O Déforges
email: odeforge@insa-rennes.fr

M Babel
email: mabel@insa-rennes.fr

N Normand
email: nnormand@polytech.univ-nantes.fr

Fast Recursive Grayscale Morphology Operators: from the Algorithm to the Pipeline Architecture

Keywords: Mathematical morphology, 8convex structuring element operators, regular dedicated pipeline architecture

This paper presents a new algorithm for an efficient computation of morphological operations for gray images and its specific hardware. The method is based on a new recursive morphological decomposition method of 8-convex structuring elements by only causal twopixel structuring elements (2PSE). Whatever the element size, erosion or/and dilation can then be performed during a unique raster-like image scan involving a fixed reduced analysis neighborhood. The resulting process offers low computation complexity combined with easy description of the element form. The dedicated hardware is generic and fully regular, built from elementary interconnected stages. It has been synthesized into an FPGA and achieves high frequency performances for any shape and size of structuring element.

Introduction

Mathematical morphology was first introduced as a method to measure binary objects, but soon became a complete theory based on set operations [Serra (1982)]. Morphology relies on the use of set operators (intersection, union, inclusion, complement) to transform an image. The transformed image usually has fewer details, but its main characteristics are still present. Once the image has been simplified, measurements can be computed to give a quantitative analysis of the image. Morphological transformation is based on a structuring element (B) characterized by its shape, size and centre location, also called origin. Each pixel in an image is compared with B by moving B so that its center hits the pixel. Depending on the type of morphological transformation, the pixel value is set to the minimal or maximal value of the pixels in the translated structuring element.

Performing a morphological operation directly from its formal definition would imply that all the neighbors present within the structuring element should be sought for each pixel. Of course, this process becomes time consuming for large elements. Fortunately, mathematical morphology properties led to the introduction of methods which partially solve this problem.

The main goal of the decomposition algorithms is to reduce the number of basic dilations needed to generate the desired structuring element. Different ways to decompose structuring elements have been investigated exploiting two fundamental properties: Minkowski additions [Xu (1991), [START_REF] Chen | Recursive erosion, dilation, opening, and closing transforms[END_REF]] and set unions of elements [Wang and Bertrand (1988), [START_REF] Ji | Erosion and dilation of binary images by arbitrary structuring elements using interval coding[END_REF]]. More particularly, Chen has defined a decomposition method using only two non-connected pixel elements. On the other hand, Xu demonstrated how any 8-convex polygon could be constructed from a set of elements included in a 4-neighborhood.

Mathematical morphology is a tool commonly used in embedded vision systems, so a great deal of work has been done to design hardware implementation. The quality of an architecture dedicated to mathematical morphology is generally measured by the ability to describe different element shapes versus hardware efficiency e.g. memory and computation requirements, or number of image scans. Of course, decomposition element methods can help the design of efficient dedicated hardware by limiting the number of operations. [START_REF] Chien | Partial-result-reuse architecture and its design technique for morphological operations with flat structuring elements[END_REF] structuring element (see Table 1). Chien architecture relies on the principle of PRR (Partial-Result-Reuse), trying to avoid redundant computations. This concept has been used earlier by [START_REF] Pitas | Fast aglorithms for running ordering and max/min calculation[END_REF] and [START_REF] Ong | A new cost-effective morphological filter chip[END_REF], but their architectures were not fully optimized. [START_REF] Coltuc | Fast computation of a class of running filters[END_REF] proposed an optimal solution but only for rectangular elements, and Chien architecture is an extension of some other shapes. A comparison between PRR and the proposed method will be later discussed.

Diamantaras and Kung (1997) architecture provides the best flexibility, supporting both non flat and arbitrarily-shaped elements. However, it is at the expense of complexity. The architecture is built as a systolic array, and one Processor Element is defined for each pixel of the element. [START_REF] Ruetz | Architectures and design techniques for real-time image processing ics[END_REF]Brodersen (1987), Sheu et al (1992) both proposed pipelined implementations deduced from decomposition element techniques, that are also limited to flat rectangular elements. The performance and capability of their approach will be further discussed in the section on "Results".

Other works concern computational efficiency for grayscale morphological operators. Van Herk (1992) proposed a very efficient method for grayscale erosions and dilations, requiring only three comparisons for 1D structuring elements, whatever the element size. It relies on a recursive forward and backward ranking of pixel values inside temporary buffers. Extension to 2D rectangular structuring elements is achieved by consecutively applying the same process to rows and columns, for a total of 6 comparisons. 2D circular symmetric filtering can also be considered by running the process along the two diagonal directions, doubling the number of operations. Soille et al (1996) introduced an extension of this algorithm to any angle 1D structuring elements. [START_REF] Gil | Efficient dilation, erosion, opening, and closing algorithms[END_REF] further improved the initial algorithm to reach 1.5 comparisons for 1D erosions and dilations, thanks to a sliding window technique. The major drawback is that this algorithm induces non regular data flow processing, as well as the generation of ordered lists, which are not compatible with efficient implementations. The other drawback is that all Van Herk-based techniques have been designed and optimized in 1D space: 2D extension requires a second pass, and is mainly restricted to rectangular elements shapes. The technique proposed in this paper performs one pass erosion or dilation, considering more generalized 2D convex shapes.

In this paper, we address the problem of grayscale morphology with a restricted class of discrete convex polygons (2). We then introduce a fast recursive technique for grayscale dilations, relying on this particular decomposition scheme. The process regularity also enables the design of a pipeline architecture (section 3). We summarize the results in section 4.

2 Fast recursive morphological operators

Grayscale morphology

Applying morphological operations on an image involve basic templates called structuring elements, used as shape parameters for the operations. The two most fundamental morphological operations are dilation and erosion. Grayscale morphology is a natural extension of binary morphology, using the Umbra and Top operators. In most of the cases grayscale morphology is generally restricted to plane structuring elements leading to simple dilation and erosion operations based on M in and M ax functions.

A gray image A is a function whose domain is a subset of the two dimensional digitized space Z × Z. For any point p ∈ Z × Z, the translation of A by p is defined by

(A) p = {a + p | ∀a ∈ A} (1)
and the erosion and dilation by the structuring element B are respectively given by

A B = min {(A) p |p ∈ B} = min p∈B (A) p , A ⊕ B = max {(A) p |p ∈ B} = max p∈B (A) p .
(2) Some useful properties must be presented to justify the algorithm proposed later.

Translation. The dilation by the translated of a structuring element can be easier to compute than the dilation by the element itself. The following properties demonstrate the equivalence of operations (except for the translation):

A (B) p = (A) p B = (A B) p A ⊕ (B) p = (A) p ⊕ B = (A ⊕ B) p (3)
Associativity. Dilations (respectively erosions) by a series of structuring elements are equivalent to a dilation (resp. erosion) by the Minkowski addition of all the elements of the series:

(A B) C = A (B ⊕ C) (A ⊕ B) ⊕ C = A ⊕ (B ⊕ C) (4)
This property justifies a lot of algorithms since, as long as a structuring element can be decomposed by Minkowski additions of smaller elements, it is generally advantageous from a computational point of view to dilate or erode the image successively by all these elements. This property is often used conjointly with the previous one. Hence dilating by a series of translated elements is equivalent to the combination of one global translation and dilations by the elements themselves. The number of operations per pixel can then be drastically reduced, but the major drawback is that the image has to be scanned several times.

Union of structuring elements. The other way to split a structuring element is to realize a set union of smaller elements. The dilation (resp. erosion) by a set union of structuring elements is then equal to the union of the dilations (resp. intersection of the erosions) by all the structuring elements:

A (B ∪ C) = (A B) ∩ (A C) A ⊕ (B ∪ C) = (A ⊕ B) ∪ (A ⊕ C) (5)
Opening/closing operators. Erosion and dilation operations are frequently combined to provide powerful filters. An erosion (resp. dilation) by B followed by a dilation (resp. erosion) by B, the symmetrical element of B, is called an opening (resp. closing) operation:

Opening : A • B = (A B) ⊕ B Closing : A • B = (A ⊕ B) B (6) 2.2 H-convex polygons A shape in R × R is said to be H-convex if it corresponds
to the (possibly unbounded) intersection of a set of halfplanes [START_REF] Boltyanskii | Combinatorial geometry and convexity classes[END_REF]], or, equivalently, if it contains all line segments defined by a couple of its points. A discrete set of points (in [START_REF] Eckhardt | Digital lines and digital convexity[END_REF]]. As a consequence, a discrete H-convex shape is not necessarily connected in the usual sense (4-or 8-connexity).

Z × Z) is said to be H-convex if it is equal to the intersection of a H- convex shape of R × R with Z × Z [
A bounded discrete H-convex shape is by definition the convex hull of a finite set of discrete points, hence is a polygon. In [START_REF] Normand | Convex structuring element decomposition for single scan binary mathematical morphology[END_REF]], a method to decompose arbitrary discrete H-convex polygons is presented and applied to the construction of structuring elements for binary morphology.

We call 8-convex polygon, a 8-connected discrete Hconvex polygon whose edge directions are restricted to 0 • , 45 • , 90 • and 135 • like the structuring elements used in [Xu (1991)]. In the following, we show that 8-convex polygons can be decomposed with a fixed (4) number of 2PSEs whereas this amount varies for general H-convex polygons.

Xu decomposition

Q1 Q61 Q6 Q53 Q52 Q5 Q4 Q3 Q2 Q8 Q71 Q51 Q7 Fig.

Generating set of structuring elements for 8-convex polygons

Xu proved that any 8-convex polygon can be built by dilations with a restricted set of generating structuring elements [Xu (1991)]. The 13 generating elements proposed by Xu are shown in figure 1. They constitute the minimal set to generate any 8-convex polygon, i.e. any convex polygon whose border is 8-connected. Their origin is normally the center of the cross but in order to restrict later considerations to causal relationships, we replace the elements by their translated version whose origin is the last point in a raster scan order. According to (3) it introduces only a translation of the resulting image. In the following the translated version of an element Q x will be referred to Q x . Figure 2 provides an example of a 8-convex structuring element and its Xu decomposition.

Causal Recursive Decomposition of structuring elements

We have demonstrated in [START_REF] Normand | Convex structuring element decomposition for single scan binary mathematical morphology[END_REF]] that all the Xu's decomposition elements can themselves be built as 2 Example of structuring element decomposition unions and dilations of the four Two-Pixel Structuring Elements (2PSEs) depicted in figure 3.a. By extension, it implies that these 2PSEs are also sufficient to build any 8-convex element as a series of both set unions and dilations. The main advantage of using 2PSEs is that they involve a restricted 8-connected causal neighborhood of analysis defined by four elementary translations We have established in [START_REF] Normand | Convex structuring element decomposition for single scan binary mathematical morphology[END_REF]] that any convex structuring element can be iteratively constructed: for each step of the construction, already-built elements can be joined together in any combination of translations and set unions. We have also introduced a deterministic method for convex shape decomposition.

