Tristan Roy
email: triroy@math.ucla.edu-barlaud

Éric Debreuve
email: debreuve@i3s.unice.fr

Michel Barlaud

Gilles Aubert
email: gaubert@math.unice.fr

Segmentation of a vector field: Dominant parameter and shape optimization

Keywords: Segmentation, vector field, dominant parameter, shape optimization, optical flow

Vector field segmentation methods usually belong to either of three classes: methods which segment regions homogeneous in direction and/or norm, methods which detect discontinuities in the vector field, and region growing or classification methods. The first two classes of method do not allow segmentation of complex vector fields and control of the type of fields to be segmented, respectively. The third class does not directly allow a smooth representation of the segmentation boundaries. In the particular case where the vector field actually represents an optical flow, a fourth class of methods acts as a detector of main motion. The proposed method combines a vector field model and a theoretically founded minimization approach. Compared to existing methods following the same philosophy, it relies on an intuitive, geometric way to define the model while preserving a general point of view adapted to the segmentation of potentially complex vector fields with the condition that they can be described by a finite number of parameters. The energy to be minimized is deduced from the choice of a specific class of field lines, e.g. straight lines or circles, described by the general form of their parametric equations. In that sense, the proposed method is a principled approach for segmenting parametric vector fields. The minimization problem was rewritten into a shape optimization and implemented by splinebased active contours. The algorithm was applied to the segmentation of precomputed optical flow fields given by an external, independent algorithm.

Introduction

A segmentation of a vector field is a partition of the domain of the field into subdomains such that each subdomain is as homogeneous as possible in terms of a certain homogeneity function. The standard deviation of the directions of the vectors contained in a subdomain is an example of a homogeneity function. The function may or may not be the same for each subdomain.

Segmentation of vector fields is part of a wider class of problems that can be called multichannel segmentation. Segmentation of color images is an example of such problems [START_REF] Ronfard | Region-based strategies for active contour models[END_REF][START_REF] Sapiro | Vector (self) snakes: A geometric framework for color, texture and multiscale image segmentation[END_REF]. More generally, it corresponds to the segmentation of an n-vector valued function. It can be solved by extending a single-channel method [START_REF] Mumford | Optimal approximations by piecewise smooth functions and associated variational problems[END_REF] by substituting a vectorial similarity measure, e.g. a weighted sum of the individual component similarities [START_REF] Sanchez | Displacement following of hidden objects in a video sequence[END_REF], for the scalar similarity measure. However, this straightforward extension does not allow to describe complex region characteristics. Basically, as far as vector fields are concerned, regions homogeneous in direction and/or norm can be detected. The dual approach is to detect discontinuities in the vector field [START_REF] Giachetti | Refinement of optical flow estimation and detection of motion edges[END_REF]. However, this approach is usually not suitable to determine regions necessary for subsequent processing tasks and it does not allow to control the type of fields to be segmented. Using the terminology of optical flow, the main motion can be segmented by thresholding the vector field magnitude [START_REF] Condell | Adaptative grid refinement procedures for efficient optical flow computation[END_REF] or, similarly, the image of frame-to-frame difference [START_REF] Wu | A gradient-based method for general motion estimation and segmentation[END_REF] of a video sequence. Even if the region of main motion is found based on a motion model [START_REF] Wu | A gradient-based method for general motion estimation and segmentation[END_REF], this kind of methods are better described as motion detections than motion segmentations. In order to segment complex vector fields or motions, the use of a model seems appropriate. A region growing, or merging, technique can be used to iteratively group pixels or patches together when they respect a common motion model (or vector field model). Precision can be increased while preserving coherence by implementing successive steps with increasing model complexity [START_REF] Choi | Multi-stage segmentation of optical flow field[END_REF]. However, a threshold is necessary in order to determine when to stop merg-ing. This can be avoided by replacing the merging decision with a classification algorithm [START_REF] Wang | Spatio-temporal segmentation of video data[END_REF]. Indeed, classification relates to a notion of closest distance with respect to motion candidates. Unfortunatelty, such a method is strongly constrained by the discretization of the vector field domain (pixels or patches). It might be difficult to impose only some smoothness to the segmentation boundary. A minimization approach with respect to the segmentation domains also allows to get rid of the thresholding while giving more flexibility to the representation of the domains [START_REF] Schnörr | Computation of discontinuous optical flow by domain decomposition and shape optimization[END_REF][START_REF] Cremers | Motion competition: Variational integration of motion segmentation and shape regularization[END_REF][START_REF] Roy | Vector field segmentation using active contours: Regions of vectors with the same direction[END_REF][START_REF] Cremers | Motion competition: A variational approach to piecewise parametric motion segmentation[END_REF]. These methods are adapted to optical flow segmentation (explicitly or implicitly when the input is a video sequence [START_REF] Cremers | Motion competition: Variational integration of motion segmentation and shape regularization[END_REF][START_REF] Cremers | Motion competition: A variational approach to piecewise parametric motion segmentation[END_REF]) and might not be intuitive enough when it comes to segment a vector field not directly related to a motion.

The proposed method follows the same philosophy combining a model and a theoretically founded minimization approach. Compared to existing methods, it relies on an intuitive, geometric way to define the model while preserving a general point of view adapted to the segmentation of potentially complex vector fields of R 2 with the condition that they can be described by a finite number of parameters.

Proposed approach 2.1 Context

The vector field to be segmented is denoted by E. Its components are denoted by E 1 and E 2 . The field is assumed to be smooth and normalized (the norm of E is equal to one everywhere). Two examples of vector fields are shown in Fig. 1. 1) represents the fact that the vector field is orthogonal to the normal along any field line (or, equivalently, collinear to the tangent) and Eq. ( 2) represents the fact that the norm of the tangent is never equal to zero. A typical example of vector fields in video is optical flow [START_REF] Lucas | An iterative image registration technique with an application to stereo vision[END_REF][START_REF] Horn | Detemining optical flow[END_REF][START_REF] Schnörr | Computation of discontinuous optical flow by domain decomposition and shape optimization[END_REF][START_REF] Blanc-Féraud | Motion estimation involving discontinuities in a multiresolution scheme[END_REF][START_REF] Aubert | Computing optical flow via variational techniques[END_REF][START_REF] Weickert | Variational optic flow computation with a spatio-temporal smoothness constraint[END_REF], i.e., the vector field describing the apparent motion between two successive frames of a video sequence. In this context, if W is the optical flow between frames f t and f t+1 , then E is given by

   E = W |W | f t+1 (x + W (x)) = f t (x) . ( 3
)
We will use the segmentation of precomputed optical flow fields as an illustration of the proposed method. We will assume that the optical flows are given by an external, independent algorithm.

Variational formulation

A segmentation problem can be solved in a variational context by defining an energy to be minimized. This energy is obviously a function of the boundaries of some/all of the segmentation domains (contour-based segmentation) and/or their regions (region-based segmentation). For simplicity, we will only consider the case of two domains: the object and the background. The term "object" must be understood as a region with specific characteristics and the expression "object domain" must be understood as an estimation of the actual region of the object. In the case of region-based segmentation, if both the object and the background domains are taken into account, the segmentation is said to involve region competition. When only the object domain is used, the energy is classically an integral on the object domain of a function characterizing the object that should be detected. This function is nonnegative and it is ideally equal to zero on the region of the object. It relates to a notion of homogeneity of the object in a broad sense. Our purpose is to find such functions and to provide a general method for the segmentation of vector fields of R 2 that can be described by a finite number of parameters. These functions are deduced from the choice of a specific class of field lines, e.g. straight lines or circles, described by the general form of their parametric equations. The general form of the proposed energies is

J(Ω) = Ω h(E, Ω, x) dx ( 4
)
where E is the vector field to be segmented and Ω is the unknown segmentation domain. Minimization of this type of energy requires special care since the set F of domains included in the image region, a bounded, connected, open set of R 2 , is not a vectorial space. The shape gradient approach is adapted to this problem. Although this aspect is not an original contribution of the paper, we will present it briefly in Section 2.3 since it is a prerequisite to the understanding of the actual contributions presented in Sections 3, 4, and 5.

Shape derivative

The shape derivative is a measure of the influence of a deformation of Ω on the value of (4). It is a function of Ω and of the deformation, V , a function from R to R 2 . In order to be derived, energy (4) should first be rewritten as a function of a deformation parameter τ

J(τ ) = Ω(τ ) h(E, τ, x) dx . (5)
By definition, parameter τ is such that

Ω(τ ) = T τ (Ω(τ = 0)) (6)
where Ω(τ = 0) is an initial domain and T τ is an unknown transformation that must be such that lim

τ →∞ Ω(τ ) = Ω ( 7
)
where Ω is the unique global minimizer of (4) (if this minimizer exists). Then, it was shown that [START_REF] Delfour | Shapes and geometries: Analysis, differential calculus and optimization[END_REF][START_REF] Aubert | Image segmentation using active contours: Calculus of variations or shape gradients?[END_REF][START_REF] Hintermüller | A second order shape optimization approach for image segmentation[END_REF][START_REF] Jehan-Besson | Dream 2 s: Deformable regions driven by an Eulerian accurate minimization method for image and video segmentation[END_REF] dJ dτ (τ, V ) =

Ω(τ ) ∂h ∂τ (E, τ, x) dx - ∂Ω(τ )
h(E, τ, x(s)) V (s) N τ (s) ds [START_REF] Caselles | Geodesic active contours[END_REF] where ∂Ω(τ ) is the oriented boundary of Ω(τ ), s is the arclength along Ω(τ ), and N τ is the inward unit normal of Ω(τ ). Given a domain Ω(τ ), we would like to know which deformation should be applied to Ω(τ ) in order to obtain a new domain "closer" to the domain for which ( 5) is minimal, if this domain exists and is unique. However, the first integral of (8) does not involve V explicitly. In other words, it is not readily usable. Nevertheless, function h is assumed to be such that there exists a function h such that

Ω(τ ) ∂h ∂τ (E, τ, x) dx = ∂Ω(τ )
h(E, τ, x(s)) V (s) N τ (s) ds .

(9) Actually, in some circumstances, this integral is equal to zero. It will be the case for the energies proposed in Sections 3, 4, and 5. Therefore, Section 2.4 is written for this case only.

Minimization algorithm

Since the purpose is to minimize [START_REF] Blanc-Féraud | Motion estimation involving discontinuities in a multiresolution scheme[END_REF], deformation V must be chosen such that derivative (8) is negative.

