relying on weight of S θ for the wavelet coefficient w j i , is proposed:

S θ (w j i ) = w j i   1 -39θ   i c * (w j R,i ) 2 + c * (w j Gi ) 2 + c * (w j Bi ) 2   -1    (7)
where w i j is the wavelet coefficient, j the resolution and c = 1 if i ∈ {x, y, t}, c = 2 if i ∈ {xy, xt, yt} and c = 4 if i = xyt. The soft wavelet thresholding functional W ST for a given wavelet coefficient, with threshold δ is given by:

W ST (S θ (w j i ), δ) =      S θ (w j i ) -δ if S θ (w j i ) ≥ δ 0 if |S θ (w j i )| ≤ δ S θ (w j i ) + δ if S θ (w j i ) ≤ -δ (8)

Numerical implementation

Thanks to recent advances in color image processing ( [START_REF] Duval | A projected gradient algorithm for color image decomposition[END_REF] and [START_REF] Bresson | Fast minimization of the vectorial total variation norm and applications to color image processing[END_REF]), motivating by the well known Chambolle's projection [START_REF] Chambolle | An algorithm for total variation minimization and its applications[END_REF], we are able now to use a projected gradient algorithm computing the solution of this dual formulation problem extended to time. So we dispose of an efficient way to numerically solve the different minimization problem induced by the extended functional (3), using fixed point method: P 0 = 0, and

P n+1 i,j,k = P n i,j,k + τ (∇(div(P n ) -f λ )) i,j,k max(1, P n + τ (∇(div(P n ) -f λ )) i,j,k 2 ) (9)
with τ < λ η to ensure the convergence of the algorithm [START_REF] Bresson | Fast minimization of the vectorial total variation norm and applications to color image processing[END_REF].

In order to compute the solution of problem (1) we propose the following numerical scheme adapted to the vectorial formulation (Eq.( 4) -( 9)) according to spatio-temporal point of view:

u ROF = f -P G λ (f) vROF = P G λ (f)
To solve the problem (2), following Aujol et al. [START_REF] Aujol | Image decomposition into a bounded variation component and an oscillating component[END_REF], we adapted the solution to the vectorial and spatio-temporal (Eq.( 4) -( 9)) case and propose the following algorithm:

1. Initialization: u 0 = 0, v 0 = 0 2. Iterations:

ṽA2BC = P Gη (f -u A2BC ) ũA2BC = f -u A2BC -ṽA2BC -P G λ (f -ṽA2BC ) 3. Repeated until: max(|ũ A2BC -u A2BC |, |ṽ A2BC -v A2BC |) ≤
So, to resolve the minimization problem (3) we propose to follow the numerical scheme of Aujol and Chambolle [START_REF] Aujol | Dual norms and image decomposition models[END_REF] adapted to vectorial formulation (Eq.( 4) -( 9)):

1. Initialization: u 0 = 0, v 0 = 0, w 0 = 0 2. Iterations:

wAC = P δB E (f -u AC -v AC ) = f -u AC -v AC -W ST (f -u AC -v AC , δ) ṽAC = P Gη (f -u AC -wAC ) ũAC = f -u AC -ṽAC -wAC -P G λ (f -ṽAC -wAC ) 3. Repeated until: max(|ũ AC -u AC |, |ṽ AC -v AC |, | wAC -w AC |) ≤
3 Numerical Results: Static versus Dynamic Decomposition (SD vs DD)

All images and results are computed from DynTex [START_REF] Péteri | Dyntex : A comprehensive database of dynamic textures[END_REF], a dynamic texture database which provides a diverse high-quality DT. DynTex sequences come from natural scenes presenting a wide variety of moving process. Such diversity grants user to identify and emphasize a lot of aspects in testing purpose.

Dynamic Texture Extraction

To analyze the influence of the dynamic property of the texture we present a comparison between the two methods of decomposition, SD versus DD, (both computed with same parameters). We can easily see that time impact in result: water in Fig. 3 is well regularized and fluid aspect is well represent into the v AC component. The regularization is more robust to illumination and movement constraints. Moreover, if user tunes parameters to obtain a stronger regularization, our algorithm is able to catch wider waves into spatio-temporal texture component: see around the circumference of fountain in Fig. 3, small wavelets are well catched in v AC than wider waves. It's a matter of deep in spatio-temporal texture extraction, which our algorithm is able to deal with, like shown in Fig. 4.