= Q' 4 Q' 71 Q' 5 B Fig.
p k , k ∈ {1 . . . 4} (see figure 3.b). p4 p1 p2 p3 a) b) c) E 4 E 3 E 1 E 2 B 4 B 3 B 1 B 2 B 0
Restricting elementary structuring elements to the set of four 2PSEs depicted in figure 3.b not only leads to consideration of only 8-convex elements but also limits the analysis space to an 8-connected causal neighborhood. In this case, each 8-convex B i can be recursively constructed according to:

B -1 = ∅; B 0 = {O}; ∀i > 1, B i = k∈{1..4} B Ik(i) ⊕E k , (7)
where I k (i) (I k (i) < i) denotes which previous element belonging to the family and dilated by E k is used to build B i . The empty set is represented by I k (i) = -1 and B 0 only contains the origin. {B i } i=0..N forms an increasing family of structuring elements as long as B i-1 ⊂ B i .

Going on with the previous example we see that the translated Xu elements (Q 4 , Q 71 , and Q 5 ) can be built by dilations and unions of 2PSEs:

Q 4 = E 2 , Q 71 = (E 3 ⊕ E 3 ) ∪ E 4 , Q 5 = E 1 ∪ E 3 (8)
In most of the cases (9/13), the Q are expressed by an unique set of unions, like Q 4 and Q 5 , and the recursive relationship from ( 7) is straightforward. To change over from the Xu decomposition scheme to the proposed one for the four left structuring elements, we introduce an intermediate step: let us assume that B i = B i-2 ⊕ Q 71 , according to ( 5) and ( 8)we get

B i = (B i-2 ⊕ E 3 ⊕ E 3 ) ∪ (B i-2 ⊕ E 4 ) = (B i-1 ⊕ E 3 ) ∪ (B i-2 ⊕ E 4 ) (9)
where

B i-1 = B i-2 ⊕ E 3 . From relation (7) it gives I 3 (i-1) = i-2, I 3 (i) = i-1 and I 4 (i) = i-2.
The corresponding growing family is given below, and the iterative construction is depicted in figure 3.c.

Dil. by Q 4 B 1 = B 0 ⊕ E 2 Dil. by Q 71 (step 1) B 2 = B 1 ⊕ E 3 Dil. by Q 71 (step 2) B 3 = (B 2 ⊕ E 3 ) ∪ (B 1 ⊕ E 4 ) Dil. by Q 5 B 4 = (B 3 ⊕ E 1 ) ∪ (B 3 ⊕ E 3 ) (10)
A growing structuring element family is then very easily described by providing only the four I k (i) indices per level. In practice most of the structuring elements used are symmetrical ones. In that particular case, all of them (except crosses) can be obtained when restricting formulae (7) to only elementary dilations. The simplified associated expression is given by:

B i = B i-1 ⊕ E i , E i ∈ {E 1 , E 2 , E 3 , E 4 } (11)
Only one index per level is necessary here for the decomposition description. The alternatively use of E 1 , E 2 , E 3 , E 4 leads to a circular structuring element, while limiting the construction to E 1 and E 3 generates rectangular ones.

Grayscale image recursive dilation and erosion

In [START_REF] Normand | Convex structuring element decomposition for single scan binary mathematical morphology[END_REF]], we took advantage of the previous structuring element decomposition method by introducing a fast binary erosion operator. The binary case, which presents a fixed number of operations whatever the structuring element, cannot be used to expand the grayscale morphology. Even if comparable performances cannot be achieved, we propose here a fast algorithm for grayscale erosion and dilation operators. The principle will be detailed only for dilation. First, consider that as the four 2PSEs contain the origin, an elementary dilation by E k is equivalent to the union of the element and the element translated by p k :

B ⊕ E k = B ∪ (B) p k (12)
Let {B i } i=0..N be a family of growing structuring elements constructed recursively by union sets and elementary dilations according to (7). In the following D Bi states the dilation of A by B i . Based on the 2PSE decomposition a recursive computation of D Bi is possible:

D Bi = A ⊕ B i = A ⊕ k∈{1..4} B Ik(i) ⊕E k = A ⊕ k∈{1..4} B I k (i) ∪ B I k (i) p k = M ax (A) p |p ∈ k∈{1..4} B I k (i) ∪ B I k (i) p k = M ax k∈{1..4} M ax (A) p |p ∈ B I k (i) ∪ B I k (i) p k = M ax k∈{1..4} M ax (A) p |p ∈ B I k (i) , M ax (A) p |p ∈ B I k (i) p k = M ax k∈{1..4} D B I k (i) , D B I k (i) p k (13)
These results show that the dilated values when considering successively all the elements in the family, are recursively obtained by performing a maximum extraction between the values of the dilation by the elements involved to build the current one, at the current position and at positions p k in the 8-connected causal neighborhood. The process is fully regular because it uses a raster scan of the image.

In this scheme, resources requirements are linear. have to be recorded, so the number of stored elements is given by the number of decomposition levels. When only p 1 is involved for one level, the intermediate value corresponds to the previous computed one, and an unique variable is necessary to store the result.

In other cases, a line-memory per level is necessary to store intermediate results from the previous line.

Figure 4 shows the progressive steps to get dilated values up to the structuring element B 3 from figure 3. D B3 computation requires five comparisons, and D B4 eight.

When B is a structuring element composed of only dilations (eq. ( 11) for symmetrical elements), the operations at each stage are reduced to an unique comparison between the dilated values by the previous element at the current position and at position p i , p i ∈ {p 1 , p 2 , p 3 , p 4 }, so that In this particular case, the number of operations is also given by the number of decomposition levels.

B i = B i-1 ∪ (B i-1 ) pi ⇒ D Bi = M ax D Bi-1 , D Bi-1 pi (14)
Image borders. To handle the problem of image borders, one has generally to expand the image or to introduce conditional processing. For a dilation operation, a suitable way consists of considering values as 0 outside borders. This can be done here without preliminary image expansion as explained below.

-If L is the image width, then the line-memories have to be of size (L+2 ). -Memory-lines have to be initialized to 0 once at the beginning of the process, variables to store results in p 1 position have to be initialized at the beginning of each line process. -The position values for pixels in the image are considered in the range [1..L]. Thus, for pixel position 1 (resp. L) and for p 2 (resp. p 4 ) neighborhood, linememory is accessed at address 0 (resp. L + 1) and provides 0 as a dilated value.

Considering 0 values outside the borders has of course an impact on results near these borders. Assuming that that D Bi = 0 if B i is totally outside the image, it modifies equation ( 13):

D Bi = M ax k∈{1..4}    D B I k (i) , D B I k (i) p k if B I k (i) ⊂ A D B I k (i) otherwise.    (15)
From the previous formulae, one can see that the process automatically adapts the shape of the structuring element near borders and that the actual structuring element expression is:

B i = k∈{1..4} B I k (i) ∪ B I k (i) p k , with I k (i) = argmax (j ∈ {0 . . . I k (i)} |B j ⊂ A) I k (i) = argmax j ∈ {0 . . . I k (i)} | (B j ) p k ⊂ A (16)
To sum up for this part, the proposed method enables an efficient software implementation of dilation and erosion operators with the following main features:

an unique scan of the image whatever the shape and size of the structuring element, an easy and systematic way to describe any 8-convex structuring element, limited resources requirements in terms of both memory and computation, automatic adaptation to borders without image expansion nor conditional processing.

Further improvements could be made, in particular concerning the number of comparisons. In [START_REF] Chien | Partial-result-reuse architecture and its design technique for morphological operations with flat structuring elements[END_REF], Chien introduced the concept of Partial-Result-Reuse (PRR). The idea is to limit the number of operations by avoiding redundant comparisons as much as possible and by exploiting already computed data. The example of a 1D structuring element of 8 pixels (called B 4 ) given by Chien is illustrated figure 5. B 4 is recursively constructed by gathering non-overlapped structuring elements. Intermediate results have to be stored at different distances from the current position : 1 for B 0 , 2 for B 1 and 4 for B 3 , giving a total of 7 recorded values.

B 0 B 1 B 2
Step 1

Step 2

Step 3

B 3 B 0 B 1 B 2
Step 1

Step 2

Step 3 B 3 Fig. 5 PRR principle for a 1D structuring element of 8 pixels If we compare this with our approach, the same strucelement would be built according to :

B 7 = B 6 ∪ (B 6 ) p1 . . . B 1 = B 0 ∪ (B 0 ) p1 (17)
The dilation is performed here after 7 comparisons. Actually, 1D dilations are the worst case for our method in terms of size of the element versus number of comparisons. Concerning memory, both methods require the storage of 7 values but our method involves an unique distance (equal to 1) whereas the depth for PRR is dependant on the element decomposition.

The PRR concept has been extended to square and circular elements. Square representation is the simplest shape leading to the greatest computational reduction. For circular elements (13 pixel elements), Chien's heuristic method requires 6 computation steps, whereas the proposed method requires 8. However, the authors do not propose a systematic method of decomposition. For a dichotomic approach, the problem can be reformulated by trying to find the minimum element B j of the family (where j ∈ [0 . . . i -1]) such that B i can be expressed as

B i = B j ∪ (B j ) T ×pi T ∈ {1 . . . i 2 }, p i ∈ {p 1 , p 2 , p 3 , p 4 } (18)
The 1D case is simple. Formula (17) can first be expanded, and, by factoring process, it is easy to find that there are four possible solutions meeting the condition set out in (17). Among them, B 3 is the minimum element.

B 7 = B 6 ∪ (B 6 ) p1 B 7 = B 5 ∪ (B 5 ) 2p1 B 7 = B 4 ∪ (B 4 ) 3p1 B 7 = B 3 ∪ (B 3 ) 4p1 (19)
The approach can be extended to 2D space but only for simple element shapes. For more complex elements, PRR is more difficult to determine, and as mentioned before, the benefit of PRR decreases when the complexity of the pattern increases.

Opening and closing Opening and closing operations imply the need to consider symmetrical elements. If B is causal, B is necessarily anti-causal, which would lead to an inverse scan of the image in order to preserve the sequential feature of the process. A better solution takes advantage of the property (3) by defining B T = ( B) p B , with p B a fixed vector such that B T is causal. Then the use of B T instead of B simply requires a global image translation of p B once processing has been completed. For a regular data flow it consists merely of introducing a constant delay in the output flow.

When B is symmetrical, deduced from (11), then B and B have the same shape, and the same structuring element can be used.

Specific pipeline architecture

The regularity of the algorithm and the restriction of the relationship area to the 8-connected causal neighborhood enable the design of a fast generic and regular specific architecture, requiring few resources in terms of memory and logic. A dilation implementation suited for an FPGA target is presented in the following paragraphs.

The whole architecture is synchronized by a global clock with a data flow rate of one input and output pixel data for each clock pulse. Each elementary stage computing D Bi from previous values, requires two types of components: a memory module to store and to provide previous results and a combinatory operative part to realize the maximum extraction.

For the following, let T be the clock period, D Bi (n) be the dilation value by the structuring element B i at time n × T (n th pixel), L be the size of an image row and M be the number of bits to encode a value.

