Nicolas Normand
email: nicolas.normand@polytech.univ-nantes.fr

Pierre Évenou
email: pierre.evenou@polytech.univ-nantes.fr

Medial Axis LUT Computation for Chamfer Norms Using H-Polytopes

Chamfer distances are discrete distances based on the propagation of local distances, or weights defined in a mask. The medial axis, i.e. the centers of the maximal disks (disks which are not contained in any other disk), is a powerful tool for shape representation and analysis. The extraction of maximal disks is performed in the general case with comparison tests involving look-up tables representing the covering relation of disks in a local neighborhood. Although look-up table values can be computed efficiently [1], the computation of the look-up table neighborhood tend to be very time-consuming. By using polytope [2] descriptions of the chamfer disks, the necessary operations to extract the look-up tables are greatly reduced.

Introduction

The distance transform DT X of a binary image X is a function that maps each point x with its distance to the background i.e. with the radius of the largest open disk centered in x included in the image. Such a disk is said to be maximal if no other disk included in X contains it. The set of centers of maximal disks, the medial axis, is a convenient description of binary images for many applications ranging from image coding to shape recognition. Its attractive properties are reversibility and (relative) compactness.

Algorithms for computing the distance transform are known for various discrete distances [START_REF] Rosenfeld | Sequential operations in digital picture processing[END_REF][START_REF] Borgefors | Distance transformations in arbitrary dimensions[END_REF][START_REF] Coeurjolly | Optimal separable algorithms to compute the reverse euclidean distance transformation and discrete medial axis in arbitrary dimension[END_REF][START_REF] Strand | Weighted distances based on neighborhood sequences[END_REF]. In this paper, we will focus on chamfer (or weighted) distances. The classical medial axis extraction method is based on the removal of non maximal disks in the distance transform. It is thus mandatory to describe the covering relation of disks, or at least the transitive reduction of this relation. For simple distances this knowledge is summarized in a local maximum criterion [START_REF] Rosenfeld | Sequential operations in digital picture processing[END_REF]. The most general method for chamfer distances uses look-up tables for that purpose [START_REF] Borgefors | Centres of maximal discs in the 5-7-11 distance transforms[END_REF].

In this paper we propose a method to both compute the look-up tables and the look-up table mask based on geometric properties of the balls of chamfer norms. Basic notions, definitions and known results about chamfer disks and medial axis look-up tables are recalled in section 2. Then section 3 justifies the use of polytope formalism in our context and presents the principles of the method. In section 4, algorithms for the 2D case are given.

Chamfer Medial Axis

Chamfer Distances

Definition 1 (Discrete distance, metric and norm). Consider a function d : Z n × Z n → N and the following properties ∀x, y, z ∈ Z n , ∀λ ∈ Z:

1. positive definiteness d(x, y) ≥ 0 and d(x, y) = 0 ⇔ x = y , 2. symmetry d(x, y) = d(y, x) , 3. triangle inequality d(x, z) ≤ d(x, y) + d(y, z) , 4. translation invariance d(x + z, y + z) = d(x, y) , 5. positive homogeneity d(λx, λy) = |λ| • d(x, y) .
d is called a distance if it verifies conditions 1 and 2, a metric with conditions 1 to 3 and a norm if it also satisfies conditions 4 and 5.

Most discrete distances are built from a definition of neighborhood and connected paths (path-generated distances), the distance from x to y being equal to the length of the shortest path between the two points [START_REF] Rosenfeld | Distances functions on digital pictures[END_REF]. Distance functions differ by the way path lengths are measured: as the number of displacements in the path for simple distances like d 4 and d 8 , as a weighted sum of displacements for chamfer distances [START_REF] Borgefors | Distance transformations in arbitrary dimensions[END_REF] or by the displacements allowed at each step for neighborhood sequence distances [START_REF] Rosenfeld | Distances functions on digital pictures[END_REF][START_REF] Borgefors | Distance transformations in arbitrary dimensions[END_REF], or even by a mixed approach of weighted neighborhood sequence paths [START_REF] Strand | Weighted distances based on neighborhood sequences[END_REF].

For a given distance d, the closed ball B c and open ball B o of center c and radius r are the sets of points of Z n :

B o (c, r) = {p : d(c, p) < r} B c (c, r) = {p : d(c, p) ≤ r} . ( 1
)
Since the codomain of d is N:

∀r ∈ N, d(c, p) ≤ r ⇔ d(c, p) < r + 1. So: ∀r ∈ N, B c (c, r) = B o (c, r + 1) . ( 2
)
In the following, the notation B will be used to refer to closed balls.

Definition 2 (Chamfer mask [START_REF] Thiel | Géométrie des distances de chanfrein[END_REF]). A weighting M = ( -→ v ; w) is a vector -→ v of Z n associated with a weight w (or local distance). A chamfer mask M is a central-symmetric set of weightings having positive weights and non-null displacements, and containing at least one basis of

Z n : M = {M i ∈ Z n × N * } 1≤i≤m
The grid Z n is symmetric with respect to the hyperplanes normal to the axes and to the bisectors (G-symmetry). This divides Z n in 2 n .n! sub-spaces (8 octants for Z 2 ). Chamfer masks are usually G-symmetric so that weightings may only be given in the sub-space 0 ≤ x n ≤ . . . ≤ x 1 .

Paths between two points x and y can be produced by chaining displacements. The length of a path is the sum of the weights associated with the displacements and the distance between x and y is the length of the shortest path.

Definition 3 (Chamfer distance [START_REF] Thiel | Géométrie des distances de chanfrein[END_REF]).

Let M = {( -→ v i , w i ) ∈ Z n × N * } 1≤i≤m
be a chamfer mask. The chamfer (or weighted) distance between two points x and y is:

d(x, y) = min λ i w i : x + λ i -→ v i = y, λ i ∈ N, 1 ≤ i ≤ m . ( 3
)
Any chamfer masks define a metric [START_REF] Verwer | Local distances for distance transformations in two and three dimensions[END_REF]. However a chamfer mask only generates a norm when some conditions on the mask neighbors and on the corresponding weights permits a triangulation of the ball in influence cones [START_REF] Thiel | Géométrie des distances de chanfrein[END_REF][START_REF] Rémy | Normes de chanfrein et axe médian dans le volume discret[END_REF]. When a mask defines a norm then all its balls are convex.

Geometry of the Chamfer Ball

We can deduce from (1) and (3) a recursive construction of chamfer balls:

B(O, r) = B(O, r -1) ∪ 0≤i≤m B(O + -→ v i , r -w i ) . ( 4
)
Definition 4 (Influence cone [START_REF] Thiel | Les distances de chanfrein en analyse d'images : fondements et applications[END_REF]).

Let M = {( -→ v i , w i ) ∈ Z n × N * } 1≤i≤m
be a chamfer mask generating a norm. An influence cone is a cone from the origin spanned by a subset of the mask vectors { -→ v i , -→ v j , -→ v k , . . .} in which only the weightings M i , M j , M k , . . . of the mask are involved in the computation of the distance from O to any point of the cone.