We present, in Fig. 5, a part of a decomposed sequence of flowing water under wood bridge. We can see the static aspect of u AC component, regularized in space and in time. It seems to be freezed, although texture component, v AC , present a real dynamic. Only moving things or objects presenting dynamicity are taken in account in the temporal divergence part of v AC component.

The third part of texture component play a key role in our process, movement information is well captured. In this way we obtain the dynamicity present in video through oscillations along time dimension. These results will be useful for future work on dynamic texture characterization.

Video denoising

We present in this section some elements of comparison between the different models in denoising purpose. Fig. 6 to 9 present the decomposition of one image, pick up from a sequence of eight images, noised separately (and decomposed conjointly in the DD process). We present the result of denoising process of the different models. One can easily see that DD clearly outperform SD whatever the model. Moreover the AC functional give the better results even if the texture component catch a part of noise in the decomposition process. In the case of very noisy sequence, (σ = 50, SNR=10.85 and PSNR=17.49), computed with exactly the same parameters and the same sequence, we have: -ROF: for SD we obtain a SNR=16.40 a PSNR=22.90 and for DD we have respectively 21.04 and 27.55.

-A2BC: for SD we obtain a SNR=18.32 a PSNR=24.83 and for DD we have respectively 25.23 and 31.73.

-AC: for SD we obtain a SNR=20.58 a PSNR=27.12 and for DD we have respectively 26.91 and 33.42.

We have investigate in the present paper the spatio-temporal extension of color decomposition models through projection approach. One can see that taking into account variations along the temporal axe appears as a useful and innovative way to catch the dynamic texture present in videos. Moreover this extension provide a useful tool for video denoising, indeed dynamic model clearly outperform classical decomposition, reinforcing the denoising ability of T V based models. So, we provide an efficient tool to separate dynamic texture, structure and noise in image sequence.

.

Fig. 1 .

 1 Fig. 1. Grayscale image decomposition with AC functional. From top to bottom and left to right: Texture component (vAC), original noisy image, noise component (wAC) and regular or geometrical part (uAC).

Fig. 2 .

 2 Fig. 2. 2D+t block of two dynamic textures. Here, a dynamic texture is seen as a data cube. One can clearly see at (1) leaves blowing in wind: a spatial texture in movement, at (2) flowing water and wavelets: spatio-temporal texture and at (3) spatio-temporal low frequencies dues to periodic anemone contractions.

Fig. 3 .

 3 Fig. 3. Left: the geometrical component, uAC, in SD (top) and its texture component vAC (bottom). Center and top: the original image, bottom: Difference between the static and dynamic vAC components. Right: the geometrical component in DD (top) and its dynamic texture vAC (bottom).

Fig. 4 .

 4 Fig. 4. First row: original image from sequence 6487310 of DynTex, the ROI and a visualization of ROI. Second row: SD of ROI with λ = 140, η = 140 from left to right uAC component and its visualization, since vAC component and its visualization. Third row, DD of the ROI with λ = 10 and η = 10 (weak regularization and texture extraction). Fourth row, DD of the ROI with λ = 140 and η = 140 (strong regularization and texture extraction). We can see than SD is not able to deal with dynamicity, moreover, in DD user can catch different scales of waves in the decomposition process.

Fig. 5 .

 5 Fig. 5. From left to right, the geometrical component, uAC, in classic color decomposition (top) and its texture and noise component vAC (bottom). The original image (center and top), the time influence in calculus of divergence of vAC component (center and bottom). Then our new dynamic decomposition components (right).

Fig. 6 .

 6 Fig. 6. From left to right, an original image from the sequence 64cc610 of DynTex, the noised version (gaussian noise with void mean and σ = 20, SNR=14.53, PSNR=21.18), and a visualization of ROI.

Fig. 7 .

 7 Fig. 7. ROF decomposition with λ = 25. Top: SD, bottom: DD. Left the vROF component, center uROF and right a visualization of ROI. We have a SNR equal to 23.03 and a PSNR equal to 29.54 with SD and respectively 29.09 and 35.59 with DD.

Fig. 8 .

 8 Fig. 8. A2BC decomposition with λ = 10 and η = 25. Top: SD, bottom: DD. Left the vA2BC component, center uA2BC and right a visualization of ROI. We have a SNR equal to 24.56 and a PSNR equal to 31.12 with SD and respectively 29.57 and 36.08 with DD.

Fig. 9 .