Memory module

For easier systematic design of the architecture, a generic memory module has been designed. The knowledge of a previous result in the {p 2 , p 3 , p 4 } neighborhood implies the storage of one row at each step of the decomposition. We have chosen a "dual port RAM" memory solution allowing both a reading and a writing access during one clock cycle. The following temporal relationships avoid the need for multiple address management process in each of the various stages:

(D Bi (n))) p1 = D Bi (n -1) (D Bi (n)) p4 = D Bi (n -(L -1)) (D Bi (n)) p3 = D Bi (n -L) = (D Bi (n -1)) p4 (D Bi (n)) p2 = D Bi (n -(L + 1)) = (D Bi (n -1)) p3 (20)
Figure 6 presents the general solution for the memory module. For this example we have considered L = 7 (3 bits to encode the address) and M =4. The storage D Bi (n-1) requires only a simple register. The memory is addressed at a fixed relative position ((n + 2) M odulo L) to provide (D Bi ) p4 at the next clock pulse. Then the dilated values at the two other positions are available by a temporal pipeline through two registers. The reading and writing addresses have a constant difference (equal to 3) which can be also seen as a temporal delay in terms of clock cycle. Therefore the simplest architectural solution consists of the implementation of three registers. Of course all the resources displayed in the memory module are not always necessary. For instance, when only p 1 among the neighborhood is involved it requires only one register. However, these kinds of optimizations are generally automatically performed by synthesis tools removing all unused resources.

Image borders

As mentioned before, image borders induce specific processing which can break the data flow regularity of the architecture. To avoid such problem, a solution has been designed. Only one extra storage element is required for the memory-lines, and as borders for both the p 2 and p 4 neighborhoods. Figure 7 illustrates the behavior of the output memory module on right and left borders. To flush the pipeline at the end of each row, and to load values at the beginning, an additional clock pulse is performed. Then a control signal (wren) is generated to avoid to overwrite stored value (0) at address L, and to reset value in p 1 . The global architecture (see fig. 8) is composed of a simple global control (counter) providing the address and the set of pipelined elementary stage of dilation. The configuration of each stage consists of determining the number of elementary dilations and connecting the output of each memory module according to the B i definition. The "Max extraction module" designates an elementary subtractor whose carry-out output drives the selection input of a multiplexer. The temporal performances of the sys-Dilation by B 2 Fig. 9 Expected values for a dilation by B2 tem are determined by the maximal data path time fixing the minimal period. Thus the maximal frequency is dependant on (N D + N U ). A conventional solution relies on the introduction of additional registers between stages to reduce the critical data path. This method produces a constant latency in terms of clock pulses between the inputs and outputs but leads to a faster clock frequency, independently of the number of stages. These registers are represented by gray boxes in the global architecture diagram. In practice, registers are available in FPGA at the output of all logic modules. So additional registers for a pipeline implementation are only required for data paths with no operation between stages such as the address of the counter or the propagation of B 2 towards B 4 .

Global architecture

Figure 9 shows expected results for a dilation of an input image by B 2 , and figure 10 gives the results of the simulation.

Implementation

Comparative implementation efficiency with existing architectures is summarized in Table 1. Except for the Diamantaras solution, all the other techniques have been optimized for only rectangular elements. For memory and cycle requirements, the proposed method achieves the same performances as the best solutions. Only Chien architecture provides a better solution in terms of comparators, but as mentioned before, the PRR concept is essentially efficient for 1D and rectangular elements. Moreover, PRR and the are not incompatible : additional improvements in order to reduce the number of operations, for simple shapes, can be a posteriori realized after the decomposition definition.

The proposed architecture has been synthesized into FPGAs. As the architecture requires few resources, a medium FPGA such as an APEX 20KC (Altera) used in these experiments is sufficient to support dilations by large structuring elements. Maximum clock rate is dependant only on one combinatorial stage and thus dependant only on M . For M = 8 the maximum clock frequency is 50 MHz. For instance, a 512 × 512 grayscale image is computed during one pass in 5 ms whatever the structuring element shape and size.

Conclusion

In this paper, we have presented an algorithm enabling the design of fast mathematical grayscale morphology operations. We have proposed a systematic way to describe any 8-convex structuring elements through a generic decomposition process using 2-pixel structuring elements. Then these 8-convex structuring elements are recursively constructed by elementary dilations and union sets, based on a fixed and close neighborhood. This decomposition system has been used to define the efficient implementation of elementary morphological operators in both software and hardware. Computation complexity is directly correlated to the number of elementary dilations involved in the decomposition.

Fig. 3

 3 Fig. 3 2PSE decomposition (a) Elementary elements, (b) corresponding translations and (c) example of growing structuring family

  -Complexity : Processing requirements concern Max operators. If N D and N U are respectively the total number of elementary dilations and unions used to build B i , then the algorithm requires only N D + N U comparisons to perform the dilation of an image by B i whatever the shape of the structuring element. -Memory : Only intermediate results {D Bi } i=0..N

Fig. 4

 4 Fig. 4 Recursive dilation process

Fig. 6

 6 Fig. 6 Generic elementary memory module

  Fig. 7 Border processing

Fig. 8

 8 Fig. 8 Global architecture

Fig. 10

 10 Fig. 10 Simulation results for a dilation by B2

  proposed a comparative table among existing architectures for a dilation by a 7 × 7

	Architecture	Support non	Support non	Comparator	Number of	Cycles required
		flat	rectangular		delay
		structuring	structuring	counts	elements	per frames
		element	element
	Pitas	No	No	8	7L + 7	L(H+7)+6
	Coltuc	No	No	6	6L +6	L(H+6)+5
	Ong	Yes	No	7	[6L +6]+1	7LH
	Diamantaras	Yes	Yes	48	[6L +42]	L(H+6)
	Ruetz	No	No	12	6L +6	L(H+6)+5
	Sheu	Yes	No	13	6L + 20	7LH + 8
	Chien	No	Partially	6	6L+6	L(H+6)+5
	This work	No	Yes	12	6L+6	L(H+6)+5

Table 1

 1 Flexibility and implementation efficiency comparison between state-of-the-art architectures for a dilation by a 7 × 7 square structuring element


Nada Matta
email: nada.matta@utt.fr

Oswaldo Castillo

Navetty Charles Delaunay
email: navetty@hotmail.com

Learning from Profession Knowledge: Application on Knitting

Keywords: Knowledge Management, profession memory, corporate memory, learning, Knitting

Knowledge Management is a global process in companies. It includes all the processes that allow capitalization, sharing and evolution of the Knowledge Capital of the firm, generally recognized as a critical resource of the organization. Several approaches have been defined to capitalize knowledge but few of them study how to learn from this knowledge. We present in this paper an approach that helps to enhance learning from profession knowledge in an organisation. We apply our approach on knitting industry.

Introduction

Knowledge management is currently defined as a process of identification, formalization, disseminating and use of knowledge in order to promote creativity and innovation in companies. This process (Figure 1. ) takes into account the transformation and the evolution of tacit to explicit knowledge [START_REF] Nonaka | The knowledge-Creating Company: How Japanese Companies Create the Dynamics of Innovation[END_REF] and of individual to collective knowledge [START_REF] Castillo | CSAO: a guide to construction of practical learning system[END_REF]. Knowledge Management [START_REF] Nonaka | The knowledge-Creating Company: How Japanese Companies Create the Dynamics of Innovation[END_REF].

Methods stemming from the knowledge engineering (such as MASK, REX, KOD, etc.) and the computer supported cooperative work (such as QOC, DIPA, etc.) [START_REF] Dieng-Kuntz | Méthodes et outils pour la gestion des connaissances. 2 eme[END_REF] were used in order to capitalize and make explicit knowledge in an organization. These methods allow defining corporate memories. A corporate memory is defined as the « explicit and persistent representation of the knowledge and the information in an organization » [START_REF] Dieng-Kuntz | Méthodes et outils pour la gestion des connaissances. 2 eme[END_REF], [START_REF] Van Heijst | Using Explicit Ontologies in KBS Development[END_REF]. We can distinguish several types of memories: profession memory, project memory and organization memory.

The sharing and the appropriation of the corporate memories are still real blocking points within organizations. The methods of knowledge management are not sufficient to allow an effective appropriation of the knowledge by the actors of the company. However, the objective of a capitalization of the knowledge is indeed the sharing and the re-use of an experience with the aim of optimizing the process of organizational learning.

The technologies of information and the communication (i.e. semantic web) can constitute a support for sharing. However, Knowledge sharing must be guided if we want to supply the good information at the right moment.

Furthermore, the distribution of information is not enough to guarantee the re-use of the knowledge. One condition to reuse knowledge is to be assimilated by the actor that is integrated into his experiences and generate appropriate knowledge at any time in the action [START_REF] Tounkara | L'appropriation des connaissances avec MASK[END_REF], [START_REF] Castillo | CSAO: a guide to construction of practical learning system[END_REF]. IEEE proceedings of the 5 th International Conference on Signal-Image Technology and Internet based Systems, Marakesh, November, 2009

We define an approach based on knowledge management and educational engineering in order to enhance profession knowledge learning in a company. This approach has been applied in textile factory in order to enhance knitting learning. We present in this paper the memory we defined (profession memory in knitting) and the approach we followed to enhance learning in Textile Company.

Profession memory

We define a profession memory as the externalisation of the knowledge produced in and for a given domain [START_REF] Castillo | CSAO: a guide to construction of practical learning system[END_REF]. It represents problem solving. The techniques of knowledge engineering allow the formalization of this type of memory. Several approaches, such as MASK, REX, CommonKADS, or KOD [START_REF] Dieng-Kuntz | Méthodes et outils pour la gestion des connaissances. 2 eme[END_REF], use techniques of knowledge engineering to extract knowledge, to formalize it in conceptual models, where the knowledge that guides problem solving is made explicit. The structure of this type of memory describes generally: the definition of the problem and the process, the problem solving methods, as well as a description of the concepts manipulated in problem solving. This representation of knowledge can lean on graphic completed presentations (for example, the models of problem solving methods of CommonKADS [START_REF] Breuker | Common-KADS Library for expertise modelling Reusable problem solving components[END_REF] and MASK [START_REF] Ermine | La gestion de connaissances[END_REF]) and with textual explanations as it can be as problem experience forms indexed with trees of concepts as it is the case of the elements of experience of REX [START_REF] Malvache | Mastering Corporate Experience with the REX Method[END_REF].

Using profession memory for learning

According to Gurteen, "sharing is not just about giving. It is about: soliciting feedback, asking questions, telling people what you plan to do before doing it, asking other people for help, asking someone to work with you in some way -however small, telling people what you are doing and more importantly why you are doing it, asking people what they think -asking them for advise, asking people what would they do differently, not just sharing information but know-how and knowwhy" [START_REF] Gurteen | Creating a knowledge sharing culture[END_REF].

Generally, a profession memory is organized to represent know-how in a given domain. It is rather about a practical knowledge acquired from the experience. The descriptions of the context, the problem solving methods as well as the evolution of the activity are not sufficient to establish a complete training on a profession. The learning techniques which we intend to define address essentially actors in a domain, aiming to learn some experience of an expert in this domain.