As noted in Section 2.3, the first integral of (8) will be equal to zero in our case. Clearly, the following expression satisfies this condition

V τ = h(E, τ, x) N τ . (10)
Applying successive deformations [START_REF] Chopp | Computing minimal surfaces via level set curvature flow[END_REF] to Ω(τ ) for increasing values of τ , domain Ω (seen as a function of τ ) should converge toward the unique global minimizer of (4) (if this minimizer exists). This is typically an active contour process [START_REF] Kass | Snakes: Active contour models[END_REF][START_REF] Caselles | A geometric model for active contours[END_REF][START_REF] Ronfard | Region-based strategies for active contour models[END_REF][START_REF] Caselles | Geodesic active contours[END_REF] whose evolution equation is, noting that V τ is equal to ∂Γ ∂τ where Γ is a short notation for ∂Ω,

   ∂Γ ∂τ (τ ) = h(E, τ, x) N τ Γ(τ = 0) = Γ 0 (11)
where Γ 0 is an initial guess, or estimate, of the segmentation contour. Therefore, the minimization of (4) can be achieved by iteratively deforming a contour according to [START_REF] Condell | Adaptative grid refinement procedures for efficient optical flow computation[END_REF] until ( 8) is equal to zero.

3 Segmentation of translation domains

Translation domain

Let C be the unit circle of R 2 and let C + be the subset of C defined by (cos(θ), sin(θ)), θ ∈]π/2, π/2].

A region of the domain of E containing only field lines that are straight lines is called a translation domain (See Fig. 1). Equivalently, a domain Ω of F is a translation domain if and only if there exists a unique parameter a(Ω) of C + such that for any field line L verifying L ∩ Ω = ∅ there exists a unique constant c(L) of R such that

a(Ω) • x = c(L), x ∈ L ∩ Ω . ( 12
)
Vector a(Ω) is called the translation parameter of Ω.

It is actually the normal to the straight field lines. In our context, the specific value of c(L) is of no importance and plays the role of an unnecessary parameter. Therefore, a condition equivalent to [START_REF] Cremers | Motion competition: Variational integration of motion segmentation and shape regularization[END_REF] while not involving c(L) should be found. Let us parametrize L by p. For any

x(p) of L ∩ Ω a(Ω) • x(p) = c(L) (13) ⇐⇒ a 1 (Ω)x 1 (p) + a 2 (Ω)x 2 (p) = c(L) . ( 14
)
The derivative of ( 14) with respect to p combined with (1) leads to the following system of equations

a 1 (Ω) a 2 (Ω) E 2 (x(p)) -E 1 (x(p)) ẋ(p) = M ẋ(p) = 0 . ( 15
)
Combining ( 2) and ( 15), it can be concluded that the determinant of M must be equal to zero. Therefore, a domain Ω of F is a translation domain if and only if there exists a unique a(Ω) of C + such that for any x of Ω a(Ω)

• E(x) = 0 . ( 16
)

Dominant translation parameter

From ( 16), it can be deduced that if Ω is a translation domain then a(Ω) is a global minimizer of

K tra Ω : C + → R α → Ω (α • E(x)) 2 dx . (17)
Indeed, K tra Ω is clearly nonnegative and K tra Ω (a(Ω)) is equal to zero. Moreover, this minimizer is unique in C + since any candidate must satisfy [START_REF] Gomes | Reconciling distance functions and level sets[END_REF], i.e., must be orthogonal to E. Therefore, it must be collinear to a(Ω), i.e., equal to a(Ω) in C + .

Looking from a domain point of view (as opposed to the parameter point of view), it can also be deduced from [START_REF] Gomes | Reconciling distance functions and level sets[END_REF] that Ω minimizes the following energy

G tra : F → R ω → ω (a(ω) • E(x)) 2 dx ( 18
)
with a(ω) defined as the unique global minimizer of K tra ω . However, there is no guarantee that for any ω of F there exists a unique global minimizer of K tra ω and that K tra ω is equal to zero for this minimizer. In other words, ω might not have a translation parameter (namely, if it is not a translation domain). Nevertheless, as long as K tra ω has a unique global minimizer â(ω), this minimizer is called the dominant translation parameter of ω.

Practical formulation

It is clear that any subdomain of a translation domain is a minimizer of energy [START_REF] Horn | Detemining optical flow[END_REF]. As a consequence, the problem of finding translation domains should be reformulated in finding translation domains having an area as big as possible. This is equivalent to minimizing G tra (Ω) while maximizing the area of Ω, i.e., minimizing the following energy

J tra : F tra → R Ω → Ω (â(Ω) • E(x)) 2 dx -µ Ω dx (19)
where F tra is the set of domains of F having a dominant translation parameter, â(Ω) is the dominant translation parameter of Ω, and µ is a positive constant.

Certainly, the minimization algorithm will require computation of â(Ω). However, this latter aspect will be presented later in Section 3.5 when it will be clear where exactly this parameter is needed in the overall process.

Shape derivative

Following the methodology of Section 2.3, energy [START_REF] Jacob | A unifying approach and interface for spline-based snakes[END_REF] is rewritten as a function of an evolution parameter τ

J tra (τ ) = Ω(τ ) (â(τ ) • E(x)) 2 dx -µ Ω(τ )
dx [START_REF] Jehan-Besson | Dream 2 s: Deformable regions driven by an Eulerian accurate minimization method for image and video segmentation[END_REF] where â(τ ) is a short notation for â(Ω(τ )). Applying result [START_REF] Caselles | Geodesic active contours[END_REF] to each integral, the shape derivative of ( 20) can be computed

dJ tra dτ (τ, V ) = Ω(τ ) d dτ (â(τ ) • E(x)) 2 dx - ∂Ω(τ ) (â(τ ) • E(s)) 2 V (s) • N τ (s) ds - Ω(τ ) dµ dτ dx + ∂Ω(τ ) µ V (s) • N τ (s) ds (21)
= dâ dτ (τ )

Ω(τ ) d dα (α • E(x)) 2 | â(τ ) dx - ∂Ω(τ ) (â(τ ) • E(s)) 2 V (s) • N τ (s) ds + ∂Ω(τ ) µ V (s) • N τ (s) ds (22) = dâ dτ (τ ) d dα Ω(τ ) (α • E(x)) 2 dx | â(τ ) - ∂Ω(τ ) (â(τ ) • E(s)) 2 -µ V (s) • N τ (s) ds (23) = dâ dτ (τ ) dK tra Ω(τ ) dα | â(τ ) - ∂Ω(τ ) (â(τ ) • E(s)) 2 -µ V (s) • N τ (s) ds (24)
where E(s) is a short notation for E(x(s)). Expression [START_REF] Osher | Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations[END_REF] still contains a surface integral. As explained in Section 2.3, the shape derivative is readily usable if it contains only a line integral. Parameter â(τ ) minimizes K tra Ω(τ ) on C + . However, since K tra Ω(τ ) is an even function, â(τ ) also minimizes it on C. Vectors of C respect the following constraint

φ(α) = |α| 2 -1 = 0 . (25)
Therefore, there exists a Lagrange multiplier λ τ such that dK tra Ω(τ )

dα | â(τ ) = λ τ dφ dα | â(τ ) . (26)
Using ( 26) in [START_REF] Osher | Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations[END_REF], the shape derivative of ( 20) is

dJ tra dτ (τ, V ) = λ τ dâ dτ (τ ) dφ dα | â(τ ) - ∂Ω(τ ) (â(τ ) • E(s)) 2 -µ V (s) • N τ (s) ds (27) = λ τ d dτ φ(â(τ )) - ∂Ω(τ ) (â(τ ) • E(s)) 2 -µ V (s) • N τ (s) ds . (28)
Since φ(â(τ )) is constant with respect to τ , the final expression of the shape derivative of ( 20) is

dJ tra dτ (τ, V ) = - ∂Ω(τ ) (â(τ ) • E(s)) 2 -µ V (s)•N τ (s) ds . (29)

Minimization algorithm

As presented in Section 2.3, the minimization of ( 19) can be achieved by iterating an active contour evolution equation deduced from shape derivative [START_REF] Sapiro | Vector (self) snakes: A geometric framework for color, texture and multiscale image segmentation[END_REF]. The evolution equation is

∂Γ ∂τ (τ ) = (â(τ ) • E) 2 -µ N τ ( 30
)
where Γ(τ = 0) is an initial estimate of the segmentation contour. Dominant translation parameter â(τ ) of Ω(τ ) must be computed in order to implement evolution [START_REF] Schnörr | Computation of discontinuous optical flow by domain decomposition and shape optimization[END_REF]. In Section 3.2, â(τ ) was expressed as the unique global minimizer of K tra Ω(τ ) . Although expression ( 17) is not readily usable, it can be rewritten as follows

K tra Ω(τ ) (α) = α T Ω(τ ) E(x) E(x) T dx α (31) = α T Q τ α . ( 32
)
Matrix Q τ is clearly symmetric and real. Therefore, it has real eigenvalues δ 1 and δ 2 . Without loss of generality, it is assumed that δ 1 ≤ δ 2 . Rewritting [START_REF] Wang | Spatio-temporal segmentation of video data[END_REF] in an orthonormal basis of eigenvectors, we have

K tra Ω(τ ) (α) = δ 1 α 2 1 + δ 2 α 2 2 (33) = (δ 1 -δ 2 ) α 2 1 + δ 2 (34)
since we look for α in C + (α 2 1 +α 2 2 = 1). If δ 1 and δ 2 are equal, K tra Ω(τ ) is constant on C + . Therefore, a necessary and sufficient condition of uniqueness of â(τ ) is that Q τ has two distinct eigenvalues. In other words, Q τ must not be equal to the identity matrix times a constant. In this case, K tra Ω(τ ) is minimum when α 1 is equal to one, which implies that α 2 is equal to zero. Since α has been expressed in a basis of eigenvectors, it is actually equal to the eigenvector of C + associated with eigenvalue δ 1 . In brief, â(τ ) is the unique solution in C + of

Q τ α = δ min α ( 35
)
where δ min is the lowest eigenvalue of Q τ .

In Appendix A, we propose a sufficient condition of nonexistence of a dominant translation parameter as an illustration of the case where Q τ is equal to the identity matrix times a constant.