 9 Fig. 9. AC decomposition with λ = 10, η = 15 and δ = 45. Top: SD, bottom: DD. Left the vAC component, center uAC and right a visualization of ROI. We have a SNR equal to 25.10 and a PSNR equal to 31.60 with SD and respectively 30.24 and 36.74 with DD.


Mathieu Lugiez

Michel Ménard

Abdallah El-Hamidi

DYNAMIC COLOR TEXTURE MODELING AND COLOR VIDEO DECOMPOSITION USING BOUNDED VARIATION AND OSCILLATORY FUNCTIONS

Dynamic, or temporal, texture is a spatially repetitive, timevarying visual pattern that forms an image sequence with a certain temporal stationarity. Important tasks are thus the detection, segmentation and perceptual characterization of Dynamic Texture (DT). Following recent work, color image decomposition appears as a good way to reach these different aims, however, to our best knowledge, no proposed model is currently able to deal with temporal aspect, inherent to color image sequences. The major contribution of this paper is to adapt static decomposition model to time aspect in order to deal with videos and color image sequences. In this paper we propose an extended decomposition model which splits a video into two components, a first one containing geometrical information, the structure of the sequence and a second one dynamic color texture and noise. Examples for color video decomposition and characterization of real dynamic present in texture component will be presented.

Introduction

Motivation

A new issue in texture analysis is its extension to the temporal domain, a field known as Dynamic Texture (DT) analysis. In DT, the notion of self-similarity central to conventional image textures is extended to the spatiotemporal domain. DT are typically result from processes such as of water flows, smoke, fire, a flag blowing in the wind, a moving escalator, or a walking crowd. Important tasks are thus the detection, segmentation and perceptual characterization of dynamic textures. The ultimate goal is to be able to support video queries based on the recognition of the actual natural and artificial dynamic texture processes.

Following recent work, color image decomposition into geometrical and texture components appears as a good way to reach this aim in extracting meaningful information, i.e texture component, independently of geometrical information.

In this way, we propose to extend spatial color decomposition model to spatiotemporal domain, and attempt to highlight time influence present in video to characterize dynamic texture. To the best of our knowledge, no such time adaptation is currently available.

Overview of the paper

The aim of this work is to extend a model, which decompose color image into two components, a first one containing geometrical structure U and a second one V, holding textural information and noise. So, we aim to deal with color image sequences in extending to time existing reliable model. Moreover, through decomposition of texture component, we attempt to determinate time impact, showing up dynamicity present in sequences, which will be suit for future work on dynamic texture.

In the first place of this paper we introduce the extended minimization functional problem and the associate discrete framework in which we place ourself and which is an appropriate one in image sequence processing. In a second part we present the extended time decomposition model and subsequently its color implementation and choice of parameters. Then, in the last part, we show some significant results and attempt to present time weight in dynamic texture extraction through color image sequence.

Time extension of decomposition model

Decomposing an image into meaningful component appears as one of major aims in recent development in image processing. The first goal was image restoration and demising; but following the ideas of Yves Meyer, in total variation minimization framework of L. Rudin, S. OSher and E. Fatemi, image decomposition into geometrical and oscillatory (i.e texture) component appears as usefull and very interesting way in computer vision and image analysis. There is a very large literature and recent advances on image decomposition models, image regularization and texture extraction and modelling. So, we only cite, among many others, most recent works which we appear like most relevant and usefull paper. In this way, reader can refer to the work of Stark et al. [START_REF] Starck | Image decomposition via the combination of sparse representations and a variational approach[END_REF], Aujol et al. [START_REF] Aujol | Image decomposition into a bounded variation component and an oscillating component[END_REF], [START_REF] Aujol | Structure-texture image decomposition -modeling, algorithms, and parameter selection[END_REF], Aujol and Chambolle [START_REF] Aujol | Dual norms and image decomposition models[END_REF], Aujol and Ha Kang [START_REF] Aujol | Color image decomposition and restoration[END_REF] and Vese and Osher [START_REF] Vese | Modeling textures with total variation minimization and oscillating patterns in image processing[END_REF], [START_REF] Vese | Image denoising and decomposition with total variation minimization and oscillatory functions[END_REF], [START_REF] Vese | Color texture modeling and color image decomposition in a variational-PDE approach[END_REF] to cover the most recent and relevant advances.