In each influence cone, the discrete gradient of the distance function is constant and equal to [START_REF] Thiel | Géométrie des distances de chanfrein[END_REF]:

(w i , w j , w k , . . .) • ( -→ v i | -→ v j | -→ v k | . . .) -1 , (5)
where

( -→ v i | -→ v j | -→ v k | . . .)
stands for the column matrix of the vectors spanning the cone. The distance d C (O, p) from the origin to any point p of this cone C is then:

d C (O, p) = (w i , w j , w k , . . .) • -→ v i -→ v j -→ v k . . . -1 • p . ( 6
)
For instance, with chamfer norm d 5,7,11 , the point (3, 1) is in the cone spanned by the vectors a = (1, 0) and c = (2, 1) and the weights involved are 5 and 11.

The distance between the origin and the point (3, 1) is then:

d Ca,c (O, (3, 1)) = 5 11 • 1 2 0 1 -1 • 3 1 = 5 1 • 3 1 = 16 .

Chamfer Medial Axis

For simple distances d 4 and d 8 , the medial axis extraction can be performed by the detection of local maxima in the distance map [START_REF] Pfaltz | Computer representation of planar regions by their skeletons[END_REF]. Chamfer distances raise a first complication even for small masks as soon as the weights are not unitary. Since all possible values of distance are not achievable, two different radii r and r may correspond to the same set of discrete points. The radii r and r are said to be equivalent. Since the distance transform labels pixels with the greatest equivalent radius, criterions based on radius difference fail to recognize equivalent disks as being covered by other disks. In the case of 3 × 3 2D masks or 3 × 3 × 3 3D masks, a simple relabeling of distance map values with the smallest equivalent radius is sufficient [START_REF] Arcelli | Finding local maxima in a pseudo-Euclidian distance transform[END_REF][START_REF] Svensson | Digital distance transforms in 3d images using information from neighbourhoods up to 5×5×5[END_REF]. However this method fails for greater masks and the most general method for medial axis extraction from the distance map involves look-up tables (LUT) that represent for each radius r and displacement

-→ v i , the minimal open ball covering B o (O, r 1 ,) in direction -→ v i [7]: Lut-→ v i (r 1 ) = min{r 2 : B o (O + -→ v i , r 1 ) ⊆ B o (O, r 2 )} .
Equivalently using closed balls (considering ( 2)): Medial Axis LUT Coefficients. A general method for LUT coefficient computation was given by Rémy and Thiel [START_REF] Thiel | Les distances de chanfrein en analyse d'images : fondements et applications[END_REF][START_REF] Rémy | Computing 3D medial axis for chamfer distances[END_REF][START_REF] Thiel | Géométrie des distances de chanfrein[END_REF]. The idea is that the disk covering relation can be extracted directly from values of distance to the origin. If d(O, p) = r 1 and d(O, p + -→ v i ) = r 2 , we can deduce the following:

Lut-→ v i (r 1 ) = 1 + min{r 2 : B(O + -→ v i , r 1 -1) ⊆ B(O, r 2 )} . ( 7
p ∈ B(O, r 1 ) = B o (O, r 1 + 1) , p + -→ v i / ∈ B(O, r 2 -1) = B o (O, r 2 ) , hence B o (O + -→ v i , r 1 + 1) ⊂ B o (O, r 2 ) and Lut-→ v i (r 1 + 1) > r 2 . If ∀p, d(O, p) ≤ r 1 ⇒ d(O, p + -→ v i ) ≤ r 2 then Lut-→ v i (r 1 + 1) = r 2 + 1. Finally: Lut-→ v i (r) = 1 + max(d(O, p + -→ v i ) : d(O, p) < r).
This method only requires one scan of the distance function for each displacement -→ v i . Moreover, the visited area may be restricted according to the symmetries of the chamfer mask. The order of complexity is about O(mL n ) if we limit the computation of the distance function to a L n image.

Medial Axis LUT Mask. Thiel observed that the chamfer mask is not adequate to compute the LUT and introduced a LUT Mask M Lut (R) for that purpose [12, p. 81]. M Lut (R) is the minimal test neighbourhood sufficient to detect the medial axis for shapes whose inner radius (the radius of a greatest ball) is less than or equal to R. For instance, with d 14,20,31,44 : Lut A mask incompleteness produces extra points in the medial axis (undetected ball coverings). A general method for both detecting and validating M Lut is based on the computation of the medial axis of all disks. When M Lut is complete, the medial axis is restricted to the center of the disk, when extra points remains, they are added to M Lut . This neighborhood determination was proven to work in any dimension n ≥ 2. However it is time consuming even when taking advantage of the mask symmetries.

3 Method Basics [START_REF] Ziegler | Lectures on Polytopes (Graduate Texts in Mathematics)[END_REF] Definition 5 (Polyhedron). A convex polyhedron is the intersection of a finite set of half-hyperplanes.

General H-Polytopes

Definition 6 (Polytope).

A polytope is the convex hull of a finite set of points.

Theorem 1 (Weyl-Minkowski). A subset of Euclidean space is a polytope if and only if it is a bounded convex polyhedron.

As a result, a polytope in R n can be represented either as the convex hull of its k vertices (V-description): P = conv({p i } 1≤i≤k ) or by a set of l half-planes (H-description):

P = {x : Ax ≤ y} , ( 8
)
where A is a l × n matrix, y a vector of n values that we name H-coefficients of P . Having two vectors -→ u and -→ v , we denote -→ u ≤ -→ v if and only if ∀i, -→ u i ≤ -→ v i .

Definition 7 (Discrete polytope). A discret polytope is the intersection of a polytope with Z n .

Minimal Representation. Many operations on R n polytopes in either V or H representation often require a minimal representation. The redundancy removal is the removing of unnecessary vertices or inequalities in polytopes. Since our purpose is mainly to compare H-polytopes defined with the same matrix A, no inequality removal is needed. However, for some operations, H-representations of discrete polytopes must be minimal in terms of H-coefficients.

Definition 8 (Minimal parameter representation). We call minimal parameter H-representation of a discrete polytope P , denoted H-representation, a

H-representation P = {x : Ax ≤ y} such that y is minimal:

P = {x ∈ Z n : Ax ≤ y} and ∀i ∈ [1..l], ∃x ∈ Z n : A i x = y i , ( 9
)
where A i means line i in matrix A. H case, the three equalities are verified for the same point [START_REF] Strand | Weighted distances based on neighborhood sequences[END_REF][START_REF] Rosenfeld | Sequential operations in digital picture processing[END_REF]. Notice that although coefficient values are minimal, this representation is still redundant: the second inequality could be removed.

The H function which gives the minimal parameters for a given polytope P is introduced for convenience: H(P ) = max{Ax : x ∈ P }. {x : Ax ≤ H(P )} is the H-representation of P = {x : Ax ≤ y}. Figure 1 depicts two representations of the same polytope P in Z 2 .