Given the proposed iterative method, there is a concern about the choice of an initial contour Γ(τ = 0). Indeed, it must be such that Ω(τ = 0) is a domain of F tra . Furthermore, this property should hold for all τ positive, until convergence. As explained in Appendix C, a sufficient condition is that Ω(τ = 0) contains domain Ω tra , minimizer of ( 19), entirely while not exceeding an area of 1 + √ 2

2 times the area of Ω tra . Naturally, since Ω tra is unknown, it implies the impossibility of a "blind" initialization. Nevertheless, it is not uncommon to have such constraints in the field of segmentation using active contours.

Segmentation of rotation domains 4.1 Rotation domain

A region of the domain of E containing only field lines that are circles is called a rotation domain (See Fig. 1). Equivalently, a domain Ω of F is a rotation domain if and only if there exists a unique parameter a(Ω) of R 2 such that for any field line

L verifying L ∩ Ω = ∅ there exists a unique constant c(L) of R such that |x -a(Ω)| 2 = c(L), x ∈ L ∩ Ω . ( 36
)
Vector a(Ω) is called the rotation parameter of Ω. It is actually the center of the circular field lines. In our context, the specific value of c(L) is of no importance and plays the role of an unnecessary parameter. Therefore, a condition equivalent to (36) while not involving c(L) should be found. Let us parametrize L by p.

For any x(p) of L ∩ Ω |x(p) -a(Ω)| 2 = c(L) (37) ⇐⇒ (x 1 (p) -a 1 ) 2 + (x 2 (p) -a 2 ) 2 = c(L) (38)
The derivative of (38) with respect to p combined with (1) leads to the following system of equations

x 1 (p) -a 1 x 2 (p) -a 2 E 2 (x(p)) -E 1 (x(p)) ẋ(p) = M ẋ(p) = 0 .
(39) Combining ( 2) and (39), it can be concluded that the determinant of M must be equal to zero. Therefore, a domain Ω of F is a rotation domain if and only if there exists a unique a(Ω) of R 2 such that for any x of Ω (xa(Ω)) • E(x) = 0 .

(40)

Dominant rotation parameter

From (40), it can be deduced that if Ω is a rotation domain then a(Ω) is a global minimizer of

K rot Ω : R 2 → R α → Ω ((x -α) • E(x)) 2 dx . (41)
Indeed, K rot Ω is clearly nonnegative and K rot Ω (a(Ω)) is equal to zero. Moreover, this minimizer is unique in R 2 (See Section 4.5).

Looking from a domain point of view (as opposed to the parameter point of view), it can also be deduced from (40) that Ω minimizes the following energy

G rot : F → R ω → ω ((x -a(ω)) • E(x)) 2 dx (42)
with a(ω) defined as the unique global minimizer of K rot ω . However, there is no guarantee that for any ω of F there exists a unique global minimizer of K rot ω and that K rot ω is equal to zero for this minimizer. In other words, ω might not have a rotation parameter (namely, if it is not a rotation domain). Nevertheless, as long as K rot ω has a unique global minimizer â(ω), this minimizer is called the dominant rotation parameter of ω.

Practical formulation

It is clear that any subdomain of a rotation domain is a minimizer of energy (42). As a consequence, the problem of finding rotation domains should be reformulated in finding rotation domains having an area as big as possible. This is equivalent to minimizing G rot (Ω) while maximizing the area of Ω, i.e., minimizing the following energy

J rot : F rot → R Ω → Ω ((x -â(Ω)) • E(x)) 2 dx -µ Ω dx (43)
where F rot is the set of domains of F having a dominant rotation parameter, â(Ω) is the dominant rotation parameter of Ω, and µ is a positive constant.

Certainly, the minimization algorithm will require computation of â(Ω). However, this latter aspect will be presented later in Section 4.5 when it will be clear where exactly this parameter is needed in the overall process.

Shape derivative

Following the methodology of Section 2.3, energy (43) is rewritten as a function of an evolution parameter τ

J rot (τ ) = Ω(τ ) ((x -â(τ )) • E(x)) 2 dx -µ Ω(τ ) dx (44)
where â(τ ) is a short notation for â(Ω(τ )). Applying result [START_REF] Caselles | Geodesic active contours[END_REF] to each integral, the shape derivative of (44) can be computed in a way similar to the development presented in Section 3.4. The only difference is that, in the rotation case, parameter â(τ ) minimizes K rot Ω(τ ) on R 2 . In other words, there are no constraints on â(τ ). As a consequence, it can be concluded that dK rot Ω(τ ) /dα| â(τ ) is equal to zero without involving a Lagrange multiplier. The shape derivative of (44) is

dJ rot dτ (τ, V ) = - ∂Ω(τ ) ((x(s) -â(τ )) • E(s)) 2 -µ V (s) • N τ (s) ds . ( 45
)

Minimization algorithm

As presented in Section 2.3, the minimization of (43) can be achieved by iterating an active contour evolution equation deduced from shape derivative (45). The evolution equation is

∂Γ ∂τ (τ ) = ((x -â(τ )) • E) 2 -µ N τ ( 46
)
where Γ(τ = 0) is an initial estimate of the segmentation contour. Dominant rotation parameter â(τ ) of Ω(τ ) must be computed in order to implement evolution (46). In Section 4.2, â(τ ) was expressed as the unique global minimizer of K rot Ω(τ ) . It can easily be shown that

H K rot Ω(τ ) = 2 Q τ ( 47
)
where H K rot Ω(τ ) is the Hessian matrix of K rot Ω(τ ) and Q τ is the matrix in [START_REF] Wang | Spatio-temporal segmentation of video data[END_REF]. Considering [START_REF] Hintermüller | A second order shape optimization approach for image segmentation[END_REF] and [START_REF] Wang | Spatio-temporal segmentation of video data[END_REF], it is clear that Q τ is nonnegative. As a consequence, K rot Ω(τ ) is convex. However, its strict convexity cannot be inferred yet.

If â(τ ) actually exists, it is such that

dK rot Ω(τ ) dα | â(τ ) = 0 (48)
Although expression (48) is not readily usable, it can be rewritten as follows

Q τ â(τ ) = Ω(τ ) (x • E(x)) E(x)dx . (49)
Therefore, a necessary and sufficient condition of existence and uniqueness of â(τ ) is that the determinant of Q τ is not equal to zero. In this case, â(τ ) is the unique solution in R 2 of (49), namely

â(τ ) = Q -1 τ Ω(τ ) (x • E(x)) E(x)dx . ( 50
)
The uniqueness of the solution of (48) implies the strict convexity of K rot Ω(τ ) . Therefore, it can be concluded that â(τ ) is indeed the unique global minimizer of K rot Ω(τ ) . In Appendix B, we propose a necessary and sufficient condition of nonexistence of a dominant rotation parameter as an illustration of the case where the determinant Q τ is equal to zero.

Given the proposed iterative method, there is a concern about the choice of an initial contour Γ(τ = 0). Indeed, it must be such that Ω(τ = 0) is a domain of F rot . Furthermore, this property should hold for all τ positive, until convergence. As explained in Appendix D, a sufficient condition is that Ω(τ = 0) contains domain Ω rot , minimizer of (43), entirely. Naturally, Ω rot is unknown. However, this constraint can easily be respected by choosing Ω(τ = 0) equal to the entire image domain.

Segmentation of g-domains

Sections 3 and 4 deal with the segmentation of specific vector fields: translation fields and rotation fields, respectively. In this section, we extend the proposed method in order to define a general framework for the segmentation of vector fields of R 2 that can be described by a finite number of parameters. As a result, we propose a step-by-step procedure from the definition of a vector field model (by means of a field line equation) to the segmentation algorithm.

Set of constraints and g-domain

A subset of R n , ∆, is called a set of constraints if it is equal to R n or if it is given by the following set of intersections ∆ = O ∩ (∩ m i=1 ∆ i ) ( 51
)
where O is a subset of R n and ∆ i is defined by

∆ i = {α ∈ R n , φ i (α) = 0} (52)
where each φ i is a function C 1 from O to R and, for j = i, φ i and φ j are linearly independent. A domain Ω of F is called a g-domain, where g is a smooth function from R 2 × R n to R, if and only if there exists a unique parameter a(Ω) of a set of constraints ∆ such that for any field line L verifying L ∩ Ω = ∅ there exists a unique constant c(L) of R such that

g(a(Ω), x) = c(L), x ∈ L ∩ Ω . ( 53
)
Vector a(Ω) is called the g-parameter of Ω.

Translation domains and rotation domains are clearly two particular cases of g-domains with, respectively,

∆ tra = C + = O ∩ ∆ 1 g tra (a(Ω), x) = a(Ω) • x , ( 54
)
where O is the set of x of R 2 such that x 1 ≥ 0 and x 2 > -1 and ∆ 1 is defined by φ 1 (α) = |α| 2 -1, and

∆ rot = R 2 g rot (a(Ω), x) = |x -a(Ω)| 2 . (55)
In our context, the specific value of c(L) in ( 53) is of no importance and plays the role of an unnecessary parameter. Therefore, a condition equivalent to (53) while not involving c(L) should be found. Let us parametrize L by p. For any x(p) of L ∩ Ω g(a(Ω), x(p)) = c(L) .

(56)

The derivative of (56) with respect to p combined with (1) leads to the following system of equations

∂g ∂x1 (a(Ω), x(p)) ∂g ∂x2 (a(Ω), x(p)) E 2 (x(p)) -E 1 (x(p)) ẋ(p) = M ẋ(p) (57) = 0 . (58)
Combining ( 2) and (58), it can be concluded that the determinant of M must be equal to zero. Therefore, a domain Ω of F is a g-domain if and only if there exists a unique a(Ω) of ∆ such that for any x of Ω

∇ x g(a(Ω), x) • E(x) = 0 (59)
where ∇ x g is the vector of components ∂g ∂xi .

Dominant g-parameter

From (59), it can be deduced that if Ω is a g-domain then a(Ω) is a global minimizer of

K g Ω : ∆ → R α → Ω (∇ x g(α, x) • E(x)) 2 dx . (60)
Indeed, K g Ω is clearly nonnegative and K g Ω (a(Ω)) is equal to zero.

Looking from a domain point of view (as opposed to the parameter point of view), it can also be deduced from (59) that Ω minimizes the following energy

G g : F → R ω → ω (∇ x g(a(ω), x) • E(x)) 2 dx (61)
with a(ω) defined as the unique global minimizer of K g ω . However, there is no guarantee that for any ω of F there exists a unique global minimizer of K g ω and that K g ω is equal to zero for this minimizer. In other words, ω might not have a g-parameter (namely, if it is not a g-domain). Nevertheless, as long as K g ω has a unique global minimizer â(ω), this minimizer is called the dominant g-parameter of ω.