Decomposition model and functional spaces

In order to decompose image sequences in suitable components we propose to extend the Osher-Vese [START_REF] Vese | Color texture modeling and color image decomposition in a variational-PDE approach[END_REF] color decomposition model. Their approach derived from Meyer decomposition model [START_REF] Meyer | Oscillating patterns in image processing and nonlinear evolution equations[END_REF]. Its rely on space of functions of bounded variation, BV and Banach space, G. Authors propose to minimize the following discretized functional:

inf (u,v)∈BV ×G/f =u+v F (u, v) = |∇u| + λ v G (1)
BV is a good space to model functions with discontinuities along lines and curves, that can represent and preserve edges in an image. For the reader convenience we recall here its definition [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF]: Definition 1. BV (Ω) is the subspace functions u ∈ L 1 (Ω) such that the following quantity, called the total variation of u, is finite:

J(u) = sup Ω u(x)div(ξ(x))dx (2) such that ξ ∈ C 1 c (Ω, R 2 ), ||ξ|| L ∞ (Ω) ≤ 1
In order to improve representation of oscillating patterns, Meyer in [START_REF] Meyer | Oscillating patterns in image processing and nonlinear evolution equations[END_REF] has introduced the Banach space, G, which modeling signals with large oscillations, in particular textures and noise. A function belonging to G may have large oscillations and nevertheless have a small norm. Thus the norm on G is well-adapted to capture the oscillations of a function in an energy minimization method. We recall here the definition of G: Definition 2. G is the Banach space composed of the distributions f which can be written f = ∂ 1 g 1 + ∂ 2 g 2 = div(g) with g 1 and g 2 in L ∞ (Ω). On G, the following norm is associate:

v G = inf g L ∞ (Ω,R 2 ) /v = div(g), g = (g1, g2), |g(x)| = (|g1| 2 + |g2| 2 )(x) (3)

Spatiotemporal structure and discretization

To take into account the spatiotemporal structure, we consider a video as an 3-D image [START_REF] Aubert | Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations[END_REF], i.e a volume, so that we can apply 2-D image algorithms extended to the 3-D case.

We assume that we have a given image sequence f ∈ L 2 (Ω), where Ω is an open and bounded domain on R 3 , with Lipschitz boundary. In order to recover u and v from f , we propose:

-An extended discrete version of gradient vector |∇u| given by:

(∇u) i,j,k = (∇u) 1 i,j,k , (∇u) 2 i,j,k , (∇u) 3 i,j,k (4)
(∇u) 1 i,j,k = u i+1,j,k -u i,j,k if i < N 0 if i = N (∇u) 2 i,j,k = u i,j,k -u i,j+1,k if j < N 0 if j = N (∇u) 3 i,j,k = u i,j,k -u i,j,k-1 if k < N 0 if k = N
-An extended discrete total variation definition:

J(u) = (∇u) 1 i,j,k + (∇u) 2 i,j,k + α(∇u) 3 i,j,k (5)
we introduce the α constant to maintain homogeneity between space and time component. It's mainly for numerical implementation, to avoid discretization problem due to quantization step, which be different along space and time dimension. In practice, we often set it to one, but user can adapt it to less, more or in function of frame per second, or quickness of movement present in sequence, to ensure most reliability and homogeneity.

-An adapted definition of G, inspired by [START_REF] Vese | Color texture modeling and color image decomposition in a variational-PDE approach[END_REF] for the vector case, extended to the third dimension:

Definition 3. Let G denote the Banach space consisting of all generalized vector-valued functions v(x, y, t) = (v R (x, y, t), v G (x, y, t), v B (x, y, t)) which can be written as:

v(x, y, t) = (div gR, div gG, div gB) (6) g1,c, g2,c, g3,c ∈ L ∞ (R 3 ), c = R, G, B ,
induced by the norm v * defined as the lower bound of all L ∞ norms of functions | g|, where

| g| = | g R | 2 + | g G | 2 + | g B | 2 = c=R,G,B ((g 1,c ) 2 + (g 2,c ) 2 + (g 3,c
) 2 ), and where the infinitum is computed over all decompositions (6) of v.

Description of the extended decomposition model

For the reader convenience we only use, by now, useful indices in our formulas to avoid overloaded notation, so we will use c indices to denote color channel (c ∈ R, G, B) and 1, 2 and 3 exponents to represent the dimension to which objects belong.