H-Polytope Translation. Let P = {x : Ax ≤ y} be a H-polytope. The translated of P by -→ v which is also the Minkowski sum of P and { -→ v } is:

(P )-→ v = P ⊕ { -→ v } = {x + -→ v : Ax ≤ y} = {x : Ax ≤ y + A -→ v } . ( 10
)
The translation of a minimal representation gives a minimal representation.

Covering Test. Let P = {x : Ax ≤ y} and Q = {x : Ax ≤ z} be two polyhedrons represented by the same matrix A but different sets of H-coefficients y and z. P is a subset of Q if (sufficient condition):

y ≤ z ⇒ P ⊆ Q . ( 11
)
Furthermore, if the H-description of the enclosed polyhedron has minimal coefficients, the condition is also necessary:

y = H(P ) ≤ z ⇔ P ⊆ Q . ( 12
)

Chamfer H-Polytopes

Describing balls of chamfer norms as polygons in 2D and polyhedra in higher dimensions is not new [START_REF] Verwer | Local distances for distance transformations in two and three dimensions[END_REF]. Thiel and others have extensively studied chamfer ball geometry from this point of view [START_REF] Thiel | Les distances de chanfrein en analyse d'images : fondements et applications[END_REF][START_REF] Rémy | Normes de chanfrein et axe médian dans le volume discret[END_REF][START_REF] Borgefors | Weighted digital distance transforms in four dimensions[END_REF]. Our purpose is to introduce properties specific to the H-representation of these convex balls.

Proposition 1 (Direct distance formulation).

d(O, x) = max 1≤i≤l (d Ci (x)) ( 13
)
where l is the number of cones, d Ci is the distance function in the i th cone. Note that this formula does not require to determine in which cone lies x.

Proof. Equation ( 6 This applies to the vertices of the unitary ball

-→ v i wi : d C ( -→ v i wi ) ≤ 1. By linearity of d C , d C ( -→ v i ) ≤ w i and d C ( i λ i -→ v i ) = i λ i d C ( -→ v i ) ≤ i λ i w i . d C ( i λ i -→ v i ) is always less than or equal to the length of the path i λ i -→ v i .
Chamfer Balls H-Representation. The H-representation of chamfer balls is directly derived from (13):

x ∈ B(O, r) ⇔ max 1≤i≤l {d Ci (x)} ≤ r ⇔ A M • x ≤ y ( 14
)
where A M is a H-representation matrix depending only on the chamfer mask M.

The number of rows in A M is equal to the number l of influence cones, each line of the matrix A M is computed with (5) and y is a column vector whose values are r. For instance, the H-representation matrix for d 5,7,11 is A M = 5 1 4 3 where (5 1) and ( 43) are the distance gradients in the two cones and

B(O, r) = x ∈ Z n : 4 3 5 1 • x ≤ r r .
Proposition 2 (Furthest point). Let A M be the matrix defined by the chamfer mask M generating a norm. The furthest point from the origin in the Hpolytope P = {x : A M • x ≤ y} is at a distance equal to the greatest component of y.

Proof. By construction of A M , ( 13) is equivalent to

d(O, x) = max i {A Mi • x}. max x∈P {d(O, x)} = max x∈P { max 1≤i≤l {A Mi • x}} = max{ H i (P )}
Proposition 3 (Minimal covering ball). The radius of the minimal ball centered in O that contains all points of a discrete H-polytope P represented by the matrix A M and the vector y is equal to the greatest component of y.

Proof. The smallest ball that covers the polytope P must cover its furthest point from the origin.

min{r

∈ N : P ⊆ B(O, r)} = max x∈P {d(O, x)} ( 15
)
Note that if P is not centered in O, the simplification due to symmetries do not hold and the full set of H-coefficients is needed, unless we ensure that the H-coefficents for the hyperplanes in the working sub-space are greater than Hcoefficients for the corresponding symmetric cones. This is the case when a Gsymmetric polytope is translated by a vector in the sub-space.

Definition 9 (Covering function). We call covering function of a set X of points of Z n the function C X which assigns to each point p of Z n , the radius of the minimal ball centered in p covering X:

C X : 2 Z n × Z n → N X , p → min{r : X ⊆ B(p,
C B(O,r) (p) = max{ H(B(p, r)} = max{ H(B(O, r)) + A M • p} . ( 16
)
One can notice that the covering function of the zero radius disk is equal to the distance function, as is the distance transform of the complement of this disk:

C B(O,0) (p) = DT Z n \{O} (p) = d(p, O) = d(O, p) .
Definition 10 (Covering cone). A covering cone C o,U in C X is a cone defined by a vertex o and a subset U of the chamfer mask neighbor set with det(U ) = ±1,

C o,U = {o + λ i -→ u i , λ i ∈ N, -→ u i ∈ U }, such that: ∀p ∈ C o,U , ∀ -→ u ∈ U, B(p, C X (p)) B(p + -→ u , C X (p + -→ u )) .

. l] and a point o such that

A Mj • -→ u i , ∀ -→ u i ∈ U and H j (B(o, r)) are maximal then C o,U is a covering cone in C B(O,r) .
Proof. Let j be the row number of a maximal component of H(B(o, r)) and

A M • -→ u i , ∀ -→ u i ∈ U ,
C B(O,r) (p) = max H (B(o, r)) + λ i A M -→ u i = H j (B(o, r)) + λ i A Mj -→ u i .

Results

An implementation of these algorithms was developed in C language. It produces output in the same format as the reference algorithm [START_REF] Rémy | Medial axis for chamfer distances: computing look-up tables and neighbourhoods in 2d or 3d[END_REF] so that outputs can be compared character-to-character. Tests were done on various chamfer masks and different maximal radii. The results are almost always identical except for insignificant cases close to the maximal radius for which covering radii exceed the maximum. Other differences may occur in the order weightings are added to M Lut (sorted with respect to the covered radius or to the covering radius).

The run times of both reference and proposed algorithms are given in Table 1 for various sizes an distances.

Conclusion and Future Works

In this paper methods to compute both the chamfer LUT and chamfer LUT mask were presented. Speed gains from the reference algorithm [START_REF] Rémy | Medial axis for chamfer distances: computing look-up tables and neighbourhoods in 2d or 3d[END_REF] are attributable to the representation of chamfer balls as H-polytopes. This description allows to avoid the use of weight propagation in the image domain and permits a constant time covering test by the direct computation of covering radii. Although not thoroughly tested, we think that LUT value computation is faster with our method due to the smaller size of the test space (linear with the radius of the maximal ball). For M Lut determination, results show must faster computation especially for large radii. This is due to the reduced search space eliminating covering cones and the constant time covering test.

While applications always using the same mask can use precomputed M Lut and LUT, other applications that potentially use several masks, adaptive masks, variable input image size can benefit from these algorithms. A faster computation of M Lut is also highly interesting to explore chamfer mask properties. Beyond improved run times, the H-polytope representation helped to prove new properties of chamfer masks. And a new formula of distance which doesn't need to find in which cone lies a point was given.

Whereas the underlying theory (H-representation of balls, translation, covering test and covering cone) does not depend on the dimension, the algorithms were given only for dimension 2. In the 2D case, vectors can be ordered by angle so two consecutive vectors define a cone. Higher dimensions require a Farey triangulation of chamfer balls.