Practical formulation

It is clear that any subdomain of a g-domain is a minimizer of energy (61). As a consequence, the problem of finding g-domains should be reformulated in finding g-domains having an area as big as possible. This is equivalent to minimizing G g (Ω) while maximizing the area of Ω, i.e., minimizing the following energy

J g : F g → R Ω → Ω (∇ x g(â(Ω), x) • E(x)) 2 dx -µ Ω dx (62)
where F g is the set of domains of F having a dominant g-parameter, â(Ω) is the dominant g-parameter of Ω, and µ is a positive constant.

Shape derivative

Following the methodology of Section 2.3, energy (62) is rewritten as a function of an evolution parameter τ

J g (τ ) = Ω(τ ) (∇ x g(â(τ ), x) • E(x)) 2 dx -µ Ω(τ ) dx (63)
where â(τ ) is a short notation for â(Ω(τ )). Applying result [START_REF] Caselles | Geodesic active contours[END_REF] to each integral and following a development similar to the translation case (See Section 3.4), the shape derivative of (63) can be computed

dJ g dτ (τ, V ) = dâ dτ (τ ) dK g Ω(τ ) dα | â(τ ) - ∂Ω(τ ) (∇ x g(â(τ ), x(s)) • E(s)) 2 -µ V (s) • N τ (s) ds . (64)
Expression (64) still contains a surface integral. As explained in Section 2.3, the shape derivative is readily usable if it contains only a line integral. Parameter â(τ ) minimizes K g Ω(τ ) on ∆ (See Section 5.1). Therefore, there exists Lagrange multipliers λ i τ such that dK g

Ω(τ ) dα | â(τ ) = m i=1 λ i τ dφ i dα | â(τ ) . (65)
Using ( 65) in (64) and following a development similar to the translation case (See Section 3.4), the final expression of the shape derivative of ( 63) is

dJ g dτ (τ, V ) = - ∂Ω(τ ) (∇ x g(â(τ ), x(s)) • E(s)) 2 -µ V (s) • N τ (s) ds . ( 66
)

Minimization algorithm

As presented in Section 2.3, the minimization of ( 62) can be achieved by iterating an active contour evolution equation deduced from shape derivative (66). The evolution equation is

∂Γ ∂τ (τ ) = (∇ x g(â(τ ), x) • E) 2 -µ N τ (67)
where Γ(τ = 0) is an initial estimate of the segmentation contour. Dominant g-parameter â(τ ) of Ω(τ ) must be computed as the unique global minimizer of K g Ω(τ )

(See Section 5.2) in order to implement evolution (67).

Experiments

The proposed algorithm was applied to vector fields obtained in two ways: synthetic fields and dense optical flow fields.

Synthetic fields and optical flow

Synthetic fields are 100× 100-pixel images composed of two regions. Each region can contain either a random field, a translation field, or a rotation field. These fields were obtained by assigning a two-dimensional vector with a norm equal to one to each pixel. The random field was obtained by drawing random numbers independently for each component and normalizing the resulting vector.

The optical flow fields between two consecutive frames of a video sequence was computed by a block matching technique similar to [START_REF] Lucas | An iterative image registration technique with an application to stereo vision[END_REF]. For each pixel of the first frame, a surrounding, centered block of 21×21 pixels was extracted and its best match in the second frame in terms of the zero-mean normalized sum of squared differences (ZNSSD) was found using a fast, suboptimal diamond search procedure [START_REF] Zhu | A new diamond search algorithm for fast block matching motion estimation[END_REF]. In order to limit the computation time to a reasonable value, the search procedure was restricted to a maximum amplitude of 4 pixels in the four directions. The vector of displacement from the original block to its matching block was assigned to the pixel at the center of the original block. In order to apply the proposed segmentation method, two processing steps were necessary. (i) Vectors equal to zero were replaced with vectors with random directions so that no global coherence was added to the field. (ii) The vectors were normalized.

Implementation

If the region, respectively object, of interest is assumed to be a translation domain, to have a motion of translation, evolution (30) should be implemented. Similarly, evolution (46) should be implemented if rotation is suspected. These active contour processes can be implemented with an implicit representation allowing changes of topology of the active contour such as level sets [START_REF] Barles | Remarks on a flame propagation model[END_REF][START_REF] Osher | Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations[END_REF][START_REF] Chopp | Computing minimal surfaces via level set curvature flow[END_REF][START_REF] Adalsteinsson | The fast construction of extension velocities in level set methods[END_REF][START_REF] Gomes | Reconciling distance functions and level sets[END_REF]. Since the vector fields used are assumed to contain only one object, changes of topology were not an issue. Therefore, we chose an explicit representation based on evolving spline curves [START_REF] Unser | B-spline signal processing: Part I -Theory[END_REF][START_REF] Brigger | B-spline snakes: A flexible tool for parametric contour detection[END_REF][START_REF] Jacob | A unifying approach and interface for spline-based snakes[END_REF][START_REF] Precioso | Smoothing B-spline active contour for fast and robust image and video segmentation[END_REF]. For translation, the segmentation algorithm can be described synthetically as follows 1. Choose weight µ of the area constraint 2. Choose an initial domain Ω(τ = 0) according to Appendix C

Compute dominant translation parameter â(τ )

of Ω(τ ) as the solution of ( 35)

4. Compute energy [START_REF] Jacob | A unifying approach and interface for spline-based snakes[END_REF] of Ω(τ ) 5. For τ positive, if the energy of Ω(τ ) is equal to the energy of Ω(τ -1), then the algorithm has converged and Ω(τ ) is the solution of the segmentation, otherwise go to step 6

6. Update Ω according to a discretization of evolution ( 30)

Γ(τ + 1) = Γ(τ ) + α (â(τ ) • E) 2 -µ N τ ( 68
)
where α is chosen heuristically in order to "ensure" stability of the evolution

α = 1 / max x∈Γ(τ ) (â(τ ) • E(x)) 2 -µ . ( 69
)
7. Go back to step 3

The boundary Γ was represented by a spline passing through samples regularly spaced at a distance equivalent to 5 pixels. The discrete representation of Ω was the mask of pixels of Γ. Evolution (68) was applied to the samples of Γ(τ ) and Γ(τ + 1) was computed as the spline passing through the moved samples. To maintain a distance of 5 pixels between the samples, periodic resampling was necessary.

Convergence was not tested exactly as stated in step 5, especially since the choice of α could cause endless oscillations around the solution. Instead, the algorithm was assumed to have converged when the energy was roughly constant during the last few iterations.

Translation domain

Since the background is not taken into account by the segmentation process, it was interesting to see the behavior of the algorithm when segmenting a translation domain on a translation background (See Figs. 2 and3) and a rotation background (See Fig. 4). If the ini-Figure 2: Segmentation of a translation domain on a translation background: (from left to right and top to bottom) the initial contour superimposed on the image of the first component of the vector field, the final contour on the same image, and the final contour on the vector field. tial domain contains a reasonably higher portion of the object than the portion of the background, then the translation domain should be detected accurately (See Fig. 2). However, if the initial domain contains more background than object, then the notions of object and background naturally exchange for each other in the computation of the dominant translation parameter (See Fig. 3). In the three tests on synthetic translation domains (Figs. 2, 3, and4), weight µ of the area constraint was taken equal to 0.1.

A real-world, standard, sport sequence was also used to test the proposed algorithm, as if part of a larger processing chain. The optical flow between frames 15 and 16 of sequence "football" was computed as explained in Section 6.1. The sequence is in the CIF format: it is composed of 352 × 288-pixel frames. The main flow is due to the motion of the player in the center of the field of view. The segmentation of the optical flow is shown in Figs. 5 and6. As far as the optical flow is concerned, the segmentation seems accurate (See Fig. 5 and the image on the left in Fig. 6). When superimposing the segmentation on frame 15 (See the image on the right in Fig. 6), the contour does not outline the player very precisely. Indeed, the optical is not accurate at the boundary of objects and in homogeneous or nearly homogeneous regions. It is a consequence of the block matching method when a block contains parts of different moving objects and when the correlation function is too flat and has several local minima, respectively. However, the purpose of this experiment was to test the proposed algorithm on a realistic optical flow, i.e. quite complex and noisy. In that respect, the result is satisfying. With sequence "football", weight µ of the area constraint was taken equal to 0.05, which is consistent with the value used with the synthetic vector fields while being a little smaller to account for noise. Indeed, a high value of µ increases the tolerance on the error of the segmentation term of evolution [START_REF] Schnörr | Computation of discontinuous optical flow by domain decomposition and shape optimization[END_REF], implying a higher probability to include incorrect vectors in the segmentation.

Rotation domain

Similarly to the translation case, segmentation of a rotation domain was evaluated with several backgrounds: a translation background (See Figs. 7 and8), a rotation background (See Fig. 9), and a random background (See Fig. 10). Two elements can be noticed on Figs. 7 and8. First, since a rotation domain potentially contains vectors in all directions, there is an inevitable match with the direction of the translation in the background. Therefore, some vectors of the background close to the object and which happen to have a direction plausible in terms of the rotation might be included in the segmentation. Second, the segmentation is different whether the initial contour is or outside the object. Nevertheless, it has the same shape and behavior regarding translation vectors in the neighborhood of the object. This tends to show that, in this configuration of object and background vector fields, energy (43) is flat around the solution and has several local minima. With a rotation background having a rotation center far enough from the rotation center of the object, the segmentation is satisfying (See Fig. 9). Although an extensive study of the robustness and accuracy of the proposed method is out of the scope of this paper, it can be expected that segmentation becomes less accurate as the rotation centers of the background and the object get closer. The last experiment with a synthetic rotation domain was done with a random background. The segmentation shown in Fig. 10 seems accurate. In the four tests on synthetic rotation domains (Figs. 7, 8, 9 and 10), weight µ of the area constraint was taken equal to 10. This value, significantly higher than the values used for translation domains, tends to show that rotation energy (43) requires more tolerance on its segmentation term to segment the whole object than translation energy [START_REF] Jacob | A unifying approach and interface for spline-based snakes[END_REF] does.

A real-world, somewhat artificial sequence was also used to test the proposed algorithm, as if part of a larger processing chain. A sponge was rotated manually by a few degrees around its center on a still background. A sequence composed of two 640 × 480-pixel frames was made and the optical flow between these frames was computed (See Fig. 11). The segmentation is shown in Fig. 12. Since the optical flow in the upper part of the background is mostly equal to zero, it was replaced with a random field as explained in Section 6.1. The lower part of the optical flow of the background was already randomly distributed. The segmentation of the optical flow, superimposed on the first frame of the sequence, appears to outline the sponge accurately.