Description of the model

We propose the following minimization problem inspired by [START_REF] Vese | Modeling textures with total variation minimization and oscillating patterns in image processing[END_REF] and [START_REF] Vese | Color texture modeling and color image decomposition in a variational-PDE approach[END_REF], for each color channel:

inf u,g 1 ,g 2 ,g 3 Gp(u, g1, g2, g3) = |∇u| + λ |f -u -∂xg1 -∂yg2 -∂tg3| 2 dxdydt +µ ( g 2 1 + g 2 2 + g 2 3 ) p dxdydt 1 p (7)
where λ, µ are tuning parameters (see next section for more useful details). Let reintroduce that u = (u R , u G , u B ), and

g i = ( g i,R , g i,G , g i,B ), i ∈ {1, 2, 3}.
Formally minimizing the above energy equation with respect to u, g 1 , g 2 , g 3 , yields the following Euler-Lagrange equation for each color channel:

u = f -∂xg1 -∂yg2 -∂tg3 + 1 2λ div ∇u |∇u| (8) µ( g 2 1 + g 2 2 + g 2 3 ) 1-p ( g 2 1 + g 2 2 + g 2 3 ) p-2 g1 = 2λ ∂ ∂x (u -f ) + ∂ 2 xx g1 + ∂ 2 xy g2 + ∂ 2 xt g3 (9)
µ( g 2 1 + g 2 2 + g 2 3 ) 1-p ( g 2 1 + g 2 2 + g 2 3 ) p-2 g2 = 2λ ∂ ∂y (u -f ) + ∂ 2 yx g1 + ∂ 2 yy g2 + ∂ 2 yt g3 (10)
µ( g 2 1 + g 2 2 + g 2 3 ) 1-p ( g 2 1 + g 2 2 + g 2 3 ) p-2 g3 = 2λ ∂ ∂t (u -f ) + ∂ 2 xt g1 + ∂ 2 yt g2 + ∂ 2 tt g3 (11)
To simplify the presentation, let us introduce the notation:

H(g1, g2, g3) = ( g 2 1 + g 2 2 + g 2 3 ) 1-p ( g 2 1 + g 2 2 + g 2 3 ) p-2

Numerical implementation

For the reader convenience, we present here a numerical scheme to solve our minimization problem. So to solve equations ( 8) to [START_REF] Aubert | Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations[END_REF] we use a semi-implicit finite differences scheme and an iterative algorithm, based on fixed point iteration [START_REF] Vese | Modeling textures with total variation minimization and oscillating patterns in image processing[END_REF]. We got, in order to compute gradient:

|∇u| • = |∇u| i,j,k = ui+1 -ui-1 2h 2 + uj+1 -uj-1 2h 2 + u k+1 -u k-1 2h 2 (12)
so, belong first dimension (x) we got:

       |∇u| i+ 1 2 ,j,k = u i+1 -u h 2 + u j+1 -u j-1 2h 2 + u k+1 -u k-1 2h 2 |∇u| i-1 2 ,j,k = u-u i-1 h 2 + u i-1,j+1 -u i-1,j-1 2h 2 + u i-1,k+1 -u i-1,k-1 2h
2 [START_REF] Lugiez | Spatiotemporal extension of color decomposition model and dynamic color structure-texture extraction[END_REF] in the same way, we compute gradient belong y and t dimension.

We are now in position to introduced complete numerical scheme to solving Euler-Lagrange equations:

u n+1 = 1 1 + 1 3λh 2 1 |∇| i+ 1 2 + 1 |∇| i-1 2 + 1 |∇| j+ 1 2 + 1 |∇| j-1 2 + 1 |∇| k+ 1 2 + 1 |∇| k-1 2 (14) *   f -∂xg1 -∂yg2 -∂tg3 + 1 3λh 2   u n i+1 |∇| i+ 1 2 + u n i-1 |∇| i-1 2 + u n j+1 |∇| j+ 1 2 + u n j-1 |∇| j-1 2 + u n k+1 |∇| k+ 1 2 + u n k-1 |∇| k-1 2     (15) g n+1 1 = 3λ µH(g n 1 , g n 2 , g n 3 ) + 6λ h 2 ∂xu n -∂xf + g n 1 i+1 + g n 1 i-1 h 2 + ∂ 2 xy g n 2 + ∂ 2 xt g n 3 (16) g n+1 2 = 3λ µH(g n 1 , g n 2 , g n 3 ) + 6λ h 2 ∂yu n -∂yf + ∂ 2 xy g n 1 + g n 2 j+1 + g n 1 j-1 h 2 + ∂ 2 yt g n 3
(17)