Let us note that an expertise is represented as conceptual models with models (model of task, activity, domain, etc.). These very useful models for the knowledge extraction are not easily accessible for knowledge appropriation. It is then important to reorganize theses models to answer the various needs of the organization actors. The CSAO approach we defined propose to reorganise models defined in a profession memory in order to facilitate their appropriation by actors.

CSAO: Profession learning approach

Based on corporate memory life cycle proposed in [START_REF] Dieng-Kuntz | Méthodes et outils pour la gestion des connaissances. 2 eme[END_REF], we propose a profession memory life cycle (Figure 2. ) that considers needs identification, knowledge formalization and dissemination, profession memory use and updating.

Identify needs

We suggest localizing knowledge to be formalized, sources and carriers of this knowledge. It is also necessary to find the future users of knowledge and the various types of users if there is. The main activity in this phase is the definition of a knowledge map. According to Aubertin [START_REF] Aubertin | Cartographie des connaissances critiques[END_REF] knowledge map allows supplying a structure of cognitive resources of the organization. Three approaches can be used to organize these resources: procedural classification (according to company processes), functional classification (pressed usually on an organization chart), and abstract classification or by domains (information organized about matters, objects and purposes). Knowledge map design forces the experts involved to sharing the knowledge that they possess, and dissemination of this map of knowledge allows to know the type of knowledge produced in companies and knowledge workers.

Formalize

To formalize a profession memory, we propose a life-cycle composed by [START_REF] Castillo | CSAO: a guide to construction of practical learning system[END_REF]:

-Knowledge elicitation: knowledge management responsible team must capture the knowledge, directly by interviewing experts and from organization documents, using knowledge engineering techniques.

-Conceptual model building: according to American Institutes for Research [START_REF]Conceptual Model[END_REF], "an application's conceptual model is the "mental map" that the designers build into it in order to present information logically". In our case expertise conceptual model is represented as: activity process, interactions in every stage of the activity process, and problems solving strategy (Figure 3. ). Concepts and constraints models are also defined. They show links between problem solving and domain knowledge. Knowledge management responsible team has to assign to every stage of the process, the group of the competences necessary.

Every competence must have assigned a value, which will be compared with the values of the competences of an employee. The profile of an employee and the educational activities had also to be defined. IEEE proceedings of the 5 th International Conference on Signal-Image Technology and Internet based Systems, Marakesh, November, 2009 Accomplishment of each one of these stages, asks strong interactions between experts and organization knowledge responsible team, and involves a big work of knowledge sharing. We propose the creation of a community of practice of experts in the organization. "A community of practice exists because it produces a shared practice as members engage in a collective process of learning" [START_REF] Wenger | Communities of Practice -Learning as a Social System[END_REF].

Update

Communities of practice, will serve to support alive the profession memory and are going to allow updating it. According to Wenger, "communities of practice are important to the functioning of any organization, but they become crucial to those that recognize knowledge as a key asset. From this perspective, an effective organization contains a constellation of interconnected communities of practice, each of them dealing with specific aspects of the company's competency-from the peculiarities of a long-standing client, to manufacturing safety, to esoteric technical inventions. Knowledge is created, shared, organized, revised, and passed on within and among these communities. In a deep sense, it is by these communities that knowledge is "owned" in practice" [START_REF] Wenger | Communities of Practice -Learning as a Social System[END_REF].

Disseminate

Learning system will be composed by an expert module, a student module, a pedagogical model and an interface module. The profession memory will be the expert module and an individual memory, representing the employee competences, will be the student module. The profession memory (expert module) have to be integrated on the employee workspace. Index and learning guides give access to knowledge at any time.

Use

One of our postulates is that the employee has to use alone the learning system. The employee will have no tutor or councilors to guide him during this learning. Our learning system, auto-training, has to have the necessary and sufficient information to allow the employee to realize the training without to consult an expert [START_REF] Castillo | CSAO: a guide to construction of practical learning system[END_REF]. The same employee will be who decide if he has already realized successfully the educational activities and will realize an auto-evaluation of the obtained answer (Figure 4. ). Operational learning system, based on employee competences, will decide if he can pass to other stage of activity process. Progression process illustration of knowledge learning [START_REF] Castillo | CSAO: a guide to construction of practical learning system[END_REF].

Applications

Textile companies have to deal with a large competition in Textile confection. They have to develop specific expertise. The French Textile Institute (IFTH) helps Textile industry to develop these expertises. It studies organizational learning techniques to help companies. One of their studies is to enhance learning from profession memory. We follow the profession memory life-cycle defined in CSAO for this aim:

Feedback

A number of points can be noted from the analysis of the applications of the CSAO approach in knitting:

 The externalization of knowledge as profession memory helped experts to formalize their knowhow and to structure it in order to emphasize the main point to To summarize our analysis, we can note that training using profession memory promote practical learning better than traditional training that takes more time to enable learner to build optimal strategy to deal with problems of his domain. The CSAO approach we use has been accepted and appreciated by even experts and IEEE proceedings of the 5 th International Conference on Signal-Image Technology and Internet based Systems, Marakesh, November, 2009 learners because that guides know-how formalization and transmission. CSAO approach must be developed in order to enhance self-learning for instance: as base of activity simulations, or to enhance organizational learning (activity communication and practice sharing).

CONCLUSION

According to Gurteen, "the most effective way to create a knowledge sharing culture -is first to start to practice. Firstly, change the culture. Secondly, put in place the knowledge sharing technology and train and educate people in its effective use. People with the appropriate knowledge sharing mindset and the appropriate knowledge sharing technology to support them will rapidly bring about a knowledge sharing culture that helps organization to better meet its business objectives" [START_REF] Gurteen | Creating a knowledge sharing culture[END_REF]. Our work proposes the accomplishment of diverse activities, which must be an example in order that most of organization's members learns to share knowledge and to use the knowledge that the experts want to share.

  Figure 1.Knowledge Management[START_REF] Nonaka | The knowledge-Creating Company: How Japanese Companies Create the Dynamics of Innovation[END_REF].

Figure 2 .

 2 Figure 2.Profession memory lifecycle[START_REF] Castillo-Navetty | A practical learning system for the French Textile and Apparel Institute[END_REF].

  .

Figure 3 .

 3 Figure 3.Conceptual model.

Figure 4 .

 4 Figure 4.Progression process illustration of knowledge learning[START_REF] Castillo | CSAO: a guide to construction of practical learning system[END_REF].

3 .

 3 Knowledge so represented is then transformed in training schemas: Learners are conducted to follow a course based on the experts' process of knitting. For each step: they have to solve a problem and follow the main steps of experts' strategy (Figure5. ).

Figure 5 .Figure 6 . 4 .

 564 Figure 5.Training schema for Integral knitting learning.

  consider in an activity: difficulties, elements to avoid, advices, etc.  Experts are very happy to participate in the reorganization of their knowledge in training structure. They need to know the concrete finality of the use of the corporate memory. They are also the only that know how to reorganize the knowledge because they know deeply their domain and their activity.  Training sessions have been organized by the IFTH in Textile companies. IFTH Teachers are very happy to have a reference documents that show the main points and the objectives of success knitting. Learner progress easily knowing the goals of steps in knitting and difficulties as expressed in the profession memory.


Jean-François Ponge
email: jean-francois.ponge@wanadoo.fr

Horizons and humus forms in beech forests of the Belgian Ardennes

Forest soil organic horizons are named on the basis of visual observations made directly in the field, thus this is often subjective. Humus form horizons were compared in thirteen beech stands (Fagus sylvatica Ehrh.) of the Belgian Ardennes (western Europe) embracing a wide range of ecological conditions, in order to find more objective bases for their classification. We used a semi-quantitative micromorphological method for the description of horizons, and a multivariate method for data analysis. These methods helped to discern objective discontinuities between Oi, Oe and Oa horizons, adding new criteria for their characterization, such as the root system of trees. Within these horizons , transitions between subhorizons are gradual, thus do not lie on clear-cut criteria. The transition between Oa and A horizons was also gradual. The composition of Oa and A horizons varies according to humus form. The vertical distribution of soil organisms and their vertical movements were considered the origin of discontinuous and continuous processes taking part in the transition from one horizon to another. The observation of horizons under a dissecting microscope may help to find more reliable bases for their nomenclature, even without the use of costly soil sections.

INTRODUCTION

Since the pioneering work of Müller (1889), who described two basic humus forms, mull and torf, beneath Danish beech forests, on the basis of microscopic observations, there have been many attempts to classify humus profiles in forest soils. Different horizons or sub-horizons are generally recognized within the O Horizon [START_REF] Brady | The nature and properties of soils[END_REF] or Ao [START_REF] Duchaufour | Pédologie. Sol, végétation, environnement[END_REF] horizon. Although discrepancies concerning the terminology of these strata exist, three main strata are recognized, Oi (entire leaves), Oe (fragmented leaves), and Oa (holorganic faecal pellets). Differences in the development of these horizons, together with structure and chemical properties of the underlying A horizon, allow recognition of three main humus forms, now called mull, moder, and mor [START_REF] Klinka | Taxonomic classification of humus forms in ecosystems of British Columbia. First approximation[END_REF][START_REF] Delecour | Les formes d'humus: identification et description[END_REF][START_REF] Green | Towards a taxonomic classification of humus forms[END_REF][START_REF] Brêthes | Classification of forest humus forms: a French proposal[END_REF][START_REF] Jabiol | Une classification morphologique et fonctionnelle des formes d'humus. Propositions du référentiel pédologique[END_REF][START_REF] Jabiol | L'humus sous toutes ses formes[END_REF]. The mor humus form, together with dysmoder (moder with a thick H stratum), has been also called raw humus [START_REF] Kubiëna | The soils of Europe. Illustrated diagnosis and systematics[END_REF][START_REF] Delecour | Essai de classification pratique des humus[END_REF]. The recognition of biological processes taking place in the development of humus profiles [START_REF] Hartmann | Waldhumusdiagnose auf biomorphologischer Grundlage[END_REF][START_REF] Zachariae | Spuren tierischer Tätigkeit im Boden des Buchenwaldes[END_REF][START_REF] Bal | Zoological ripening of soils[END_REF] suggested that some features, such as compaction of the soil matrix, deposition of faecal pellets, skeletonization of leaves or tunnelling of needles, are the direct result of soil faunal activity.

Thus, as for most biological processes, the transition from one horizon to another should be considered discontinuous, due to tolerance limits and food and habitat preferences of soil organisms.

Characterization of organic horizons and humus forms on the basis of morphological features is common; however, the existence of clear-cut changes between one horizon and another has been poorly demonstrated through quantitative or semi-quantitative morphological data. [START_REF] Federer | Subjectivity in the separation of organic horizons of the forest floor[END_REF] pointed out that simple remeasurements of thickness of forest soil horizons may lead to false conclusions if done by two individuals. Discrepancies between field and laboratory observable features of horizons have been reported [START_REF] Bernier | Comparative study of soil organic layers in two bilberryspruce forest stands (Vaccinio-Piceetea). Relation to forest dynamics[END_REF]. This may indicate that some morphological criteria used for the definition of organic horizons are inadequate or difficult to employ with accuracy in the field. This is particularly true for the transition from Oa to A horizons and for subdivisions which have been recognized within Oi and Oe horizons [START_REF] Babel | Gliederung und Beschreibung des Humusprofils in mitteleuropäischen Wäldern[END_REF].