)

  Consider for instance the d 5,7,11 distance [16, Fig. 14]. Lut (1,0) (10) = 12 means that B o (O, 10) ⊆ B o ((2, 1), 12) but B o (O, 10) ⊆ B o ((2, 1), 11). Or, in terms of closed balls, B(O, 9) ⊆ B((2, 1), 11) but B(O, 9) ⊆ B((2, 1), 10).

( 2 , 1 )

 21 (291) = 321 and Lut (2,1) (321) = 352 but the smallest open ball of center O covering B o ((4, 2), 291) is B o (O, 351). In this particular case, the point (4, 2) is not in the chamfer mask but should be in M Lut (R) for R greater than 350.

Fig. 1 .

 1 Fig. 1. H-representations of a discrete G-symmetrical polytope P (restricted to the first octant). Dashed lines: H-representation of P . Thick lines: b H-representation of P . In the bH case, the three equalities are verified for the same point[START_REF] Strand | Weighted distances based on neighborhood sequences[END_REF][START_REF] Rosenfeld | Sequential operations in digital picture processing[END_REF]. Notice that although coefficient values are minimal, this representation is still redundant: the second inequality could be removed.

  ) states that for all the influence cones C that contain p, d C (O, p) = d(O, p). For other cones, this relation does not hold, but it is still possible to compute d C (O, p). The influence cone C corresponds to a facet of the unitary real ball supported by the hyperplane {x ∈ R n : d C (x) = 1}. Due to convexity, the unitary ball is included in the halfspace {x ∈ R n : d C (x) ≤ 1}.

  r)} . The covering function of the chamfer ball B(O, r) at point p gives the radius of the minimal ball centered in p that contains B(O, r). It is equal using central symmetry to the minimal ball centered in O covering B(p, r) and therefore it is the maximal component of the H-representation of B(p, r):

Proposition 4 .Proposition 5 .

 45 If C o,U is a covering cone in C B(O,r) then for any point q in C o,U \ U \ {o} there exists p such that:B(O, r) B(p, C B(O,r) (p)) B(q, C B(O,r) (q)) .Proof. Since q = o, there always exists another pointp in C o,U , distinct from q, such that B(p, C B(O,r) (p)) B(q, C B(O,r) (q)). B(O, r) ⊆ B(p, C B(O,r) (p)) always holds by definition of C B(O,r) . A sufficient condition for B(O, r) = B(p, C B(O,r)) is that a point p distinct from O exists in C o,U which is always the case if q is not equal to one of the generating vectors of C o,U . If there is a integer j ∈ [1 . .

  then j is a maximal component of any positive linear combination of these vectors. Let p be any point in C o,U , p = o + i λ i -→ u i , λ i ∈ N and B p the minimal ball covering B(O, r). From (16) we deduce that C B(O,r) is affine in C o,U :

Table 1 .

 1 Run times (in seconds)

	L	5,7,11	14,20,31,44	62,88,139,196,224	68,96,152,215,245, 280,314,346,413
	reference 200 0.468524 0.001109 1.388558 0.005708 6.366711 0.053448 8.867095 0.346118 H reference H reference H reference H
	500 8.315003 0.002784 25.262300 0.017302 125.293637 0.145683 177.506492 0.975298
	1000 90.670806 0.007007 268.807796 0.036611 1267.910045 0.276737 1684.583989 1.778505

http://www.cb.uu.se/~tc18/

A paper will greater details and results is being prepared for presentation in a journal. The source code for algorithms presented here is available from the IAPR technical committee on discrete geometry (TC18) 1 .

In the same way, the radius of the minimal ball centered in p + -→ u , -→ u ∈ U covering B(O, r) is:

In other words, B(p, C B(O,r) (p)) is a subset of B(p + -→ u , C B(O,r) (p + -→ u )).

LUT and M Lut Computation for 2D chamfer norms

LUT and M Lut computation methods for 2D chamfer norms are presented here.

Both are based on the minimal H-representation of the chamfer balls, from which we compute the covering function and deduce the LUT values. M vectors are ordered by angle so that each influence cone is defined by two successive angles

H-Representation of Chamfer Balls

The computation of the LUT is based on a H-representation of the chamfer norm balls. All share the same matrix A M which depends only on the chamfer mask [START_REF] Arcelli | Finding local maxima in a pseudo-Euclidian distance transform[END_REF]. H-coefficients of balls are computed iteratively from the ball of radius 0, B(O, 0) = {x : A M x = 0} using ( 4) and [START_REF] Verwer | Local distances for distance transformations in two and three dimensions[END_REF].

Each LUT value is obtained from the covering function ( 16):

LUT Mask

The algorithm starts with an empty mask and balls with increasing radii are tested for direct covering relations as in [START_REF] Rémy | Medial axis for chamfer distances: computing look-up tables and neighbourhoods in 2d or 3d[END_REF]. However, in our case, the covering relations are seen from the perspective of the covered balls (in the covering function) whereas in [START_REF] Rémy | Medial axis for chamfer distances: computing look-up tables and neighbourhoods in 2d or 3d[END_REF], they are considered from the point of view of the covering balls (in the distance map). Another difference lies in the computation of covering radii which does not require the propagation of weights thanks to a direct formula [START_REF] Rémy | Medial axis for chamfer distances: computing look-up tables and neighbourhoods in 2d or 3d[END_REF]. In order to remove useless points, all known LUT neighborhoods are checked for an indirect covering by the procedure visitPoint.

In each influence cone (O, { -→ v i , --→ v i+1 }), a set of covering cones are detected to limit the search space: a 2D covering cone ((a, a), {(1, 0), (0, 1)}) with a vertex chosen on the cone bissector [O, (1, 1)), then 1D covering cones ((b, α), {(1, 0)}), ((α, b), {(0, 1)}) for α varying from 1 to a-1 (coordinates relative to (O, { -→ v i , --→ v i+1 })). Fig. 2 shows the working sub-space partitioning, covering cones and visit order of points for the chamfer norm d 14,20,31,44 and the inner radius 20. Fig. 2. 1D (dashed lines) and 2D (filled areas) covering cones and visit order of points.

Algorithm 1: Computation of the look-up table mask

// current inner radius, increasing order for i ← 1 to m do // visit vector axes visit1DCone(MLut, r, O, -→ vi );

Procedure visit1DCone(M Lut , r, p, -→ u ) Input: MLut, inner radius r, vertex and direction of the tested cone p, -→ u for α ← 1 to findCoveringConeVertex(r, p, -→ u ) do visitPoint(MLut, r,


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DIGESTA FLOWS IN SHEEP FED POOR-QUALITY HAY SUPPLEMENTED WITH UREA AND CARBOHYDRATES

Keywords: Urea, Molasses, Maize, Ruminants, Sheep, Festuca.)