Conclusion

A methodology was proposed for the segmentation of vector fields of R 2 that can be described by a finite number of parameters. The idea is to characterize a specific vector field by its underlying field lines. Given the type of subfield to be segmented (e.g., translation or rotation), a specific model must be defined by means of the general form of the parametric equations of the field lines (straight lines or circles, respectively). The segmentation problem is expressed as the minimization with respect to the segmentation domain of an energy deduced from the field line equations. The minimization is rewritten into a shape optimization and solved by a spline-based active contour procedure requiring the choice of an initial estimate. The parameters of the field line equations are computed as part of the evolution process of the active contour (normal to the straight lines and center of the circles, respectively).

The method proved to work well for translation domains and rotation domains. Furthermore, it provides a general framework for more complex vector fields. However, it is adapted to normalized vector fields. Therefore, the necessary normalization step prior to the segmentation of an unnormalized field induces some loss of information: (i) A vector with a small norm and a random direction within a coherent region (a "noise" vector) will have, after normalization, as much influence on the computation of the dominant g-parameter as a vector with a large norm and a coherent direction; (ii) It is not possible to make the distinction between two translation domains having different norms but the same direction. As a consequence, further developments should use the "natural" vector field and take the norm of the vectors into account.

A Domain with no dominant translation parameter

Let Ω be a domain such that there exists a diffeomorphism t from R 2 to R 2 such that for any x of Ω

t(Ω) = Ω (70) | det(J t (x))| = 1 (71) E(t(x)) • E(x) = 0 (72)
where J t is the Jacobian matrix of t. The orthogonality between E(t(x)) and E(x) implies that

E 1 (t(x)) = ǫ E 2 (x), ǫ ∈ {-1, 1} (73) E 2 (t(x)) = -ǫ E 1 (x) (74) E 1 (t(x)) E 2 (t(x)) = -E 1 (x) E 2 (x) . (75)
Matrix Q Ω defined in [START_REF] Wang | Spatio-temporal segmentation of video data[END_REF] has the following form where a 1 , a 2 , and b are equal to

Q Ω = a 1 b b a 2 (76)
a i = Ω E i (x) 2 dx, i ∈ {1, 2} (77) b
= Ω E 1 (x) E 2 (x) dx . (78)
We have

a 1 = Ω E 2 (t(x)) 2 dx (79) = t(Ω) E 2 (y) 2 | det(J t (y))| dy (80) = t(Ω) E 2 (y) 2 dy (81) = Ω E 2 (y) 2 dy (82) = a 2 . (83)
Moreover, we have

b = - Ω E 1 (t(x)) E 2 (t(x)) dx (84)
Therefore, b is equal to zero. In conclusion, Q Ω is equal to the identity matrix times a constant. As shown in Section 3.5, this means that Ω has no dominant translation parameter.

For instance, a disk containing vectors rotating around the center of the disk has no dominant translation parameter. Indeed, it suffices to choose t as the rotation of π 2 around the center of the disk.

B Domain with no dominant rotation parameter

Taking the opposite of the equivalence of existence of a dominant rotation parameter proposed in Section 4.5, it can be stated that Ω has no dominant rotation pa-rameter if and only if the determinant of Q Ω is equal to zero. Explicitly, it means that

Ω E 1 (x) 2 dx Ω E 2 (x) 2 dx = Ω E 1 (x) E 2 (x) dx 2 . (89)
According to Cauchy-Schwarz, (89) is a superior-orequal inequality in general. It is an equality, like in the present case, if and only if there exists a constant λ of R such that

E 2 (x) = λ E 1 (x) (90)
for any x of Ω. However, E being normalized, for any x of Ω we also have

E 1 (x) 2 + E 2 (x) 2 = 1 . (91)
From ( 90) and (91), it can easily be deduced that E is constant within Ω. In other words, Ω is a translation domain.

Since this result was obtained by a series of equivalences, it can be concluded that a domain has no dominant rotation parameter if and only if it is a translation domain.

C Initial segmentation of a translation domain

In Section 3.5, it is shown that a domain Ω has a dominant translation parameter if and only if there is no constant λ such that

Q Ω = λ Id ( 92
)
where Id is the identity matrix or, equivalently,

Ω E 2 1 (x) dx = λ (93) Ω E 2 2 (x) dx = λ (94) Ω E 1 (x) E 2 (x) dx = 0 . ( 95
)
Vector field E being normalized, E As a conclusion, if initial domain Ω(τ ) respects conditions (99) and (100), then it has a dominant translation parameter. Intuitively, evolution (30) will make Ω(τ ) shrink until, ideally, Ω(τ = 0) = Ω tra .

Therefore, conditions (99) and (100) should hold for all τ positive, until convergence.

D Initial segmentation of a rotation domain

The opposite of the equivalence of nonexistence of a dominant rotation parameter proposed in Appendix B reads that a domain has a dominant rotation parameter if and only if it is not a translation domain. Let Ω rot be the (unknown) minimizer of (43). It is assumed that initial domain Ω(τ = 0) is such that

Ω(τ = 0) ⊃ Ω rot . (128)
Clearly, since Ω(τ = 0) contains a rotation domain, it cannot be a translation domain. Therefore, it can be concluded that it has a dominant rotation parameter. Intuitively, evolution (46) will make Ω(τ ) shrink until, ideally, Ω(τ = 0) = Ω rot .

Therefore, condition (128) should hold for all τ positive, until convergence.

Figure 1 :

 1 Figure 1: Two examples of vector fields: translation (left) and rotation (right). We recall that L is a field line of E if and only if ẋ1 (p) E 2 (x(p)) -ẋ2 (p) E 1 (x(p)) = 0 (1) and ẋ(p) = 0 (2) where x(p) is the equation of L, p belongs to a certain interval [a, b], and ẋ represents the derivative of x with respect to p. Equation (1) represents the fact that

Figure 3 :

 3 Figure 3: Segmentation (interrupted before convergence) of a translation domain on a translation background: (from left to right and top to bottom) the initial contour superimposed on the image of the first component of the vector field, the final contour on the same image, and the final contour on the vector field.

Figure 4 :

 4 Figure 4: Segmentation of a translation domain on a rotation background: (from left to right and top to bottom) the initial contour superimposed on the image of the first component of the vector field, the final contour on the same image, and the final contour on the vector field.

Figure 5 :

 5 Figure 5: Segmentation of the optical flow of frame 15 of sequence "football": the initial contour superimposed on the image of the first component of the vector field (left) and the final contour on the same image (right).

Figure 6 :

 6 Figure 6: Segmentation of the optical flow of frame 15 of sequence "football" superimposed on the vector field (left) and superimposed on frame 15 (right).

Figure 7 :

 7 Figure 7: Segmentation of a rotation domain on a translation background: (from left to right and top to bottom) the initial contour superimposed on the image of the first component of the vector field, the final contour on the same image, and the final contour on the vector field.

Figure 8 :

 8 Figure 8: Segmentation of a rotation domain on a translation background: (from left to right and top to bottom) the initial contour superimposed on the image of the first component of the vector field, the final contour on the same image, and the final contour on the vector field.

Figure 9 :

 9 Figure 9: Segmentation of a rotation domain on a rotation background: (from left to right and top to bottom) the initial contour superimposed on the image of the first component of the vector field, the final contour on the same image, and the final contour on the vector field.

Figure 10 :

 10 Figure 10: Segmentation of a rotation domain on a random background: (from left to right and top to bottom) the initial contour superimposed on the image of the first component of the vector field, the final contour on the same image, and the final contour on the vector field.

Figure 11 :

 11 Figure 11: Rotating sponge on a still background: sponge before rotation (top) and optical flow of the rotation before processing as explained in Section 6.1 (bottom).

Figure 12 :E 1 E 1

 1211 Figure 12: Segmentation of the optical flow superimposed on the image of the sponge before rotation (initialization (top) and solution (bottom)).

  Our purpose is to show that if conditions (99) and (100) are respected, then (97) or (98) is true.Since E is normalized, there exists θ, a function from R 2 to [0, 2π], such that Furthermore, Ω tra being a translation domain, there exists a constant θ tra such that for any x of Ω tra θ(x) = θ tra .

	2 2 can be replaced 1 in (94). As a result, (93) and (94) can be with 1 -E 2 combined into Ω E 2 1 (x) dx = |Ω| 2 (96) + Ω/Ω tra E 1 (x) E 2 (x) dx (113) = 1 2 sin(2θ tra ) |Ω tra | + Ω/Ω tra E 1 (x) E 2 (x) dx . (114) Combining (112) and (114), it can be concluded that Ω/Ω tra E 1 (x) 2 -1 2 dx = Ω/Ω tra E 1 (x) 2 dx -|Ω/Ω tra | 2 (116) ≤ Ω/Ω tra E 1 (x) 2 -1 2 dx (117) ≤ Ω/Ω tra sin 2 (θ(x)) -1 2 dx (118) ≤ Ω/Ω tra | cos(2θ(x))| 2 dx (119) ≤ Ω/Ω tra 1 2 dx (120) ≤ 1 2 |Ω/Ω tra | (121) < 1 2 | cos(2θ tra )| |Ω tra | . (122) It can be concluded that, in particular, Ω/Ω tra E 1 (x) 2 dx -|Ω/Ω tra | 2 = 1 2 cos(2θ tra ) |Ω tra | (123) ⇐⇒ Ω/Ω tra E 1 (x) 2 dx + (sin 2 (θ tra ) -1 2 ) |Ω tra | = 2 (124) |Ω/Ω tra | | sin(2θ Ω tra or Ω E 1 (x) E 2 (x) dx = 0 . = |Ω/Ω tra | 2 + |Ω tra | 2 Let Ω tra be the (unknown) minimizer of (19). It is (125) (98) assumed that Ω ⊃ Ω tra (99) and |Ω/Ω tra | < √ 2 2 |Ω tra | (100) where A/B is the intersection between A and the com-plement of B. Note that (100) is equivalent to |Ω| < (1 + √ 2 2 )|Ω tra | . (101) E(x) = sin(θ(x)) cos(θ(x)) . (102) (103) Finally, note that min θ∈[0,2π] {max{| sin(2θ)|, | cos(2θ)|}} = √ 2 2 . (104) Combining (100) and (104), it can be deduced that |Ω/Ω Ω/Ω tra |E 1 (x) E 2 (x)| dx (107) ≤ Ω/Ω tra | sin(2θ(x))| 2 dx (108) ≤ Ω/Ω tra 1 2 dx (109) ≤ 1 2 |Ω/Ω tra | (110) < 1 2 E 1 (x) E 2 (x) dx ⇐⇒ Ω E 1 (x) 2 dx = |Ω| 2 . (126)
	⇐⇒	Ω/Ω tra	Ω	E 2 1 (x) dx = Ω tra |Ω| 2 E 1 (x) 2 dx	(97)

where |Ω| is the measure of Ω. Therefore, Ω has a dominant translation parameter if and only

if tra | < | sin(2θ tra )| |Ω tra | (105) or |Ω/Ω tra | < | cos(2θ tra )| |Ω tra | .