In the present study, our purpose was to determine whether the transition from one organic horizon to another is a continuous or, rather, a step-by-step process, with sharp delineations in horizon properties. In doing so, we addressed the question as to whether horizons exhibit true emergent properties and have not been created by soil scientists for only classification purposes.

MATERIAL AND METHODS

Nomenclature

Comparable organic horizons have several different names depending on the taxonomic system in use. The commonly used nomenclature of [START_REF] Brêthes | Classification of forest humus forms: a French proposal[END_REF] corresponds to USDA nomenclature as follows: OL equals Oi, OF equals Oe, and OH equals Oa. These horizons correspond to L, F and H horizons, respectively, recognized by [START_REF] Hesselmann | Studier över barrskogens humustäcke[END_REF] and later refined by [START_REF] Babel | Gliederung und Beschreibung des Humusprofils in mitteleuropäischen Wäldern[END_REF].

The classification of humus forms by [START_REF] Brêthes | Classification of forest humus forms: a French proposal[END_REF] was used in this study. It is not based on a strong relationship between the features O and A horizon, as this is the case in other classifications.

For instance, a crumby A horizon (typical of mull) may coexist with a thick O horizon (typical of moder). Some humus forms, such as hemimoder and amphimull, were described by other authors as mull-moder [START_REF] Duchaufour | Pédologie. Sol, végétation, environnement[END_REF] or mull-like moder [START_REF] Kubiëna | The soils of Europe. Illustrated diagnosis and systematics[END_REF][START_REF] Delecour | Essai de classification pratique des humus[END_REF][START_REF] Delecour | Les formes d'humus: identification et description[END_REF]. A hemimoder means a moder humus form (A horizon with organic matter juxtaposed to mineral matter) without any Oa horizon.

An amphimull means a mull humus form (A horizon with organo-mineral assemblages) with a distinct Oa horizon.

The nomenclature of soil types used in this study followed the FAO-UNESCO classification [START_REF] Driessen | The major soils of the world[END_REF]. Cambisols and podzols correspond to Inceptisols and Spodosols, respectively, in USDA Soil Taxonomy [START_REF] Brady | The nature and properties of soils[END_REF]. Nomenclature of plant species followed [START_REF] Rameau | Flore forestière française. Guide écologique illustré. I. Plaines et collines[END_REF].

Study sites

Thirteen mature beech stands were selected in the Belgian Ardennes, covering a wide range of humus forms (Table 1). Beech (Fagus sylvatica) forests were chosen because of their wide distribution and the need for a wide range of environmental conditions, in particular soil and climate, without strong variation in the composition of litter. All studied sites were located on nutrient-poor geological substrates (schists, graywackes, quartzites) ranging from Cambrian to Devonian age. Altitude and related regional factors (climate, mineral richness of parent rock) were found to be the main source of variation of soil animal communities, humus forms and site quality over the studied range [START_REF] Ponge | Soil fauna and site assessment in beech stands of the Belgian Ardennes[END_REF]. Results of litter and soil chemical analyses were also reported in the aforementioned paper.

Categories of the soil matrix in humus horizons

At each site, two humus profiles were sampled for micromorphological description of horizons. These profiles were chosen to represent the range of observed within-site variation of humus forms.

Sampling was completed in June 1989. Preparation of the samples (5 x 5 cm section monoliths) was carried out according to the method of [START_REF] Bernier | Humus form dynamics during the sylvogenetic cycle in a mountain spruce forest[END_REF], except that only the top 1 cm of the A horizon was sampled. Horizons were separated in the field on the basis of variation visible to the naked eye, without reference to any a priori classification. Afterwards, they were classified into Oi, Oe, and Oa horizons according to the abovementioned field criteria, and numbered according to their order from the top to the bottom of a given horizon, i.e. Oi1, Oi2, Oi3, Oe1, Oe2, etc... All 172 horizons were immediately immersed in ethyl alcohol then transported to the laboratory. The composition of each horizon was analyzed by observing the soil matrix in alcohol under a dissecting microscope. No attempt was made to quantify the volume or mass of each category. Rather, we used the following visual semi-quantitative coding of the abundance of a given category in a given horizon: absent 0 present but scarce 1 present and common 2 present and dominant 3

A total of 185 categories were recognized. Most of them were plant litter, in varying degrees of decomposition or comminution by fauna. Animal faeces were classified according to the corresponding animal group, their degree of comminution by other animals, and their degree of connection with uneaten plant categories. Animals were counted and classified into broad groups.

Multivariate analysis

Correspondence analysis [START_REF] Greenacre | Theory and applications of correspondence analysis[END_REF] was used to give an overall picture of affinities and differences between horizons and/or categories. This method, using the chi-square distance, allows horizons and categories to be simultaneously projected on factorial axes, thus groups of samples could be directly associated with groups of categories. As was theoretically developed by [START_REF] Benzécri | L'analyse des données. II. L'analyse des correspondances[END_REF], this method has been refined for particular purposes such as community gradient analysis [detrended correspondence analysis [START_REF] Hill | Detrended correspondence analysis: an improved ordination technique[END_REF]] or the study of community-environment relationships

[canonical correspondence analysis [START_REF] Ter Braak | The analysis of vegetation-environment relationships by canonical correspondence analysis[END_REF]]. Here, our purpose was to analyze the structure of a data matrix (the composition of horizons) without any a priori hypotheses concerning their relationships with environmental factors and with more than one single factor suspected to result from the analysis.

Additional (passive) variables were used to help interpretation of the factorial axes and not to quantify causal relationships. Thus, we used the approach of original correspondence analysis, that was devised to evaluate global patterns underlying complex data matrices, and not that of the aforementioned derived methods.

Matrices were analyzed so that horizons were observations (rows) and categories were active variables (columns). Animal groups, horizon names, humus form names, phytosociological types, and altitude were added as passive variables. They had no influence on the results, but they were projected on factorial axes as the active variables.

Categories were coded and counted as indicated above. Animal groups were recorded as total counts. Horizons, humus forms and phytosociological types were coded as 1 when a given horizon (row) belonged to a given category (column), 0 when not. Altitude was recorded in meters. All variables (active and passive) were standardized with unit variance and a mean of 20. This allowed contributions to factorial axes to be proportional to factorial coordinates and data of a different nature, such as counts, semi-quantitative coding and measurements, to be included in the same analysis [START_REF] Ponge | The hetrogeneity of humus profiles and earthworm communities in a virgin beech forest[END_REF]. In the case of altitude, high elevation and low elevation were distinguished. Standardized altitude values (high elevation) were complemented to 40 in order to create a new variable, varying in an opposite sense (low elevation). The new variable had similar mean (20) and standard deviation (1). Following this procedure, already used by [START_REF] Ponge | The hetrogeneity of humus profiles and earthworm communities in a virgin beech forest[END_REF], the range of elevation values was described by two symmetrical points.

RESULTS

Composition of the humus profile

We analyzed first the total 172 horizons and 185 categories. Only axes 1 and 3 were considered for the global description of humus horizons. Axis 2 was neglected, because it isolated only one horizon (Oe1 in the second profile studied in site 16), which was characterized by the development of the root system of hair-grass (Deschampsia flexuosa) at the inside of decaying plant fragments. This development was not apparent in other humus profiles, due to scarcity of ground vegetation, and we preferred to discard the corresponding axis rather than to discard this horizon from the analysis.

The projection of horizons in the plane of axes 1 and 3 (Fig. 1) indicated that they could be classified into three groups, corresponding to Oi, Oe, and Oa+A horizons, respectively. All horizons sampled did not fall exactly in each of the three branches depicted by the plane of axes 1 and 3. This

does not indicate that Oi, Oe, and Oa+A horizons were poorly differentiated on the basis of their composition. Rather, this phenomenon indicates that a few horizons had been badly classified in the field; correspondence analysis revealed that they had more points in common with another group than with their own group. Differences between Oa and A horizons were negligible when compared with differences between Oi and Oe horizons. Oa and A horizons did not exhibit distinct branches. In addition, differences within each of the Oi, Oe or Oa horizons were negligible compared to differences between these horizons.

The projection of categories in the plane of axes 1 and 3 (Appendix 1) allowed us to characterize each horizon or group of horizons by its composition (Table 2). The Oi horizon was characterized by entire leaves of beech at varying stages of fungal conditioning and small recently fallen tree categories such as bud scales, male flowers, and twigs. Smears of faecal material, holorganic or organo-mineral, were observed at the surface of some beech leaves. A laminated mixing of leaf fragments and faeces of litter-consuming enchytraeids, which we called "sandwich material", together with plant categories such as male flowers of beech, grass stems, twigs, at some stage of decomposition, was characteristic of the Oe horizon. The presence of beech petioles and nerves, mycelial strands and living bases of wood-rush (Luzula forsteri) was also noticeable. Both Oa and A horizons were characterized by dead or senescent fine roots, living and dead woody roots, old (compacted) holorganic enchytraeid faeces, and fragments of parent rock (both intact or weathered). Charcoal was also present in this group of horizons. Sclerotia of the mycorrhizal ascomycete Cenococcum geophilum also occurred.

Living fine roots of beech and faeces of litter-consuming enchytraeids were placed in an intermediate position between the Oe horizon and the Oa+A horizons. This indicated that they were present in all of these horizons and, thus, they did not help in differentiating among horizons. Maple (Acer pseudoplatanus) and ash (Fraxinus excelsior) plant materials were different. These categories only occurred at one site (100) which was characterized by an intense earthworm activity, with abundant cast deposition within the Oi horizon (Fig. 1). The projection of corresponding samples was influenced by the presence of organo-mineral material (typical of the A horizon of mull humus) together with categories typical of the Oi horizon, which influenced in turn the position of the abovementioned ash and maple categories.

Oa and A horizons, which were considered as diagnostic horizons for the separation of mull versus moder humus forms [START_REF] Delecour | Les formes d'humus: identification et description[END_REF][START_REF] Brêthes | Classification of forest humus forms: a French proposal[END_REF], were not clearly separable on the basis of composition as revealed by the use of a dissecting microscope. In order to reveal possible variation in the composition of these horizons, we analyzed separately Oa and A horizons (partial analysis). The projection of horizons in the plane of axes 1 and 2 (Fig. 2) separated Oa and A (classified by field observation), but with a high degree of overlapping. This indicated that the distinction between Oa and A horizons that had been made in the field (Oa = holorganic, A = organo-mineral) did not reflect true composition. The projection of some passive variables such as humus form, altitude and phytosociological type (Fig. 3) helped to establish a link between Axis 2 of partial analysis and ecological conditions prevailing in the studied sites. Mull humus forms were on the negative side of Axis 2; moder humus forms were placed on the positive side of Axis 2.