Two metabolism trials were conducted with 12 yearling crossbred wethers per trial (34 and 38 kg for trials 1 and 2, respectively). The wethers, equipped with ruminal, abomasal and ileal cannulae, were randomly allotted for each trial to the following treatments: 1) bay alone or hay supplemented with 2) .9% urea, 3) 1% urea and 6.5% molasses or 4) 1% urea and 5.2% corn. Two digestive flow markers were used: Cr203 powder and Co-ethylenediaminetetraacetic acid (Co-EDTA). Urea and Co-EDTA were infused continuously into the rumen via cannula. Daily dry matter (DM) intake averaged 517 g. Urea supplementation improved N retention (P < .01). Apparent digestibility of DM, acid detergent fiber (ADF) and energy was not affected by treatment. Urea and carbohydrate supplementation increased ruminal propionic acid molar proportions (P < .05). Apparent ruminal DM digestion accounted for 41% of the total DM degraded, whereas 77.4% of the digestible ADF was degraded in the rumen. Urea supplementation increased ADF digestion in the large intestine (P < .01). Urea and carbohydrate supplementation resulted in a stepwise increase in N flowing with the liquid phase at the abomasum. Mean retention times of the solid and liquid phases of digestive contents were similar across treatments. Overall, benefits of supplementation of poor-quality fescue hay diets by small amounts of urea and readily available carbohydrates remain questionable for sheep fed at a fixed level of intake below maintenance.

Introduction

Large improvements have been accomplished in forage production and conservation; however, large quantities of poor-quality forages are still produced. Intake of such forages is generally limited, and N supplementation is essential for optimal intake [START_REF] Church | Nitrogen metabolism and requirements[END_REF]. Urea, a relatively inexpensive source of N, is widely used, but responses to urea have been variable, without any apparent explanations [START_REF] Johnson | Influence of carbohydrate solubility on non-protein nitrogen utilization in the ruminant[END_REF].

When a low-N basal diet is supplemented with urea, a source of carbohydrate is required [START_REF] Mills | Utilization of urea and growth of heifer calves with corn molasses or cane molasses as the only readily available carbohydrate in the ration[END_REF]. It is generally believed that urea-molasses supplements are beneficial in grazing management. In a grazing trial with sheep, [START_REF] Nolan | Body weight and wool production in grazing sheep given access to a supplement of urea and molasses: Intake of supplement/ response relationships[END_REF] calculated that a urea-molasses supplement was used as an energy source, and in no instance did it prevent the sheep from losing weight. In another trial with sheep fed oat straw and urea, the addition of starch increased feed intake, weight gain and wool production, but at least 10% starch was required for a net gain in tissue energy storage [START_REF] Mulholland | Effect of starch on the utilization by sheep of a straw diet supplemented with urea and minerals[END_REF]. Corn was shown to improve N retention to a greater extent than cane molasses, when fed at more than 40% of the diet [START_REF] Bell | Value of urea nitrogen in rations containing different carbohydrate feeds[END_REF].

The objectives of this study were to determine the effects of urea on the use of poorquality hay by sheep and to compare the further supplementation of sugarcane molasses or corn grain. Nitrogen and energy supplements were fed at levels equivalent to the level of intake of urea-molasses liquid supplement when self-fed in a lick tank [START_REF] Ortigues | Nitrogen supplementation to stocker cattle fed poor quality hay[END_REF].

Experimental Procedure

Animals. Two metabolism trials were conducted with 12 yearling crossbred (Western • Suffolk) wethers (avg weight, 35.9 kg) in each aMineral supplement consisting of 10 g of trace mineral salt and 10 g of defluorinated phosphate was fed daily to each wether.

bDry-matter basis.

trial. The wethers were equipped with abomasal, ileal and ruminal cannulae. The wethers were fitted, 2 mo before the trials, with a single plexiglass-polyethylene cannula 6 at the abomasum (10 cm from the pyloric valve, and exteriorized behind the last rib) and a T-shaped stainless-steel cannula 6 at the distal ileum (10 to 15 cm from the ileo-cecal juncture, and exteriorized higher up on the right flank). A single ruminal cannula made from Tygon tubing (id 1.25 cm) and glued together with cyclohexanone was subsequently implanted nonsurgically according to a modification of the procedure described by [START_REF] Faichney | A simple technique to establish a self-retaining rumen catheter suitable for long-term infusions[END_REF] for ruminal catheters. No difficulties were experienced with the abomasal and ileal cannulae, whereas ruminal cannulae allowed leakage of ruminal fluid on most wethers after 6 mo.

Wethers were blocked by weight and randomly allotted to the treatment groups for each trial with the restriction that no animal would receive the same treatment during the two trials. Wethers were kept in metabolism crates allowing separate collection of feces and urine [START_REF] Briggs | Metabolism stalls for wethers and steers[END_REF].

6 Precision Machine Co., Lincoln, NE. ~Mandostat, New York.

Diets. Four treatment diets were compared (Table 1). The basal diet consisted of chopped, poor-quality fescue-clipping hay. The supplements used were urea, sugarcane molasses, ground corn grain and corn gluten meal. Amounts of dry matter (DM) consumed were equalized across treatments on the basis of 90% ad libitum intake achieved by wethers on the basal diet.

The diets were fed in two equal portions at 12-h intervals. Urea was infused continuously into the rumen via cannulae, using multichannel cassette infusion pumps7; a sham infusion of water was administered to the control group. Chromic oxide powder (2 g/d or 1 g/feeding) was introduced into the rumen at mealtime, and Co-EDTA (2 g/d) was infused continuously into the rumen with the urea or sham infusates. At the end of the sampling period, the daily administration of markers was stopped and single larger doses of Cr203 (6 g) and Co-EDTA (6 g) were administered at feeding time to measure mean retention time.

Experimental Plan. Each trial consisted of a 6-d transition period, a 10-d preliminary period and a 7-d collection of feed, feces and urine. This was followed by a 6-d sampling of feed, feces, blood, ruminal fluid and gas, abomasal and ileal contents and a 5-d total collection of feces every 4 h. Fecal material was collected daily, dried in a forced-air oven at a maximum of ,50 C and composited. Urine was collected in 15 ml of 50% H2SO4 (v/v), subsampled and coraposited. Samples were taken of blood (via jugalar puncture), of ruminal fluid (from the ventral sac, via cannula) and of ruminal gas (through cannula, using a device similar to that described by [START_REF] Sumio | Collection of rumen gases from young calves[END_REF] )once daily, advancing the time by 4 h each day to represent every 4 h of the clock. The ruminal gas samples were stored in vacutainers previously filled with N2. Abomasal and ileal contents were sampled manually at 12-h intervals daily, advancing the time by 2 h each day to represent every 2 h of the clock. They were then composited on an equal-volume basis. Samples of digestive contents were then centrifuged at 2,000 x g for 10 min. Liquid phase, obtained by pouring off all the supernatant, was frozen at -20 C until further analyses. The residues, considered as representing the solid phase, were freeze-dried and ground through a .5-mm screen.