(106)

• First case: Inequality (105) is true. We have

Ω/Ω tra E 1 (x) E 2 (x) dx ≤ tra )| |Ω tra | .

(111)

It can be concluded that, in particular,

Ω/Ω tra E 1 (x) E 2 (x) dx = -1 2 sin(2θ tra ) |Ω tra | .

(112)

Besides

Ω E 1 (x) E 2 (x) dx = Ω E 1 (x) E 2 (x) dx = 0 .

(115)

• Second case: Inequality (106) is true. We have

E 1 (x) 2 dx +


Wahju Aniwidyaningsih

Raphaëlle Varraso

Noel Cano

Christophe Pison
email: cpison@chu-grenoble.fr

MD Pr Christophe Pison

Impact of nutritional status on body functioning in chronic obstructive pulmonary disease and how to intervene

Keywords: Nutrition disorders, COPD, inflammation, pulmonary rehabilitation, enteral feeding, oral nutrition supplement, anabolic agents, diet 21-03-08

Purpose of review

Chronic obstructive pulmonary disease (COPD) is the fifth cause of mortality in the world. This article reviews diet as a risk or protective factor for COPD, mechanisms of malnutrition, undernutrition consequences on body functioning and how to modulate nutritional status of COPD patients.

Recent findings

Different dietary factors (dietary pattern, foods, nutrients) have been associated with COPD and the course of the disease. Mechanical disadvantage, energy imbalance, disuse muscle atrophy, hypoxemia, systemic inflammation and oxidative stress have been reported to cause systemic consequences such as cachexia and compromise whole body functioning. Nutritional intervention makes it possible to modify the natural course of the disease provide that it is included in respiratory rehabilitation combining bronchodilators optimization, infection control, exercise and in some patients correction of hypogonadism.

Summary

Diet, as a modifiable risk factor, appears more as an option to prevent and modify the course of COPD. Reduction of mechanical disadvantage, physical training and anabolic agents should be used conjointly with oral nutrition supplements to overcome undernutrition and might change the prognosis of the disease in some cases. Major research challenges address the role of systemic inflammation and the best interventions for control it besides smoking cessation.

Introduction

Nutritional status has been associated with respiratory function in COPD with complex interplays between environmental and genetic factors (Fig. 1) [1•]. Studies have shown that malnutrition served as negative prognostic factor in COPD. Conversely, diet could be a protective or a harmful factor regarding the risk of incident COPD and modifying factor in established disease [START_REF] Schols | Body composition and mortality in chronic obstructive pulmonary disease[END_REF][START_REF] Vestbo | Body mass, fat-free body mass, and prognosis in patients with chronic obstructive pulmonary disease from a random population sample findings from the Copenhagen City Heart Study[END_REF][START_REF] Hallin | Nutritional status and long-term mortality in hospitalised patients with chronic obstructive pulmonary disease (COPD)[END_REF].

The question is how we could intervene with nutritional factors to improve prognosis. This review will discuss diet as a risk factor in COPD, mechanisms of malnutrition, consequences of undernutrition on body functioning and how to modulate nutritional status of these patients.

Diet as a risk factor for COPD

Cigarette smoking is the most important risk factor for COPD, but up to one third of patients with COPD have never smoked [START_REF] Viegi | Definition, epidemiology and natural history of COPD[END_REF]. These suggest that other factors, such as diet, also are involved. In the last decade, there has been a growing interest in identifying foods related to lung function or COPD symptoms [5,6,7••]. Nutritional epidemiologists propose to assess overall diet instead of the individual foods or nutrients due to interaction between nutrients [START_REF] Hu | Dietary pattern analysis: a new direction in nutritional epidemiology[END_REF]. This approach has been used in three recent studies which has used dietary patterns to assess diet in relation with respiratory phenotypes [6,7••,9]. Among the Chinese Singaporeans, a 'meat-dim sum' pattern (meat, dim sum foods, and noodle dish) was associated with an increased risk of incident cough with phlegm [START_REF] Butler | Prospective study of dietary atterns and persistent cough with phlegm among Chinese Singaporeans[END_REF]. In both a large cohort of US men and US women, a "prudent" pattern (high intake of fruits, vegetables, fish and whole grain products) was associated with a decrease risk of newly diagnosed COPD whereas a Western pattern (high intake of refined grains, cured and red meats, desserts and French fries) was associated with an increase of COPD [6,7••].

Most investigations have focused on the possibility that fruits and vegetables exert beneficial effects on lung health and a very little attention has been paid to other foods or nutrients. Recently, it 21-03-08 has been suggested that dietary fibre was independently associated with better lung function and reduced prevalence COPD [START_REF] Kan | Dietary fiber, lung function, and chronic obstructive pulmonary disease in the atherosclerosis risk in communities study[END_REF][START_REF] Romieu | Nutrition and lung health[END_REF]. Previous studies have mostly focused on the potentially protective effects of foods rich in antioxidants, and not on foods with a potential deleterious effect. However, it was recently reported both in three longitudinal surveys [7••,12,13] and in a cross-sectional survey [START_REF] Jiang | Cured meat consumption, lung function, and chronic obstructive pulmonary disease among United States dults[END_REF] that frequent consumption of cured meats was associated with higher prevalence and incidence of COPD. All these studies provide additional support to the hypothesis that diet, a modifiable risk factor, is associated with respiratory diseases. Nutritional epidemiology may play an important role in advancing our understanding of the relevant pathophysiological mechanisms involved in the development and progression of respiratory diseases.

Epidemiology and mechanisms of nutritional depletion in COPD

Malnutrition in COPD is one of negative predictive factors and mechanisms of nutritional depletion are multifactorial processes that will be described in the next following paragraphs.

Epidemiology

Weight loss in COPD was already recognized as a clinical finding and it is more frequent in emphysematous patients as contrary to the blue bloaters [START_REF] Wouters | Nutrition and metabolism in COPD[END_REF]. The prevalence of nutritional depletion was about 20-35% in outpatients COPD and up to 70% in patients with acute respiratory failure or in patients awaiting a lung transplantation [START_REF] Hallin | Nutritional status and long-term mortality in hospitalised patients with chronic obstructive pulmonary disease (COPD)[END_REF][START_REF] Budweiser | Nutritional depletion and its relationship to respiratory impairment in patients with chronic respiratory failure due to COPD or restrictive thoracic diseases[END_REF][START_REF] Vermeeren | Prevalence of nutritional depletion in a large out-patient population of patients with COPD[END_REF][START_REF] Schwebel | Prevalence and consequences of nutritional depletion in lung transplant candidates[END_REF].

Mechanisms of nutrition depletion

Nutrition depletion occurs by including several factors such as energy imbalance, disuse atrophy of the muscles, hypoxemia, systemic inflammation, oxidative stress and hypogonadism in some patients.

Energy imbalance and metabolism changes

The largest component of energy expenditure (EE) is the basal metabolic rate (BMR), which is the energy needed for basic processes of life and it accounts for 60-70% of total daily EE [START_REF] Steele | Quantitating physical activity in COPD using a triaxial accelerometer[END_REF].

Increased energy expenditure due to mechanic disadvantage, metabolic inefficiency with some fibre-21-03-08 type shift in skeletal muscle from type I to type II, together with systemic inflammation may be determinants of an hypermetabolic state that may not be balanced by dietary intake [20••]. In COPD patients, resting energy expenditure (REE) has been reported to be 15-20% above predicted values due to the increased energy required for breathing [21,[START_REF] Ezzell | Malnutrition in chronic obstructive pulmonary disease[END_REF]. Total daily expenditure was also higher in COPD patients compared to healthy subjects and it may be caused by increased level of non-resting daily expenditure [START_REF] Baarends | Total free living energy expenditure in patients with severe chronic obstructive pulmonary disease[END_REF]. Energy expenditure changes due to impaired mechanical efficiency may be partially reversed by reducing hyperinflation with medication, breathing techniques or lung volume reduction. This could reduced oxygen cost for breathing and increased oxygen availability which favours carbohydrate metabolism and recovery in body composition with lower fat and and higher fatfree masses [START_REF] O'donnell | Effect of fluticasone propionate/salmeterol on lung hyperinflation and exercise endurance in COPD[END_REF][START_REF] Mineo | Resting energy expenditure and metabolic changes after lung volume reduction surgery for emphysema[END_REF][START_REF] Vaughan | A prospective analysis of the inter-relationship between lung volume reduction surgery and body mass index[END_REF].

Chronic obstructive lung disease may induce insulin resistance and changes glucose metabolism [START_REF] Sauerwein | Glucose metabolism in chronic lung disease[END_REF]. Data from the Third National Health and Nutrition Examination Survey showed that impaired glucose regulation was associated with impaired lung function [START_REF] Mckeever | Lung function and glucose metabolism: An analysis of data from the third National Health and Nutrition Examination Survey[END_REF]. Glucose plasma concentrations were similar in non-hypoxemic COPD patients and healthy subjects. Abnormal glucose metabolism may be a risk factor for cardiovascular disease and type II diabetes in COPD [START_REF] Bolton | Insulin resistance and inflammation-a further systemic complication of COPD[END_REF]. Some data showed that increased proteolysis in cystic fibrosis (CF) patients was related to impaired glucose tolerance due to resistance to the anabolic effects of insulin on proteolysis [START_REF] Hardin | Proteolysis associated with insulin resistance in cystic fibrosis[END_REF]. High catecholamine level in COPD may induce hypermetabolic response which causes increased EE and muscle-protein catabolism [START_REF] Hofford | The nutritional status in advanced emphysema associated with chronic bronchitis. A study of amino acid and catecholamine levels[END_REF][START_REF] Herndon | Reversal of catabolism by beta-blockade after severe burns[END_REF]. These findings suggested some role of insulin resistance in FFM (fat free mass) depletion.