The composition of Oa and A horizons, as indicated by the projection of categories in the plane of axes 1 and 2 of partial analysis (Appendix 2), depended on the humus form. The Oa horizon of moder humus forms (eumoder, hemimoder and dysmoder) was mostly characterized by compacted holorganic faeces of enchytraeids, different plant organs hard to decay, bundles of skeletonized beech leaf fragments, living woody roots of beech, and dead black mycorrhizae of beech produced by Cenococcum geophilum. The upper part of the A horizon of mull humus forms (oligomull, amphimull, dysmull) was mostly characterized by organo-mineral earthworm faeces, milliped faeces, intact stones, beech leaves skeletonized by macrofauna or mesofauna and categories belonging to ground vegetation such as the root system of grass species or bleached leaves of wood-sorrel (Oxalis acetosella). The Oa horizon of amphimull shared many features with the A horizon of the mull group (indicated by its position on the negative side of Axis 2). The A horizon of the moder group was placed in an intermediary position (not far from the origin, on both sides of Axis 2), thus without characteristic features distinguishing both the horizon and the humus form.

Distribution of soil animals

Animal groups found during the dissection of humus horizons (Table 3) were projected as passive variables in the plane of axes 1 and 3 of total analysis (Fig. 4). Comparison with the position of horizons (Fig. 1) revealed that the Oi horizon was very poor in fauna at the time of sampling (June), no animal group being placed far from the origin in the direction of the Oi Ihorizon (see also Table 3). The Oe horizon was characterized by mites other than Platynothrus peltifer (MIT, ORI, PHT) and springtails (SPR), the latter being present at a greater depth than the former as ascertained by the respective position of corresponding points along Axis 1. The only group that characterized the Oa+A horizons was enchytraeids (ENC). Enchytraeids mainly characterized the Oa horizon, as it appeared from the comparison of Oa and A horizons (Fig. 5, Table 3). Most fauna (here mostly mesofauna) were concentrated in the Oe and Oa horizons, the most abundant group being enchytraeids, followed at a far lower level of abundance by mites and springtails, with an increasing preference for deeper horizons at the time of sampling in the order mites < springtails < enchytraeids.

DISCUSSION AND CONCLUSION

The existence of a transitional horizon between typical Oa and A horizons, due to a progressive enrichment in mineral particles of the Oa horizon, has been suggested by [START_REF] Delecour | Essai de classification pratique des humus[END_REF][START_REF] Delecour | Les formes d'humus: identification et description[END_REF]. He even considered it as a characteristic feature of the humus forms belonging to the moder group, rather than the accumulation of holorganic faeces in Oe and Oa horizons which is commonly used to define moder humus [START_REF] Klinka | Taxonomic classification of humus forms in ecosystems of British Columbia. First approximation[END_REF][START_REF] Green | Towards a taxonomic classification of humus forms[END_REF][START_REF] Duchaufour | Pédologie. Sol, végétation, environnement[END_REF]. This feature is also used in the classification of forest humus forms by [START_REF] Brêthes | Classification of forest humus forms: a French proposal[END_REF] where the A horizon of moder humus forms is considered to be made of holorganic faeces juxtaposed to mineral particles, without true incorporation of organic matter to mineral matter. This composition can be achieved by the vertical movements of small animals carrying mineral particles onto their tegument and depositing them, together with faeces, in the course of their wandering between food sources (Oe horizon) and refuges (Oa and A horizons). In moder humus forms, where no other agent mixes organic matter with mineral matter, this behaviour is typical of animals with a sticky tegument such as enchytraeid worms [START_REF] Ponge | Succession of fungi and fauna during decomposition of needles in a small area of Scots pine litter[END_REF], whose daily vertical movements of several centimeters are well-known [START_REF] Springett | Vertical movement of Enchytraeidae (Oligochaeta) in moorland soils[END_REF]. In this case the gradual passage from the Oa to the A horizon can be interpreted as an active (biological) diffusion process, the source of organic matter being decaying litter, which is actively consumed by these animals and humified [START_REF] Ponge | Succession of fungi and fauna during decomposition of needles in a small area of Scots pine litter[END_REF], and the source of mineral matter being underlying mineral horizons. As has been observed in other studies [START_REF] Zachariae | Spuren tierischer Tätigkeit im Boden des Buchenwaldes[END_REF][START_REF] Ponge | Succession of fungi and fauna during decomposition of needles in a small area of Scots pine litter[END_REF] enchytraeids also consume faeces of other animals such as oribatid mites. As they live deeper (on average) than other litter-dwelling animals (Table 3) they progressively incorporate faeces of litter-consuming animals into their own faecal material. This may explain why the Oa horizon of moder humus forms appears mainly made of compacted enchytraeid faeces and unconsumed material rather than remains of the activity of all litter-feeding animals. They also ingest silt particles (unpublished data), and carry them onto their tegument [START_REF] Ponge | Etude écologique d'un humus forestier par l'observation d'un petit volume, premiers résultats. III. La couche F1 d'un moder sous Pinus sylvestris[END_REF]. This pattern may lead to the observed soft transition from Oa to A horizons.

The presence of clear discontinuities between Oi and Oe horizons and between Oe and Oa+A horizon, and the absence of clearcut subdivisions within them, may be explained by threshold levels in the vertical distribution of soil animals and other organisms. We cannot speculate from our data about the depth levels at which animals consume litter and defecate at any time of the year, and about all the factors which govern these patterns. Nevertheless, it can be inferred from existing literature that the vertical distribution of soil fauna is determined, among others, by i) genetically fixed behavioural features [START_REF] Stevenson | Effect of leaf shape on forest litter spiders: community organization and microhabitat selection of immature Enoplognatha ovata (Clerck) (Theridiidae)[END_REF][START_REF] Kretzschmar | Besoins biologiques des vers de terre et porosité du sol[END_REF], ii) ecophysiological requirements of animals [START_REF] Atalla | The survival of some soil mites at different humidities and their reaction to humidity gradients[END_REF][START_REF] Haukka | Growth and survival of Eisenia fetida (Sav.) (Oligochaeta: Lumbricidae) in relation to temperature, moisture and presence of Enchytraeus albidus (Henle) (Enchytraeidae)[END_REF], iii) the stage of decomposition of litter [START_REF] Hayes | Studies on the feeding preferences of some phthiracarid mites (Acari: Oribatidae)[END_REF][START_REF] Soma | Ecological studies of soil organisms with reference to the decomposition of pine needles. II. Litter feeding and breakdown by the woodlouse, Porcellio scaber[END_REF][START_REF] David | Influence de la durée du séjour dans la litière des feuilles mortes de chêne (Quercus petraea Liebl.) sur leur consommation par le Diplopode Cylindroiulus nitidus[END_REF]. For instance, if we take into account only leaf litter as a food, the depth level at which animals will consume it can be interpreted as a compromise between the need for food of a high nutritional value such as freshly fallen litter [START_REF] Soma | Ecological studies of soil organisms with reference to the decomposition of pine needles. II. Litter feeding and breakdown by the woodlouse, Porcellio scaber[END_REF], the toxicity or repellence of some compounds present in undecayed litter [START_REF] Satchell | Selection of leaf litter by Lumbricus terrestris[END_REF], and the search for better micro-climate conditions [START_REF] Joosse | Ecological aspects of aggregation in Collembola[END_REF]). If we examine diagnostic features of Oi and Oe horizons as ascertained in our analysis (Table 2) it appears that the passage from Oi to Oe horizon is characterized by the deposition of faeces of small litter-consuming animals, such as oribatid mites and enchytraeids, these categories being near absent in the Oi horizon.

The passage from the Oe to the Oa+A horizon is marked by the appearance of the root system of beech (Table 2). This creates a new discontinuity along the humus profile. According to the humus form, the feeder root system of beech will develop throughout accumulated organo-mineral earthworm faeces (A horizon of mull humus) or holorganic enchytraeid faeces (Oa horizon of moder humus). This common feature is probably the main reason for the absence of a clear distinction between Oa and A horizons in the global analysis (Fig. 1). From existing literature, it can be concluded that the vertical distribution of root tips is strongly influenced by humus form horizons and the humus form itself [START_REF] Meyer | Distribution of root tips and tender roots of beech[END_REF][START_REF] Persson | The distribution and productivity of fine roots in boreal forests[END_REF][START_REF] Harvey | Distribution of active ectomycorrhizal short roots in forest soils of the inland Northwest: effects of site and disturbance[END_REF][START_REF] Ponge | Etude écologique d'un humus forestier par l'observation d'un petit volume, premiers résultats. III. La couche F1 d'un moder sous Pinus sylvestris[END_REF][START_REF] Bernier | Humus form dynamics during the sylvogenetic cycle in a mountain spruce forest[END_REF]. These results point out that the upper part of the Oa horizon together with the bottom of the Oe horizon is the main micro-site for mycorrhizal root development in moder humus forms, mull humus forms being characterized by a lower number of root tips which are widely distributed throughout the A horizon. Unfortunately mechanisms which could explain these patterns are poorly understood, although the need for available water and nutrients has been advocated as a reason for the preferential development of feeder roots in carbon-rich substrates [START_REF] Persson | The distribution and productivity of fine roots in boreal forests[END_REF].

As this is now well-admitted the genesis of humus form horizons, and thus of humus forms, is placed under the influence of microflora, fauna, and plant subterranean parts, whose activity is determined by climate, parent rock and quantity and quality of vegetation [START_REF] Kubiëna | Animal activity in soils as a decisive factor in establishment of humus forms[END_REF][START_REF] Bal | Zoological ripening of soils[END_REF][START_REF] Toutain | Activité biologique des sols, modalités et lithodépendance[END_REF][START_REF] Green | Towards a taxonomic classification of humus forms[END_REF][START_REF] Brêthes | Classification of forest humus forms: a French proposal[END_REF]. The existence of horizons, not for the purpose of classification, but as a discernable entity, with clear emergent properties, can be understood if we take into account the accumulation through time of key categories with a high longevity. This is the case for dead leaves, enchytraeid faeces, earthworm faeces, or fine roots of trees. If a category present in a given horizon has a low longevity, i.e. if it is rapidly transformed into another category or if it disappears from the soil matrix through mineralization or leaching, then it cannot directly participate in the building of this horizon. Rather, it may, i) participate in the building of underlying horizons (for instance colloidal organic compounds precipitating in B horizons), ii) be taken-up by organisms, iii) diffuse in the atmosphere or in the water flow. This is the case, among others, of oribatid faeces which rapidly disappears from the moder profile when consumed by enchytraeids [START_REF] Ponge | Succession of fungi and fauna during decomposition of needles in a small area of Scots pine litter[END_REF]. This is also the case for enchytraeid faeces when ingested in turn by earthworms living in mull humus [START_REF] Bal | Zoological ripening of soils[END_REF]. The passage from one horizon to another should thus be considered as a check in a continuous process (the slow maturation of individual categories in the absence of disturbances), and a start for another continuous process of maturation. The fact that at a given depth level the bulk of categories of a given type are transformed into categories of another type (due to the activity of vertically distributed organisms) will create discontinuities visible to the naked eye and thus will help us to distinguish horizons and humus forms on the field. In the absence of these discontinuities (as between Oa and A horizons of moder humus forms) this distinction may be more difficult and field observation may lead to false conclusions, as has been always suggested by [START_REF] Bernier | Comparative study of soil organic layers in two bilberryspruce forest stands (Vaccinio-Piceetea). Relation to forest dynamics[END_REF]. In this case the identification of horizon components under a dissecting microscope and the use of counting methods [START_REF] Bernier | Comparative study of soil organic layers in two bilberryspruce forest stands (Vaccinio-Piceetea). Relation to forest dynamics[END_REF][START_REF] Bernier | Humus form dynamics during the sylvogenetic cycle in a mountain spruce forest[END_REF]) may help a better characterization of these horizons. 3. [START_REF] Thill | Typologie et potentialités forestières des hêtraies naturelles de l'Ardenne Centrale[END_REF]. Soil types according to FAO-UNESCO classification [START_REF] Driessen | The major soils of the world[END_REF]. Humus forms according to [START_REF] Brêthes | Classification of forest humus forms: a French proposal[END_REF].