Chemical Analyses. Ammonia was determined according to the [START_REF] Conway | Microdiffusion Analysis and Volumetric Error[END_REF] procedure, chromic oxide according to [START_REF] Hill | Comparison of metabolizable energy and productive energy determinations with growing chicks[END_REF], and cobalt by atomic absorption s after a nitric-perchloric acid digestioga. Energy was determined with an adiabatic bomb calorimeter 9, and methane and carbon dioxide by procedures described by [START_REF] Schauer | Metabolism of formate in Methanobacterium formicisum[END_REF].

In samples of the collection period, N was analyzed by the Kjeldahl procedure (AOAC, 19.30); N was analyzed for the sampling period by Nesslerization [START_REF] Umbreit | Manometric Techniques[END_REF]. Nonprotein-N was separated from protein-N by trichloroacetic acid precipitation. The method of [START_REF] Coulombe | A new simple semimicro method for colorimetric determination of urea[END_REF] was used to determine urea. Acid detergent fiber (ADF) and lignin were analyzed according to [START_REF] Goering | Forage Fiber Analyses: Apparatus, Reagents, Procedures and Some Applications[END_REF]. Volatile fatty acids (VFA) were measured by gas chromatography by procedures described by [START_REF] Panditharatne | Yield, chemical composition and digestibility by sheep of orchardgrass fertilized with different rates of nitrogen and sulfur or associated red clover[END_REF].

Calculations. Fecal excretion during the sampling period was calculated from the Cr203 co~acentrations in feces corrected for 100% fecal recovery. The flow rates of liquid and solid phases were calculated according to [START_REF] Armentano | Method for calculating digesta flow and apparent absorption of nutrients from non-representative samples of digesta[END_REF], assuming the wethers were in a steady state over a 24-h SSpectrophotometer 403, Perkin Elmer Corp., Norwalk, CT.

9Parr Instrument Co., Moline, IL.

period. In these conditions, the instantaneous rate of change in the amount of marker over time at the point of sampling is equal to zero; it is also equal to the difference between the quantity of marker arriving at the point of sampiing and that leaving. The equations were as follows:

ARI = F s [Cr2Oals + FL [Cr203]L and AR2 = F s [Col s + FL [Col L;
therefore,

(ARI" [Col s --AR2" [Cr2Oa]s) FL -([Cols" [Cr203lL --[ColL" [Cr203ls) and (AR2 --FL" [Col L) FS - [Co] S
The terms ARI and AR2 are the rates (gg marker/min) of continuous administration of Cr203 and Co, respectively; F S and F L refer to flow (g digesta phase/min) of the solid and liquid phases, respectively; [Cr203]s and [Cr203]L refer to concentrations of Cr203 in the solid and liquid phases, respectively; and [Col s and [CO]L refer to concentration of Co in the solid and liquid phases, respectively (~g marker/g digesta phase). Marker concentrations in abomasal and ileal solid and liquid phases were adjusted based on the difference in total fecal recovery of Cr and Co from 100%. The flow rate of any constituent was assumed to be equal to the sum of the flow rates of that constituent in the solid and liquid phases as follows: flow of nutrient = FL(nutrient)L + Fs(nutrient) S. The mean retention time was calculated as described by [START_REF] Faichney | The effect of formaldehyde treatment of a concentrate diet on the passage of solute and particulate markers through the gastro-intestinal tract of sheep[END_REF].

Statistical Analyses. The ruminal and blood variables were analyzed according to a splitplot design with time trend, block effect nested within trial as the split-plot factor. The model included trial, treatment, block (within trial), time, trial x treatment, treatment • time and treatment x block. Balance trial and the flow data were analyzed according to the following model: trial, block (within trial), treatment and trial • treatment. In both models the error mean square was that for trial x treatment x block interaction. Because of missing observations for two sheep caused by large feed refusals, the data were analyzed by SAS (1982), using the least-squares procedure described by [START_REF] Harvey | Least-squared analysis of data with unequal subclass numbers[END_REF]. Treatment means were compared by orthogonal contrasts. Contrasts were supplemented vs. unsupplemented, urea vs urea plus carbohydrate supplementation, and molasses vs corn supplementation. Time effect was tested by orthogonal polynomials.

Results

Dry Matter Intake. The main feature of these data is that the level of intake was generally very low as determined from the ad libitum intake of wethers on the basal diet. It was observed, however, that wethers receiving the urea supplemented diets would have increased their intake, relative to the control group, if allowed to. Drv matter intake was 10% higher in the second than in the first trial, probably due to elevated DM and N contents of the hay (1.04 and 1.21% N, dry basis, in the first and second trials, respectively). This difference in intake was responsible for the trial or trial • treatment effects noted for some variables. Treatment differences were greater in the first than in the second trial, hence the interaction term. In both trials, however, effects were in the same direction, which explains the presentation of the average results only. Differences in urea feeding across diets (Table 1) resulted from variations in the volumes of urea solution effectively infused.

Weigbt Cbanges. At the beginning of trials 1 and 2, wethers weighed 33.5 kg and 38.3 kg, respectively. They lost an average of 2.8 kg and 6.4 kg during the respective trials. For the two trials, weight losses averaged 4.01, 4.23, 5.29 and 4.95 kg for the wethers fed the control diet and diets supplemented with urea, molasses and corn, respectively.

Marker Recovery. Based on total fecal collection, recovery of Cr203 and Co averaged 98.47 + 1.47% and 106.78 -+ 1.76%, respectively. Urinary recovery of Co was 3.83 -+ .24%.

Balance Trial. Apparent digestibility of DM and ADF was low and was not affected by treatments (Table 2). Apparent N digestibility for the total diet was improved by urea supplementation (P< .01). If urea digestibility is assumed to be 100%, then supplements did not bstandard error of the mean.

CTrial effect (P < .05).

dValues for the hay diet are different from values for the urea, molasses and corn supplemented diets (P < .01). eTrial X treatment interaction (P < .05). dTrial X treatment interaction (P < .05).

evalues for the hay diet are different from values for the urea, molasses, and corn supplemented diets (P < .01).

fValues for the urea supplemented diet are different from values for the molasses and corn supplemented diets (P < .05).

gvalues for the molasses supplemented diet are different from values for the corn supplemented diet (P < .05).

affect N digestibility of the hay. Nitrogen balance was negative for all treatments, but urea addition improved N retention (P < .01).

Ruminal and Blood Parameters. No treatment • time interaction was observed for any of the parameters measured, indicating that the kinetics of ruminal digestion were not altered by urea or carbohydrate supplementation. Ruminal pH was similar among treatments and was relatively stable after feeding (Table 3). Ruminal NH3-N and blood urea-N concentrations were not affected by treatments. Concentrations in these variables decreased for 6 to 10 h after feeding and increased thereafter (P < .01). Total VFA varied between 70.7 and 78 /~M/ml (Table 3). Urea and carbohydrate supplementation resulted in statistically significant stepwise increases in molar proportions of propionic acid. Branched-chain VFA were not found at all times nor in all wethers, and valeric acid was never detected. Concentrations of total VFA (Figure 1) and acetic, propionic and butyric acids decreased linearly with time after feeding (P < .01). There was a tendency, how-ever, for this decrease to start only 6 h after feeding, the time ruminal pH tended to be lowest. Proportions of methane in the rumen gas were high (Table 3) and increased linearly (P < .01) with the time after feeding (Figure 2). Methane proportion tended to be higher for the control and corn supplemented diets, but differences were not significant.