g n+1 3 = 3λ µH(g n 1 , g n 2 , g n 3 ) + 6λ h 2 ∂tu n -∂tf + ∂ 2 xt g n 1 + ∂ 2 yt g n 2 + g n 3 k+1 + g n 3 k-1 h 2 (18) (19)
where h denote the discrete step space, so in practice we set h = 1 in our algorithm, moreover in furtherance of time computing, we always use the most recent computed values. For the initialization of parameters we set:

u 0 = f , g 0 1 = -1 3λ ∂xf √ ∂xf 2 +∂yf 2 +∂tf 2 , g 0 2 = -1 3λ ∂yf √ ∂xf 2 +∂yf 2 +∂tf 2 and g 0 3 = -1 3λ ∂tf √ ∂xf 2 +∂yf 2 +∂tf 2

Choice of parameters

To obtain week regularization, we use parameters as: λ = 0.1, µ = 0.1 and we compute about 20 iterations. For strong regularization we got: λ = 0.005, µ = 0.005 iterated 50 times. So, for classic decomposition we use parameters as: λ = 0.01, µ = 0.01 iterated 20 times. For all results present in the next section we computed our decomposition on 8 images treated simultaneously as one block, but our algorithm is able to deal with bigger image block to catch lower frequences and wider range of time oscillating patterns.

Numerical results and time component

All images and results are compute from DynTex, the dynamic texture database [START_REF] Péteri | Dyntex: A comprehensive database of dynamic textures[END_REF] which provide a large and diverse database of high-quality dynamic textures. Dyntex sequences come from natural scene presenting a wide variety of moving process as flowing water, leaves blowing in wind, walking crowd... Such diversity grants user to identify and emphasize a lot of aspects in testing purpose. For more details, demonstration sequence, wider range of results and for a presentation of similar color image sequences decomposition, relying on different approaches in the same framework [START_REF] Lugiez | Spatiotemporal extension of color decomposition model and dynamic color structure-texture extraction[END_REF], please consult this URL: http://perso.univlr.fr/mlugiez.

Static vs dynamic decomposition

In order to prove that our dynamic decomposition method show more significant result than a static decomposition, we present a comparison between two methods (static and dynamic decomposition are both computed with same classic parameters). We can easily see that time impact in result, water in Fig. 1 is well regularized and fluid aspect is well represent in the V component. In Fig. 2 we can clearly see the reenforcement of moving cars texture without that static part and objects present in sequence are taken into account. Moreover the regularization is more robust to illumination and movement constrain.

Moreover, if user tunes parameters to obtain stronger regularization, our algorithm is able to catch wider waves in spatiotemporal texture component: see the circumference of fountain in Fig. 1, more regularized (in U component) than wider waves. It's a matter of deep in spatiotemporal texture extraction, which our algorithm is able to deal with.

Time impact in our decomposition

We present, in Fig. 3, a part of a decomposed sequence of flowing water under wood bridge. We can see the static aspect of U component, regularized in space and in time, seems to be freezed, although texture component, V, present a real dynamic, strengthened by time influence. Only moving things or objects presenting dynamicity are taken in account in the third part of V component (i.e g 3 ). In Fig. 4 we can distinctly see time influence, reed's branch oscillating under water flow is clearly highlight, waves present in basin's fountain are well regularized in U component, water dynamicity is totaly catch as texture. We can clearly see that is due to the third part of texture component, movement information is well captured in this component.In this way we obtain the dynamicity present in video through oscillations along time dimension. These results will be usefull for future work on dynamic texture characterization.

Fig. 1 .

 1 Fig. 1. From left to right, the geometrical component, U, in classic color decomposition (top) and its texture and noise component V (bottom). The original image (center and top), the difference between static and dynamic V component (center and bottom). Then our new dynamic decomposition components (right).

Fig. 2 .

 2 Fig. 2. From left to right, the geometrical component, U, in classic color decomposition (top) and its texture and noise component V (bottom). The original image (center and top), the difference between static and dynamic V component (center and bottom). Then our new dynamic decomposition components (right).

Fig. 3 .

 3 Fig. 3. From left to right, the geometrical component, U, in classic color decomposition (top) and its texture and noise component V (bottom). The original image (center and top), the time influence g3 part of V component (center and bottom). Then our new dynamic decomposition components (right).

Fig. 4 .

 4 Fig. 4. From left to right, the geometrical component, U, in classic color decomposition (top) and its texture and noise component V (bottom). The original image (center and top), the time influence g3 part of V component (center and bottom). Then our new dynamic decomposition components (right).