LEGENDS OF FIGURES

Entire -0.0045 0.0030 Petioles and nerves of beech filled w ith faeces of phthiracarid oribatid mites -0.0025 0.0188 Petioles and nerves of beech filled w ith faeces of sciarid dipteran larvae -0.0014 0.0091 Petioles and nerves of beech filled w ith grass roots -0.0045 0.0030 Petioles and nerves of beech brow n and tough -0.0112 -0.0027 Petioles and nerves of beech bleached -0.0018 0.0009 Sandw ich material made of beech leaf fragments and holorganic enchytraeid faeces 0.0039 0.0250 Sandw ich material made of beech leaf fragments and holorganic earthw orm faeces -0.0045 0.0034 Sandw ich material made of beech leaf fragments and holorganic oribatid faeces 0.0033 0.0106 Sandw ich material made of beech leaf fragments and organo-mineral earthw orm faeces -0.0003 0.0024 Sandw ich material made of beech leaf fragments and organo-mineral enchytraeid faeces 0.0037 -0.0025 Sandw ich material made of beech leaf fragments and holorganic sciarid faeces 0.0001 -0.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Results from total correspondence analysis of organic horizons with faunal characteristics in beech forests of the Belgian Ardennes. Projection of horizons (rows) in the plane of axes 1 and 3. Indicated names of horizons were based on field observation and were not included as main variables.

Fig. 3 .

 3 Fig. 3. Partial correspondence analysis on Oa and A horizons. Projection of passive variables in the plane of axes 1 and 2. (festuc = Luzulo-Fagetum festucetosum; melic = Melico-Fagetum festucetosum; typic = Luzulo-Fagetum typicum; vaccin = Luzulo-Fagetum vaccinietosum).

Fig. 4 .

 4 Fig. 4. Total correspondence analysis. Projection of animal groups (passive variables) in the plane of axes 1 and 3. Coding of animal groups as in Table3.

Fig. 5 .

 5 Fig. 5. Partial correspondence analysis on Oa and A horizons. Projection of animal groups (passive variables) in the plane of axes 1 and 2. Otherwise as for Fig. 4.

  Fig. 1

Fig. 3

 3 Fig. 3

Fig. 4

 4 Fig. 4

Fig. 5

 5 Fig. 5

Table 1 .

 1 Main features of the 13 studied sites. Phytosociological types according to

	Site Altitude Phytosociological type	Soil type	Humus form
	1	m	Luzulo-Fagetum festucetosum	Dystric cambisol	Dysmull
	3	m	Luzulo-Fagetum festucetosum	Dystric cambisol	Eumoder
	4	m	Luzulo-Fagetum typicum	Dystric cambisol	Dysmoder
	5	m	Luzulo-Fagetum vaccinietesosum	Dystric cambisol	Eumoder to dysmoder
	16	m	Luzulo-Fagetum vaccinietesosum	Dystric cambisol	Eumoder
	17	m	Luzulo-Fagetum typicum	Dystric cambisol	Hemimoder to eumoder
	22	m	Luzulo-Fagetum typicum	Gleyic cambisol	Eumoder to dysmoder
	24	m	Luzulo-Fagetum festucetosum	Dystric cambisol	Dysmull to dysmoder
	26	m	Luzulo-Fagetum vaccinietesosum	Leptic podzol	Dysmoder
	28	m	Luzulo-Fagetum festucetosum	Dystric cambisol	Amphimull to eumoder
	40	m	Luzulo-Fagetum vaccinietesosum	Ferric podzol	Dysmoder
	100	m	Melico-Fagetum festucetosum	Dystric cambisol	Oligomull to dysmull
	307	m	Luzulo-Fagetum vaccinietesosum	Leptic podzol	Amphimull

Table 2 .

 2 Most typical categories in the three main groups of horizons depicted by correspondence analysis

	Oi
	brown leaves of beech
	Entire bleached leaves of beech
	Intact bud scales of beech
	Intact male inflorescences of beech
	Entire variegated leaves of beech
	Intact twigs
	Holorganic faecal material smearing beech leaves
	Organo-mineral material smearing beech leaves
	Oe
	Sandwich material made of beech leaf fragments and holorganic enchytraeid faeces
	Brown decaying male inflorescences of beech
	Fragments of grass stems browsed by fauna
	Pollen
	Woodlice shells
	Petioles and nerves of beech filled with faeces of phthiracarid oribatid mites
	Bud scales of beech, entire but brown and soft
	Intact unidentified fragments of seed wings
	Intact seed coats of beech
	Twigs filled with enchytraeid holorganic faeces
	Twigs filled with oribatid holorganic faeces
	Living leaf bases of Luzula albida
	White rhizomorphs
	Oa+A
	Dead pale yellow creamy mycorrhizae of beech
	Sclerotia of Cenoccum geophilum
	Intact dead fine long roots of beech
	Compacted holorganic enchytraeid faeces
	Intact living woody roots of beech
	Strongly decayed bud scales of beech
	Well-decayed bark fragments
	Intact stones
	Decaying woody roots of beech
	Weathering stones
	Charcoal
	Dead fine long roots of beech tunnelled by fauna
	Decaying grass roots
	Dead black mycorrhizae of beech
	Compacted organo-mineral material

Table 3 .

 3 Mean number of animals which were found during the dissection of humus horizons (L x l x h = 5 x 5 x 1 cm)

	Categories

  Categories used for the description of humus horizons. Coordinates on factorial axes 1 and 3 of total correspondence analysis Categories used for the description of humus horizons. Coordinates on factorail axes 1 and 2 of partial correspondence analysis (OH and A).