Sites of Dry Matter and Acid Detergent Fiber Digestion.

As calculated by difference between the measured fecal DM output of the collection and sampling periods, sampling removed an average of 12% of the DM of the digestible contents.

Dry matter flows along the gastro-intestinal tract were similar across treatments (Table 4). Dry matter flowing with the liquid phase represented 15% and 14% of the total DM flow at the abomasurn and ileum, respectively. The apparent DM digestion in the intestines was quite important, compared with pre-intestine digestion. In fact, 41%, 36% and 23% of the apparently digestible DM was digested in the stomach, small intestine and large intestine, respectively. Flows of ADF were based solely on the solid phase (Table 5). The stomach compartments were the primary sites of ADF digestion, accounting for the degradation of 77% of the digestible ADF. Urea supplementation in- bstandard error of the mean.

CTrial effect (P < .05).

dValues for the hay diet are different from values for the urea, molasses, and corn supplemented diets (P < .01). evalues for the urea supplemented diet are different from values for the molasses and corn supplemented diets (P < .01).

fValues for the molasses supplemented diet are different from values for the corn-supplemented diet (P < .01). bstandard error of the mean.

CTrial effect (P < .01). dvalues for the hay diet are different from values for the urea, molasses, and corn supplemented diets (P < .01).

evalues for the urea supplemented diet are different from values for the molasses and corn supplemented diets (P < .01).

fValues for the molasses supplemented diet are different from values for the corn supplemented diet (P < .01).

creased (P<.01) ADF digestion in the large intestine.

Sites of Nitrogen Digestion. Total N flow at the abomasum represented 173%, 108%, 116% and 128% of N intake for the control, urea, urea-molasses and urea-corn supplemented diets, respectively (Table 6). Digestion of N in the intestines was not different (P > .05) among treatments, but the extent of N digestion in the small intestine tended to be greater for the control diet and the diet containing the urea-corn supplement.

Mean Retention Time. The mean retention time (+ standard error) of Cr203 in the entire gastro-intestinal tract was 39 + 1.5, 41 + 1.25, 38.5 + 1.5 and 38.0 + 1.5 h for the control and for diets supplemented with urea, with urea plus molasses and with urea plus corn, respectively. The mean retention time of cobalt was 32.0 +-2.0, 33.2 +-1.8, 31.0 + 2.0 and 29.5 + 2.0 h, respectively. No significant differences among treatments were observed.

Discussion

The low intake of poorly digestible forage by cannulated, growing wethers made the wethers more susceptible to stress. Indeed, wethers in the present study consumed only 29% of the NRC (1985) recommendations for digestible energy of 35 kg wethers growing at 285 g/d and 19% to 32% of the recommended crude protein levels. These intake figures do not account for the removal of digesta by sampling. The problems relating to the use of poorquality forages by ruminants are therefore linked to some extent with those problems of feeding animals below maintenance, and illustrate some of the difficulties that can be encountered in such studies.

Lignification of the hay was high, which impeded ruminal degradation. Consequently, the level of digestible energy intake was too low to permit large benefits from urea. Adding small amounts of carbohydrates was of no ap- bstandard error of the mean.

CTrial effect (P < .O1L dvalues for the hay diet are different from values for the urea, molasses, and corn supplemented diets

(P < .Ol).
evalues for the urea supplemented diet are different from values for the molasses and corn supplemented diets (P < .05). fValues for the molasses supplemented diet are different from values for the corn supplemented diet (P < .05).

parent benefit. The poor nutritional status of the wethers and the additional stress due to removal of nutrients by sampling of digesta probably increased the variability of response among animals.

Nitrogen retention was improved by urea infusion (P < .01) but not by carbohydrate addition. In a similar experiment with Pangola grass hay, [START_REF] Martin | Effect of level and form of supplemental energy and nitrogen on utilization of low quality roughage by sheep[END_REF] obtained a significant stepwise increase in N retention with addition of urea and molasses. Those authors used slightly higher levels of molasses than were used in the present study. The enhanced N retention observed here with urea supplementation contradicts the trend for higher body weight losses observed with these treatments. It is not likely that differences in weight loss could be artifacts due to differences in gut fill, because DM digestibility and rates of passage of digesta along the tract were similar across treatments, whereas differences in DM intake were only slight.

Mean retention times were similar across treatments, although disturbances in digesta flow might have occurred as a result of sampiing. Values of the mean retention time of the solid phase are similar to those quoted by [START_REF] Warner | Rate of passage of digesta through the gut of mammals and birds[END_REF] for whole digesta of sheep fed hay. [START_REF] Campling | Factors affecting the voluntary intake of food by cows. 3. The effect of urea on the voluntary intake of oat straw[END_REF] noted that urea infusions increased rate of passage of food with ad libitum feed intake. With restricted intake, the increase in rate of passage was not so large, and [START_REF] Redman | Utilization of low quality roughages: Effects of urea and protein supplements of differing solubility on digesta flows, intake and growth rate of cattle eating oaten chaff[END_REF] did not detect any effect of urea supplementation.

It can be questioned whether the trend for additional weight loss of wethers fed the urea diets resulted from an energy requirement for excreting excess N in the form of urea. Because the wethers were on a low-energy status, body reserves would be used, and it is known that NH3 detoxification is related to energy metabolism [START_REF] Visek | Effects of urea hydrolysis on cell life span and metabolism[END_REF]. Based on a hypothesis proposed by [START_REF] Krebs | The metabolic fate of amino acids[END_REF], and assuming a ratio of muscle protein to water loss of 1 to 3, it can be calculated that to excrete the excess N, the required energy derived from muscle breakdown would account for 48%, 14% and 15% of the weight loss difference between the control diet and the urea, urea-molasses and urea-corn diets, respectively. These values are minimal estimates due to the uncertainties attached to calculating the energy cost of urea synthesis and excretion in the urine [START_REF] Grisolia | On specific dynamic action, turnover and protein synthesis[END_REF][START_REF] Oldham | Interrelationships between protein yielding and energy yielding nutrients[END_REF]. Ruminal fermentations, as determined from pH, VFA concentrations and molar proportions, and methane are characteristic of poorquality forage diets [START_REF] Thomas | Manipulation of rumen fermentation[END_REF] and showed only slight changes with supplementation. The linear changes of VFA concentrations with time point to slow ruminal degradation of hay not altered by urea supplementation, and are consistent with the suggestion that with such forages the ruminal microbial population is relatively stable [START_REF] Barry | Rumen fermentation studies of two contrasting diets. 1. Some characteristics of the in vivo fermentation, with special reference to the composition of the gas phase, oxidation/reduction state and volatile fatty acid proportions[END_REF][START_REF] Thomas | The clearance rate of rumen liquid and rumen fermentation pattern in sheep receiving forage diets[END_REF]. The high acetate and methane proportions reveal the inefficient pattern of ruminal fermentations relative to the metabolism of end-products of digestion in ruminants [START_REF] Thomas | Manipulation of rumen fermentation[END_REF]. Theoretical methane production was calculated using the stoichiometry of formation of fermentation end-products (~rskov et al., 1968) and yielded values of methane proportions 5 to 13 percentage units lower than those measured experimentally. The largest discrepancies were related to a higher variability of the results and could indeed be due to the sampling technique [START_REF] Jouany | Description d'une technique permettant d'effectuer des prelevements repetes de gaz darts le rumen[END_REF].