COPD patients often have dyslipidemia related to metabolic syndrome (abdominal obesity, increased triglycerides, dyslipidemia) and it is due to some changes in lipid metabolism. [START_REF] Marquis | The metabolic syndrome in patients with chronic obstructive pulmonary disease[END_REF].

Jackobsson et al found a decrease in lipolysis and increased glucose levels in patients with advanced COPD with chronic respiratory failure, signaling a role of insulin resistance in the reduction of lipolysis 21-03-08 [START_REF] Jakobsson | Blood fuel metabolites at rest and during exercise in patients with advanced chronic obstructive pulmonary disease with and without chronic respiratory failure[END_REF]. These findings suggested that FM (fat mass) relatively preserved or increased due to decrease lipolysis.

Disuse atrophy of muscles

Patients with very severe COPD are very inactive as compare to healthy people of the same advanced age [START_REF] Pitta | Characteristics of physical activities in daily life in chronic obstructive pulmonary disease[END_REF]. Several studies have shown that these patients have an increased protein breakdown pathways, in particular the NF-kB-activated ubiquitin/proteasomal pathway and apoptosis [START_REF] Agusti | NF-kappaB activation and iNOS upregulation in skeletal muscle of patients with COPD and low body weight[END_REF].

Hypoxemia

It has been suggested that the hypoxemia of COPD is one causative factor for cachexia.

Hypoxemia may increase the generation of ROS and TNF-α which in turn may give rise to inflammatory changes leading to cachexia [START_REF] Takabatake | The relationship between chronic hypoxemia and activation of the tumor necrosis factor-alpha system in patients with chronic obstructive pulmonary disease[END_REF]. It may stimulate the sympathetic nervous system and this has been shown to produce systemic inflammation [START_REF] Andreas | Neurohumoral activation as a link to systemic manifestations of chronic lung disease[END_REF].

Systemic inflammation and oxidant stress

Systemic inflammation has important metabolic consequences and become the primary focus of research into the genesis of cachexia in COPD. The molecules receiving the greatest amount of attention are TNF-α, IL-1β, IL-6, CRP, and ROS and reactive nitrogen species (see ref.

Hypogonadism

There is evidence that insufficiency of one or more of three hormones may contribute to cachexia. Creutzberg et al have provided an overview of endocrine factors in COPD [START_REF] Creutzberg | Endocrinological disturbances in chronic obstructive pulmonary disease[END_REF]. Low levels 21-03-08 of circulating hormones do not prove cause and effect in cachexia in COPD. Inversely, anabolic hormones replacement may have a role in selected patients as reported in various publications.

Impacts of malnutrition in COPD

Impact of malnutrition in COPD on body functioning will be emphasized on body composition, on lung parenchyma, respiratory function, respiratory mechanics, physical capacity, and also morbidity and mortality.

Body composition

Patients with COPD are known to be generally underweight and have reduced FFM as showed by several studies [START_REF] Sergi | Body composition and resting energy expenditure in elderly male patients with chronic obstructive pulmonary disease[END_REF][START_REF] Schols | Weight loss is a reversible factor in the prognosis of chronic obstructive pulmonary disease[END_REF]. Study by Sergi et al showed that COPD patients had a lower FFM as compared to healthy subjects (38% versus 31%) [START_REF] Sergi | Body composition and resting energy expenditure in elderly male patients with chronic obstructive pulmonary disease[END_REF]. In a large out-patient population of patients with COPD in Netherlands, they found prevalence of normal BMI and low FFMI was 15%, and of low BMI and low FFMI 11% [START_REF] Vermeeren | Prevalence of nutritional depletion in a large out-patient population of patients with COPD[END_REF].

Lung parenchyma, respiratory function and respiratory mechanics

Massaro et al described loss of pulmonary alveoli in calorie restriction and regenerated after ad libitum refeeding [START_REF] Massaro | Calorie-related rapid onset of alveolar loss, regeneration, and changes in mouse lung gene expression[END_REF]. Undernutrition led to lung and chest wall mechanical changes, such as distorted structure of diaphragm and intercostals, reduction of surfactant and decrease in elastic fibre content of pulmonary parenchyma [START_REF] Chamberlain | Cachexia in Cancer-Zeroing in on myosin[END_REF]. Inspiratory muscle weakness and maximum inspiratory pressure generation was an independent determinant of survival in severe COPD [START_REF] Ottenheijm | Diaphragm muscle fiber dysfunction in chronic obstructive pulmonary disease: Toward a pathophysiological concept[END_REF]. The diaphragm as a main inspiratory muscle is suffered from muscle protein degradation and loss of contractile protein [START_REF] Ottenheijm | Diaphragm dysfunction in chronic obstructive pulmonary disease[END_REF]. In study by Cano et al the degree of respiratory impairment (FEV 1 , FVC and 6-min walking test) was correlated with nutritional status especially FFM [START_REF] Cano | Nutritional depletion in patients on long-term oxygen therapy and/or home mechanical ventilation[END_REF].

Physical capacity

Daily physical activity can be considered as ''the totality of voluntary movement produced by skeletal muscles during everyday functioning'' [START_REF] Pitta | Quantifying physical activity in daily life with questionnaires and motion sensors in COPD[END_REF]. It is widely known that in patients with COPD, 21-03-08 lower levels of physical activity in daily life are related to higher risk of hospital readmission and shorter survival. The effect of skeletal muscle dysfunction contributes to declining body functioning as we defined as physical activity. Physical activity greatly influences patient outcomes because level of physical activities was shown to predict the risk for hospitalization [START_REF] Watz | Extrapulmonary effects of chronic obstructive pulmonary disease on physical activity[END_REF][START_REF] Cano | C-reactive protein and body mass index predict outcome in end-stage respiratory failure[END_REF]. Overall impairment of body functioning can be described in Figure 2 as adapted from International Classification of Functioning (ICF) [START_REF]Towards a Common Language for Functioning, Disability and Health: ICF[END_REF].

Morbidity and mortality

Mortality of COPD patients and occurrence of acute exacerbation requiring hospitalization are higher in underweight patients. During hospitalization, COPD patients are most likely to loose weight because of the higher metabolic demand due to increased ventilatory disadvantage or infection and underweight patients also had a higher risk of new exacerbation [START_REF] Hallin | Nutritional status, dietary energy intake and the risk of exacerbations in patients with chronic obstructive pulmonary disease (COPD)[END_REF]. Landbo et al described that in mild to moderate COPD the best prognosis was found in normal weight or overweight subjects, whereas in severe COPD, overweight or obese patients were associated with a better survival. These patients may be somehow protected from weight loss because of higher energy reserves [START_REF] Landbo | Prognostic value of nutritional status in chronic obstructive pulmonary disease[END_REF].

How to intervene

The questions on how to intervene in these patients exist and we will discuss how and possible methods to modify disease progression or prognosis.

Nutrition requirements

The question on how much we should give nutritional repletion still remains. In the study by Planas et al, it was described that total daily energy intake of REE x 1.3 was preferable than REE x 1.7 in mild stable COPD patients. They found that administration of nutritional supplements, high in proteins, with predominance of carbohydrates over fat, and enriched in antioxidants to achieve total daily defined energy intake in patients in group REE x 1.3 was followed by a significant improvement of body weight, handgrip strength, decrease airflow limitation and increase quality of life [START_REF] Planas | Nutritional support and quality of life in stable chronic obstructive pulmonary disease (COPD) patients[END_REF]. Total

21-03-08

calories being administered as 20% proteins seemed to be the optimal in nutritional supplements for stable malnourished COPD patients. Small portion of carbohydrate and protein-rich supplementation seemed to have an impact on weight gain after 8 weeks when compared to normal size supplements of similar macronutrient composition, probably because many patients still usually took ordinary meals outside the supplementation [START_REF] Broekhuizen | Optimizing oral nutritional drink supplementation in patients with chronic obstructive pulmonary disease[END_REF]. High load of portion will compromise diaphragm movement and caused postprandial dyspnea because it will prevent the diaphragm to move easily due to gastric filling pressure [START_REF] Anker | ESPEN Guidelines on Enteral Nutrition: Cardiology and pulmonology[END_REF].

Methods

Metabolic syndrome carries a lot of consequences in managing nutrition in COPD patients because of glucose level, hypertension, atherogenic dyslipidemia or other co-morbidities such as congestive heart failure. Methods that are known for nutritional intervention in COPD patients varied between studies.

Oral nutrition supplements

Nutritional support includes food enrichments, oral nutritional supplements, tube feeding and parenteral nutrition. [START_REF] Lochs | Introductory to the ESPEN guidelines on enteral nutrition: Terminology, definitions and general topics[END_REF] There is limited evidence that wasting COPD patients will benefit only with enteral nutrition as showed by meta-analysis by Ferreira et al [START_REF] Ferreira | Nutritional supplementation for stable chronic obstructive pulmonary disease[END_REF] and reported in ESPEN recommendations in 2006 [START_REF] Anker | ESPEN Guidelines on Enteral Nutrition: Cardiology and pulmonology[END_REF]. Table I described some nutritional supplementation studies in COPD patients which were integrated or non-integrated with pulmonary rehabilitation. Refeeding in malnourished COPD patients is not easy and difficult to maintain due to many factors. The 2005 Cochrane Review on nutritional supplementation for stable COPD had some important limitations relative to the small numbers of patients included, 214 in supplement arm versus 205 in control group, the non-integration of nutritional intervention in a pulmonary rehabilitation process, the short duration of intervention between 2 weeks to 3 months, and the lack of relevant endpoints to measure the effects of nutrition supplementation as quality of life, body composition or exercise capacity [START_REF] Anker | ESPEN Guidelines on Enteral Nutrition: Cardiology and pulmonology[END_REF][START_REF] Budweiser | Weight gain in cachectic COPD patients receiving noninvasive positive-pressure ventilation[END_REF]. These limitations should encourage larger studies with a multimodality approaches and clinically relevant end-points.

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Page 10 / 28

Exercise

Physical training will also have positive effects as anabolic process as it can increase FFM and functional status although it did not directly correlated [START_REF] Remels | Peroxisome proliferator-activated receptor expression is reduced in skeletal muscle in COPD[END_REF]. Some others studies showed that physical training only will cause negative energy balance because of increase total energy expenditure, but this effect can be overcome by giving supplementation rich in carbohydrate. Glycogen storage is low in muscle of COPD patients and therefore carbohydrate can serve as better energy source [START_REF] Faager | Creatine supplementation and physical training in patients with COPD: a double blind, placebo-controlled study[END_REF].