	Categories Appendix 1. Categories Appendix 1. Continued Appendix 1. Continued Appendix 2.	Axis 1 Axis 1 Axis 2 Axis 3
	Intact w ood fragments Categories Brown leaves of beech skeletonized by macrofauna	-0.0048 -0.0002 Axis 1 -0.0319 -0.0029	Axis 3
	Decayed w ood fragments Wood fragments tunnelled by fauna Winged seed of Acer pseudoplatanus w ith intact w ing Bleached leaves of beech skeletonized by macrofauna Bleached leaves of beech skeletonized by mesofauna	0.0016 0.0096 -0.0002 0.0001 -0.0319 -0.0093 0.0036 -0.0002 -0.0319	-0.0007
	Wood fragments penetrated by grass roots Winged seed of Acer pseudoplatanus w ith skeletonized w ing Intact petioles and nerves of beech	-0.0045 -0.0062 -0.0130 -0.0013 0.0030	0.0030
	Wood fragments penetrated by beech fine roots Intact bark fragments Wingless seed of Acer pseudoplatanus Petioles and nerves of beech tunnelled by fauna Well-decayed bark fragments Bark fragments tunnelled by enchytraeids Bark fragments tunnelled by phthiracarid mites Winged seed of Fraxinus excelsior w ith intact w ing Petioles and nerves of beech filled with enchytraeid faeces Brow n entire leaves of Quercus petraea Petioles and nerves of beech filled with faeces of phthiracarid oribatid mites Sandwich material made of beech leaf fragments and holorganic enchytraeid faeces Bark fragments tunnelled by sciarid larvae Leaves of Quercus petraea skeletonized by mesofauna Sandwich material made of beech leaf fragments and holorganic earthworm faeces Bark fragments penetrated by grass roots Intact unidentified fragments of seed w ings Sandwich material made of beech leaf fragments and holorganic oribatid faeces Intact living fine long roots of beech Living fine long roots of beech brow sed by fauna Intact dead fine long roots of beech Skeletonized unidentified fragments of seed w ings Sandwich material made of beech leaf fragments and organo-mineral earthworm faeces Brow n entire needles of Picea abies Sandwich material made of beech leaf fragments and organo-mineral enchytraeid faeces Skeletonized beech leaf fragments Dead fine long roots of beech tunnelled by fauna Bleached entire needles of Picea abies Bundles of skeletonized beech leaf fragments Dead fine long roots of beech penetrated by grass roots Needles of Picea abies brow sed by fauna Brown beech leaf fragments untouched by fauna Dead fine long roots of beech, voided Living w oody roots of beech Living w oody roots of beech brow sed by fauna Seed w ings of Picea abies Intact bud scales of beech Brow n rhizomorphs Bud scales of beech, entire but brown and soft Strongly decayed bud scales of beech Decaying w oody roots of beech White rhizomorphs Brown decaying male inflorescences of beech	0.0025 0.0072 -0.0012 -0.0100 0.0135 -0.0063 -0.0060 0.0179 -0.0147 -0.0018 0.0094 0.0001 -0.0053 -0.0051 0.0069 -0.0076 -0.0094 -0.0090 -0.0343 0.0089 0.0075 -0.0022 -0.0049 -0.0054 -0.0001 0.0046 -0.0045 -0.0050 -0.0383 -0.0025 0.0030 0.0231 0.0184 0.0123 -0.0061 0.0238 -0.0027 -0.0031 -0.0038 -0.0010 -0.0002 -0.0008 0.0047 0.0114 -0.0178 0.0145 -0.0108 0.0002 0.0201 -0.0119 0.0031 -0.0002 -0.0051 -0.0081 -0.0069 0.0046 -0.0075 0.0237 -0.0031 0.0069 -0.0053 -0.0214 -0.0201 -0.0047 -0.0292 -0.0056 0.0016 0.0205 -0.0058 0.0160 -0.0133 -0.0023 -0.0246 0.0038	0.0051 -0.0192 -0.0067 -0.0013 0.0176 0.0036 -0.0002 -0.0034 0.0086 -0.0032 0.0075 0.0130
	Living pale yellow creamy mycorrhizae of beech Yellow rhizomorphs Intact seed coats of beech	0.0221 -0.0105 0.0028 0.0065 0.0136	0.0074
	Pale yellow creamy mycorrhizae of beech brow sed by fauna Dead pale yellow creamy mycorrhizae of beech Living orange brow n mycorrhizae of beech w ith w oolly mycelium Orange brow n mycorrhizae of beech w ith w oolly mycelium brow sed by fauna Dead rhizomorphs of Armillaria Seed coats of beech tunnelled by phthiracarid mites Dead rhizomorphs of Armillaria tunnelled by fauna Seed coats of beech tunnelled by enchytraeids Sclerotia of Cenoccum geophilum Intact fragments of beech burr Soft fragments of beech burr	0.0181 0.0260 0.0106 0.0016 0.0020 0.0100 0.0054 0.0003 -0.0139 0.0078 0.0018 0.0100 -0.0015 0.0051 0.0245 -0.0152 -0.0048 0.0041 0.0022	-0.0074 -0.0074 -0.0136
	Dead orange brow n mycorrhizae of beech w ith w oolly mycelium Lichens Soft fragments of beech burr tunnelled by oribatid mites	0.0128 -0.0090 -0.0106 0.0136 -0.0068	-0.0053
	Living black mycorrhizae of beech (produced by Cenoccum geophilum) Living black mycorrhizae of beech brow sed by fauna Dead black mycorrhizae of beech Living yellow mycorrhizae of beech w ith w oolly mycelium Intact caterpillar faeces Soft fragments of beech burr tunnelled by enchytraeids Caterpillar faeces tunnelled by phthiracarid mites Beech cupules tunnelled by fauna Intact slug faeces Twigs decayed by white-rot Twig fragments tunnelled by fauna	0.0181 0.0019 0.0141 0.0028 -0.0137 0.0012 0.0197 0.0104 0.0043 -0.0087 0.0074 0.0056 0.0051 0.0050 -0.0142 0.0000 -0.0066 -0.0013 0.0229	0.0099 0.0078 -0.0045
	Living shoots of Polytrichum formosum Slug faeces tunnelled by enchytraeids Bark remnants of twigs	-0.0067 -0.0089 0.0008 -0.0122 0.0087	-0.0049
	Skeletonized beech leaf fragments Bundles of skeletonized beech leaf fragments Brow n beech leaf fragments untouched by fauna Intact bud scales of beech Bud scales of beech, entire but brow n and soft Strongly decayed bud scales of beech Intact male inflorescences of beech Brow n decaying male inflorescences of beech Pollen mass Intact seed coats of beech Seed coats of beech tunnelled by phthiracarid mites Seed coats of beech tunnelled by enchytraeids Seed coats of beech tunnelled by sciarid larvae Fragments of stems of Polytrichum formosum, red and tough Fragments of stems of Polytrichum formosum, voided Slug faeces tunnelled by sciarid larvae Twigs filled with enchytraeid holorganic faeces Dead stem bases of Polytrichum formosum Decaying stem bases of Polytrichum formosum Living shoots of Scleropodium purum Intact holorganic earthw orm faeces Twigs filled with enchytraeid organo-mineral faeces Holorganic earthw orm faeces tunnelled by enchytraeids Twigs filled with sciarid holorganic faeces Twigs filled with oribatid holorganic faeces Dead shoots of Scleropodium purum Unidentified holorganic faeces Twigs penetrated by beech roots Living shoots of Leucobryum glaucum Intact organo-mineral earthw orm faeces Decayed wood fragments Dead shoots of Leucobryum glaucum Dead moss, undetermined Intact leaves of Luzula forsteri Compacted organo-mineral earthw orm faeces Wood fragments tunnelled by fauna Organo-mineral earthw orm faeces tunnelled by enchytraeids Wood fragments penetrated by beech fine roots Intact bark fragments Bleached leaves of Luzula forsteri Holorganic w oodlice faeces Well-decayed bark fragments Living leaf bases of Luzula forsteri Holorganic milliped faeces Bark fragments tunnelled by enchytraeids Decaying leaf bases of Luzula forsteri Intact leaves of Deschampsia flexuosa Decaying leaves of Deschampsia flexuosa Holorganic milliped faeces tunnelled by enchytraeids Bark fragments tunnelled by phthiracarid mites Holorganic milliped faeces tunnelled by phthiracarid mites Intact living fine long roots of beech Living fine long roots of beech browsed by fauna Living leaf bases of Deschampsia flexuosa Holorganic cranefly faeces Intact dead fine long roots of beech Seed coats of beech penetrated by roots Decaying leaf bases of Deschampsia flexuosa Intact holorganic sciarid faeces Dead fine long roots of beech tunnelled by fauna Intact fragments of beech burr Soft fragments of beech burr Soft fragments of beech burr tunnelled by oribatid mites Soft fragments of beech burr tunnelled by enchytraeids Intact inflorescences of Deschampsia flexuosa Decaying inflorescences of Deschampsia flexuosa Living grass roots Compacted holorganic sciarid faeces Dead fine long roots of beech penetrated by grass roots Intact holorganic enchytraeid faeces Dead fine long roots of beech, voided Living woody roots of beech Decaying grass roots Compacted holorganic enchytraeid faeces Living woody roots of beech browsed by fauna Soft fragments of beech burr tunnelled by sciarid larvae Intact grass stems Organo-mineral enchytraeid faeces Decaying woody roots of beech Soft fragments of beech burr tunnelled by springtails Soft fragments of beech burr penetrated by grass roots Beech cupules tunnelled by fauna Fragments of grass stems brow sed by fauna Fragments of decaying grass roots Intact leaves of Vaccinium myrtillus Compacted organo-mineral enchytraeid faeces Living pale yellow creamy mycorrhizae of beech Compacted organic-dominant organo-mineral material Pale yellow creamy mycorrhizae of beech browsed by fauna Dead pale yellow creamy mycorrhizae of beech Intact beech gallnuts Skeletonized leaves of Vaccinium myrtillus Compacted organo-mineral material Living orange brown mycorrhizae of beech with woolly mycelium Intact tw igs Roots of Vaccinium myrtillus Compacted mineral-dominant organo-mineral material Dead orange brown mycorrhizae of beech with woolly mycelium Tw igs decayed by w hite-rot Tw ig fragments tunnelled by fauna Bark remnants of tw igs Tw igs filled w ith enchytraeid holorganic faeces Tw igs filled w ith enchytraeid organo-mineral faeces Tw igs filled w ith sciarid holorganic faeces Tw igs filled w ith oribatid holorganic faeces Tw igs penetrated by beech roots Continued. Living rhizomes of Vaccinium myrtillus Decaying rhizomes of Vaccinium myrtillus Bleached leaves of Oxalis acetosella Brow n entire leaves of Acer pseudoplatanus Brow n leaves of Acer pseudoplatanus skeletonized by macrofauna Unidentified mineral assemblages Living black mycorrhizae of beech (produced by Cenoccum geophilum) Charcoal Dead black mycorrhizae of beech Snail shells Living shoots of Polytrichum formosum Fragments of stems of Polytrichum formosum, voided Bleached leaves of Acer pseudoplatanus Woodlice shells Dead stem bases of Polytrichum formosum Bleached leaves of Acer pseudoplatanus skeletonized by macrofauna Leaves of Acer pseudoplatanus skeletonized by mesofauna Continued. Intact stones Dead shoots of Leucobryum glaucum Weathering stones Bleached leaves of Luzula forsteri Weathering stones impregnated w ith organic matter Living leaf bases of Luzula forsteri Continued	0007 -0.0046 0.0060 -0.0019 -0.0037 0.0191 -0.0043 -0.0120 0.0236 0.0204 0.0175 0.0077 0.0037 0.0044 -0.0089 0.0032 0.0098 0.0055 0.0141 -0.0299 0.0008 0.0217 -0.0232 -0.0057 -0.0022 0.0038 0.0006 0.0051 -0.0009 0.0051 -0.0119 0.0159 0.0095 0.0024 -0.0001 0.0047 -0.0024 -0.0073 -0.0124 -0.0289 -0.0002 0.0007 0.0002 0.0068 -0.0287 0.0078 0.0111 0.0042 -0.0056 -0.0294 0.0045 0.0043 -0.0064 -0.0025 0.0010 -0.0055 -0.0054 0.0003 -0.0012 -0.0057 -0.0035 -0.0060 0.0088 -0.0168 -0.0104 0.0042 0.0024 -0.0066 -0.0140 0.0038 -0.0057 -0.0034 -0.0120 0.0091 0.0086 -0.0070 0.0000 0.0048 -0.0090 0.0079 0.0136 0.0047 -0.0024 -0.0074 -0.0030 -0.0135 0.0036 -0.0022 -0.0101 0.0036 -0.0212 -0.0033 -0.0021 0.0103 0.0017 -0.0037 0.0004 0.0061 -0.0014 -0.0003 0.0025 0.0028 0.0120 -0.0012 -0.0044 0.0045 0.0073 0.0004 -0.0098 0.0019 0.0099 0.0028 0.0025 0.0108 0.0068 -0.0058 -0.0045 -0.0098 -0.0003 0.0090 0.0074 0.0049 -0.0025 0.0141 0.0047 -0.0033 0.0028 0.0182 -0.0022 0.0144 -0.0083 0.0219 0.0040 0.0103 0.0069 0.0029 0.0008 -0.0136 0.0078 0.0113 0.0113 0.0069 0.0012 0.0043 0.0033 -0.0048 0.0006 -0.0022 0.0204 -0.0062 -0.0026 -0.0075 -0.0063 0.0070 -0.0172 -0.0035 0.0066 -0.0016 0.0125 0.0040 0.0047 -0.0024 0.0051 -0.0101 -0.0034 0.0114 -0.0052 0.0069 0.0010 -0.0225 0.0053 0.0013 0.0040 0.0058 0.0065 -0.0033 -0.0087 -0.0031 -0.0074 0.0106 0.0079 0.0072 0.0115 0.0205 0.0102 -0.0024 0.0015 0.0101 0.0085 0.0138 0.0129 0.0009 0.0016 0.0018 0.0072 -0.0001 0.0013 -0.0039 -0.0051 -0.0198 -0.0051 -0.0016 -0.0079 0.0027 0.0132 0.0057 0.0152 -0.0028 -0.0054 -0.0088 0.0010 -0.0105 -0.0198 -0.0053 -0.0022 0.0037 -0.0076 -0.0192 -0.0051 -0.0198 -0.0018 -0.0002 0.0161 -0.0012 -0.0043 0.0147 -0.0012 -0.0063 0.0048 0.0045 -0.0002	-0.0049 0.0118 0.0057 -0.0037 0.0036 -0.0016 0.0034 0.0033 0.0049 0.0010 0.0010 0.0048 -0.0007 -0.0069 0.0081 -0.0120 -0.0044 -0.0117 -0.0064 -0.0120 -0.0085 -0.0011 -0.0153 0.0098 0.0204 -0.0103 -0.0128 -0.0096

ACKNOWLEDGEMENTS

The author is greatly indebted to Pr. Dr. F. Delecour (Faculté des Sciences Agronomiques, Gembloux, Belgium) for the selection of study sites.

Categories

Axis 1 Axis 2