Apparent pre-intestinal DM digestion was less than DM digestion in the whole intestines. Because of the low intake and the low digestibility of the diets, the relative importance of endogenous plus microbial DM is magnified. Assuming that the DM of the abomasal liquid phase is mostly of microbial or endogenous origin [START_REF] Hogan | The digestion of pasture plants by sheep. 4. The digestion of Phalaris Tuberosa at different stages of maturity[END_REF][START_REF] Poncet | Utilisation du cerium-141 comme marqueur de la phase solide des contenus digestifs chez les ruminants[END_REF] or is composed of solutes, all the remaining nondegraded dietary DM was recovered in the solid phase. In this case, the rumen accounted for the degradation of 67% of the total DM digestion. This figure is closer to the measured value of 77% of the ADF digestion taking place in the rumen; to a large extent the difference probably was caused by the presence of microbes in the solid phase [START_REF] Czerkawski | Degradation of solid feeds in the rumen: Spatial distribution of microbial activity and its consequences[END_REF].

No S was supplemented, so N to S ratios were 59 to 1, 89 to 1, 32 to 1 and 93 to 1 for the control, urea, urea-molasses and urea-corn supplemented diets, respectively. [START_REF] Bird | Sulphur metabolism and excretion studies in ruminants. 9. Sulphur, nitrogen and energy utilization by sheep fed a sulphur-deficient and a sulphate supplemented, roughagebased diet[END_REF] showed that with S-deficient, urea-oat hulls diets, ruminal degradation of organic matter was limited, compared with S supplemented diets. It is not certain, however, that in the present study S was the primary limiting factor, especially because it was demonstrated with sheep that when the level of digestible energy was sufficient urea supplementation still enhanced microbial growth, even with N to S ratios of up to 60 to 1 [START_REF] Elliott | The effect of urea and urea plus sodium sulphate on microbial protein production in the rumen of sheep given diets high in alkali-treated barley-straw[END_REF].

The marker, Cr203, is known to migrate independently of the solid phase [START_REF] Corbett | Excretion of chromium sesquioxide administered as a component of paper to sheep[END_REF], and this may have been accentuated in the present study in which digestive contents were very fluid because of the forage diet and low intake. It was also noted that sampling removed a large portion of digestive contents. These factors contributed to the variability in the results and might explain the high DM and ADF digestion in the small intestine, which is contradictory with previous results [START_REF] Ulyatt | Structure and function of the large intestine of ruminants[END_REF]. Urea infusion increased N flow in the liquid phase at the abomasum, but the protein N values of the liquid phase were low, as expected from the small extent of ruminal fermentation.

It appears that growing wethers cannulated at three sites of the digestive tract are questionable models to study the use of poorquality forages. Cannulated wethers appear to be more susceptible to stress and to have a smaller capacity for intake than their intact pairs. Therefore, when the use of poor-quality forage diets by ruminants is studied with cannulated wethers, situations of net underfeeding can occur, as was the case in the present study, in which the techniques used were not precise enough to distinguish between the dietary, microbial and endogenous components of digesta. This study shows the need to separate the mechanisms of digestion of poorquality feeds and the effects of undernutrition experimentally.

Urea improved N retention; however, it was fed in excessive quantities relative to the energy available, and therefore it was costly to excrete. Urea and carbohydrates were supplemented in proportions similar to those observed with ad libitum intake of hay and liquid supplement [START_REF] Ortigues | Nitrogen supplementation to stocker cattle fed poor quality hay[END_REF]. The beneficial effects of low levels of readily available carbohydrates are still questionable.

Figure 1 .

 1 Figure 1. Postprandial changes in ruminal concentrations of total volatile fatty acids in wethers. Standard error for treatment • time interaction is 4.01.

Figure 2 .

 2 Figure 2. Postprandial changes in proportion of methane in ruminal gas in wethers. Standard error for treatment X time interaction is 3.07.

TABLE 1 .

 1 COMPOSITION OF THE DIETS a

	Supplement
	Urea +	Urea +

TABLE 2

 2

	. APPARENT DIGESTIBILITY OF DRY MATTER, ACID DETERGENT FIBER,
		NITROGEN AND ENERGY, AND NITROGEN BALANCE a
				Supplement
				Urea +	Urea +
	Item	None	Urea	molasses	corn	SE b
	Dry matter intake, g/d c	517	519	509	523	1.3
	Apparent digestibility, %c
	Dry matter	44.9	45.1	47.5	46.9	.45
	Acid detergent fiber	44.0	44.2	44.7	43.1	.58
	Crude protein de	25.2	52.1	51.4	53.3	.93
	Energy e	43.6	45.0	46.8	46.0	.62
	Nitrogen balance, g/d
	Intake cd	5.8	9.0	9.0	9.5	.09
	Fecal c	4.3	4.4	4.4	4.4	.05
	Urinary cd	3.8	5.0	5.6	5.2	.20
	Retention d	--2.3	--.4	--.9	--.1	.23
	aLeast-squares means.

TABLE 3 .

 3 RUM1NAL AND BLOOD MEASUREMENTS OF WETHERS FED POOR-QUALITY HAY a

	979

c[CHJ(CO2 + CH4)] X 100.

TABLE 4

 4

		. DRY MATTER FLOW AND DIGESTION IN WETHERS
			FED POOR-QUALITY HAY a
				Supplement
				Urea +	Urea +
	Item	None	Urea	molasses	corn	SE b
	lntake, g/d cdef	523	521	509	519	.9
	Flow, g/d
	Abomasum
	Solid phase	352	362	331	357	7
	Liquid phase c	59	62	63	70	4
	Ileum
	Solid phase	273	284	277	284	6
	Liquid phase c	40	43	42	50	2
	Feces	283	260	261	266	4
	Digested
	Pre-intestinally
	g/d	113	98	115	92	7
	% of intake	21.3	18.9	22.5	17.6	1.3
	Small intestine
	g/d	97	96	75	94	6
	% of intake	18.6	18.4	14.7	18.2	1.1
	Large intestine
	g/d c	35	67	57	70	5.2
	% of intake c	6.6	12.7	11.1	13.3	1.0
	aLeast-squares m cans.

TABLE 5 .

 5 ACID DETERGENT FIBER FLOW AND DIGESTION IN WETHERS FED POOR-QUALITY HAY a

	981

TABLE 6 .

 6 NITROGEN FLOW AND DIGESTION IN WETHERS FED POOR-QUALITY HAY a

	Supplement
	Urea +	Urea +

J. Anim.Sci. 1988. 66:975-985