Anabolic agents

Anorexia is one of many factors contributes to cachexia in COPD. The use of appetite stimulating therapy in cachexia COPD patients is now getting more attention [START_REF] Fuld | Creatine supplementation during pulmonary rehabilitation in chronic obstructive pulmonary disease[END_REF]. Progestogens (megestrol acetate) can be used to treat anorexia-cachexia. This agent can stimulate appetite and has antagonist effects of pro-inflammatory cytokines [START_REF] Matsuyama | Effects of Omega-3 Polyunsaturated Fatty Acids on Inflammatory Markers in COPD[END_REF]. Androgenic hormones has been reported with some successes when combined with exercise, but problems of extra physiologic doses remain [START_REF] Pison | Effects of home pulmonary rehabilitation in patients with chronic respiratory failure and nutritional depletion[END_REF].

(Table II)

Multimodality approaches

A holistic approaches such as the combination of adequate nutritional support with reduction of mechanical insufficiency (with drugs, lung volume reduction, breathing technique), physical activity, anabolic agents or appetite stimulants and non-invasive ventilation will give these patients potential improvements strategy [START_REF] O'donnell | Effect of fluticasone propionate/salmeterol on lung hyperinflation and exercise endurance in COPD[END_REF][START_REF] Mineo | Resting energy expenditure and metabolic changes after lung volume reduction surgery for emphysema[END_REF][START_REF] Creutzberg | A role for anabolic steroids in the rehabilitation of patients with COPD? A double-blind, placebo-controlled, randomized trial[END_REF].

New tools

Studies also have shown encouraging effects of n-3 polyunsaturated fatty acids (PUFAs) on cytokine release, markers of immune function, on exercise capacity and FFM of COPD patients [START_REF] Yeh S-S, Deguzman | Reversal of COPD-Associated Weight Loss Using the Anabolic Agent Oxandrolone*[END_REF].

Therefore nutrition support with PUFA can be considered to give positive effect for these patients.

Oxygen supplementation could reduce dynamic hyperinflation at rest and during or after exercise due to changes in ventilatory pattern [START_REF] Ferreira | The influence of 6 months of oral anabolic steroids on body mass and respiratory muscles in undernourished COPD patients[END_REF]. A significant weight gain was observed in malnourished COPD patients after NPPV initiation due to reduce hyperinflation which can help diaphragm to work more

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Page 11 / 28 efficiently and avoid dyspnea when eating (gastric filling) [START_REF] Burdet | Administration of growth hormone to underweight patients with chronic obstructive pulmonary disease. A prospective, randomized, controlled study[END_REF]. A better knowledge of molecular steps and targets involved in systemic consequences of COPD is of paramount importance to have better treatments. As an example, it appeared that the peroxisome proliferator-activated receptors (PPARs) are involved in cachexia, decreased oxidative muscle metabolism, oxidative stress and systemic inflammation suggesting role for PPARs agonists [START_REF] Agusti | Dynamic hyperinflation and pulmonary inflammation: a potentially relevant relationship[END_REF].

Conclusion and non-answered questions

Nutritional depletion has been widely reported in COPD patients. Diet may also play a role both in the development and the progression of the disease. Holistic management for these patients should include nutritional modulation beside reduction of mechanical disadvantage with long-acting bronchodilators, lung volume reduction, physical activity, anabolic agents or appetite stimulants and non-invasive ventilation. Furthermore, metabolic syndrome must be considered in the management of COPD in order to avoid any co-morbidities complication.

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Legends of figures

  20 for review) and may play a role in cachexia. COPD has been proposed in this context a new syndrome called "chronic systemic inflammatory syndrome" including at least 3 out 6 following factors: age older than 40, more than 10 year-smoking history, symptoms and pulmonary functions compatible with COPD, chronic heart failure, insulin resistance, and increased plasma CRP [39••,40]. There may be also a link between inflammation and leptin levels. Nevertheless, major gaps persist related to environmental and genetic factors underlying cachexia susceptibility, to sequential molecular steps [20••].

Figure 1 .Figure 2 . 28 Fig. 2 Abbreviations

 12282 Figure 1. Gene-environnement interactions on body composition and the risk of COPD

Table I . Nutritional supplementation studies with more than 2 weeks supplementation in COPD patients, integrated (filled in grey) and non integrated with rehabilitation programme (includes only studies after year 1990)

 I

	21-03-08 21-03-08 21-03-08		Page 25 / 28 Page 26 / 28 Page 27 / 28
		failure at home, on outpatients	randomized,	pulmonary rehabilitation + androgens +	exacerbation rate, survival forces, 6MWT
	Fuenzalida,	NIV and/or LTOT Nutritionally	controlled study, at Prospective,	oral nutritional supplements, 560 9, 5/4, oral supplementation, 1080	Inclusions completed, results pending Body weight, anthropometry, immune response
	Author, year 1990[66]	Patient's depleted,	Study design home controlled study	n, supplementation group/control, kcal/day versus health education alone kcal/day, no pulmonary rehabilitation	Judgements criteria Improvement of imuunity status
	Vermeeren et	characteristics, Nutritionally hospitalized then	Randomized	intervention 57, Energy and protein-rich nutritional	Results Body composition, respiratory and skeletal muscle strength, lung
	al, 2004[62]	settings depleted COPD outpatients	double-blind,	supplements during hospitalization for	function and symptoms
	Faager et al, Whittaker,	COPD patients with patients, hopitalized Nutritionally	Randomized, placebo-controlled Prospective,	23, 13/10, creatine supplementation & an acute exacerbation 10, 6 / 4, nasogastric tube	Physical performance, lung function test No significant improvements in lung function or muscle strength Body weight, respiratory muscle forces, hand grip, lung function
	2006[58] 1990[67]	FEV1 < 70% , patients depleted,	double-blind, study controlled study	exercise training supplementation + 1000 kcal versus +	No significant improvement Improvement of body weight and respiratory muscle forces
	Steiner,	outpatients Nutritionally hospitalized	placebo-controlled Prospective,	85, 42 / 43, oral supplementation, 570 100 kcal, 16 days, no rehabilitation	Body weight, body composition, quality of life (CRQ),
	study controlled study Abbreviations: PImax: maximal inspiratory mouth pressure; PEmax: maximal expiratory mouth pressure; MVV : Maximal Voluntary Ventilation ; 2003[63] depleted COPD kcal / j, pulmonary rehabilitation quadriceps muscle forces, hand grip, shuttle test
	Fuld et al, 6MWT : 6 minutes walking test ; REE : Resting Energy Expenditure Moderate to severe Randomized 38, 20/18, creatine nutritional patients, oupatients	Fat-free mass, peripheral muscle strength and endurance, health Improvement of shuttle test and quality of life
	2005[59] Creutzberg,	COPD, outpatients Nutritionally	double-blind, Prospective,	supplementation, pulmonary 69 / 28, oral supplementation 570	status Body weight, body composition, lung function, hand grip,
	2003[64]	depleted COPD	placebo-controlled controlled study	rehabilitation kcal/day, pulmonary rehabilitation	Significant improvement in all parameters, except exercise respiratory muscle forces, maximal exercice test, quality of life
		patients, oupatients	study		capacity Improvements of body weight, fat free mass, hand grip,
					respiratory muscle forces
	Matsuyama et	COPD patients	Randomized,	64, 32/32, nutritional support with	Leukotriene B4 levels, tumor necrosis factor-alpha and
	al, 2005[60] Rogers,	FEV 1 < 60% Malnourished	double-blind Prospective,	omega-3 PUFA-rich diets Oral supplementation, no pulmonary	interleukin-8 levels Body weight, respiratory muscle forces, hand grip, lung function,
	1992[65]	COPD patients,	controlled study controlled study	rehabilitation	Decreased significantly in the n-3 group 6MWT
		hospitalized then			Improvement of body weight, hand grip, respiratory muscle
	Pison, 2004[61] Chronic respiratory 3-month	122, 60/62, health education +	6MWD, quality of life, body composition, exercise capacity,

Table II . Effects of anabolic agents in COPD patients integrated (filled in grey) and non-integrated with rehabilitation programme

 II

	21-03-08	Page 28 / 28
	Author, year Ferreira,	Patient's characteristics, settings BMI below 20 kg/m 2 , the maximal	Study design Prospective, randomized,	n, supplementation 17, 7/10, 250 mg of testosterone	Judgements criteria Body mass index (BMI), lean body mass,
	1998[77]	inspiratory pressure (PImax) was	controlled, double-blind	group/control, intervention IM at baseline and 12 mg of oral	Results anthropometric measures, respiratory
	Casaburi,	FEV 1 of 60% predicted or less, FEV 1 below 60% of the predicted value,	Prospective, randomized, study	47, 100 mg/week testosterone stanozolol a day for 27 weeks	Body weight, body composition, muscle strength, and functional exercise
	2004[74]	to vital capacity ratio of 60% or less, outpatient	controlled, double-blind Pulmonary rehabilitation	enanthate in sesame oil	respiratory muscle function capacity
		outpatients	study, rehabilitation in 2 out		Significant increased lean body mass Increases in BMI, lean body mass, and
			4 groups		anthropometric measures of arm and thigh
					circumference
	Creutzberg, Burdet,	FEV 1 < 70% predicted, FEV 1 increase FEV1/FVC ratio ≤ 70% predicted,	Prospective, randomized, Prospective, randomized,	63, 33/30, 50 mg of nandrolone 16, 8/8, subcutaneous injection of	Body composition, muscle function, Nutritional status, resting metabolism,
	2003[75] 1997[78]	≤ 10% after inhalation ß 2 -agonist, FEV 1 increase ≤ 10% after albuterol	controlled, double-blind double blind, controlled	decanoate IM 0.15 IU/kg rhGH	exercise capacity, health status, muscle strength, exercise tolerance,
		outpatient inhalation, BMI ≤ ideal body weight,	study study		erythropoietic parameters, laboratory dyspnea
		outpatient	Pulmonary rehabilitation Pulmonary rehabilitation		parameters Increases lean body mass in underweight
					Improvements in muscle function and patients
					exercise capacity
	Yeh, 2002[76]	FEV 1 < 50% predicted, FEV 1 /FVC	Prospective, open-label, 4-	128, oxandrolone, 10 mg bid	Body weight, body composition,
		ratio < 70%, weight 90% ideal body	month clinical trial		spirometry, 6MWT
		weight, outpatient			Significant weight gain