Deise Santana

Minh-Tan Pham

Sébastien Lefèvre

Watershed-based attribute profiles for pixel classification of remote sensing data

We combine two well-established mathematical morphology notions: watershed segmentation and morphological attribute profile (AP), a multilevel feature extraction method commonly applied to the analysis of remote sensing images. To convey spatial-spectral features of remote sensing images, APs were initially defined as sequences of filtering operators on the max-and min-trees computed from the original data. Since its appearance, the notion of APs has been extended to other hierarchical representations including tree-of-shapes and partition trees such as α-tree and ω-tree. In this article, we propose a novel extension of APs to hierarchical watersheds. Furthermore, we extend the proposed approach to consider prior knowledge from training samples, leading to a more meaningful hierarchy. More precisely, in the construction of hierarchical watersheds, we combine the original data with the semantic knowledge provided by labeled training pixels. We illustrate the relevance of the proposed method with an application in land cover classification using optical remote sensing images, showing that the new profiles outperform various existing features.

1. for any x and x in V , if {x, x } ∈ E then {x, x } ∈ E ; and 2. there is no edge e = {y, y } ∈ E such that y ∈ V \ V and y ∈ V .

Let G = (V, E, w) be a graph and let G = (V , E , w) be a connected subgraph of G. If the weight of any edge in E is equal to a constant k and if w(e) > k for any edge e = {x, y} such that x ∈ V and y ∈ V \ V , then G is a (local) minimum of G.

Introduction

Mathematical morphology has a long history with the processing and analysis of remote sensing images, as attested by earlier surveys on this topic [START_REF] Soille | Advances in mathematical morphology applied to geoscience and remote sensing[END_REF]. In particular, in the past decade, special attention has been given to a multi-level feature extraction method, known as Attribute Profile (AP) [START_REF] Dalla Mura | Morphological attribute profiles for the analysis of very high resolution images[END_REF], which relies on hierarchical image representations to convey spatial-spectral features of remote sensing images.

In this article, we study the relevance of hierarchical watersheds for remote sensing applications. Our contributions are two-fold: [START_REF] Aksoy | Performance evaluation of building detection and digital surface model extraction algorithms: Outcomes of the PRRS 2008 algorithm performance contest[END_REF] the introduction of the Watershed-AP, which is an extension of AP to hierarchical watersheds; and (2) an investigation on the use of prior knowledge in the construction of hierarchical watersheds for the classification of remote sensing images.

This article is organized as follows. In section 2, we recall the definitions of graphs, hierarchical watersheds and AP, and we review the literature on prior knowledge for image processing. Section 3 introduces the Watershed-AP and our method to integrate semantic knowledge in its construction. Finally, experiments with remote sensing images are given in section 4.

Background notions

In this section, we first review graphs and hierarchical watersheds. Then, we review the literature on the use of prior knowledge and markers in the field of image processing. Finally, we recall the definition of Attribute Profile (AP).

Graphs and hierarchical watersheds

Watershed segmentation was proposed in the late 70's and, since then, this concept has been extended to several frameworks and implemented through a variety of algorithms. The intuition behind the various definitions of the watershed segmentation derive from the topographic definition of watersheds: dividing lines between catchment basins, which are, in their turn, areas where collected precipitation flows into the same regional minimum. These notions can be extended to gray-scale images and graphs, leading to different definitions of watershed segmentation. In this paper, we focus on watershed-cuts and hierarchical watersheds defined in the context of edge-weighted graphs, as formalized in [START_REF] Cousty | Watershed cuts: Minimum spanning forests and the drop of water principle[END_REF][START_REF] Cousty | Constructive links between some morphological hierarchies on edge-weighted graphs[END_REF]. In the remainder of this section, we present the notions of graphs, hierarchies of partitions and hierarchical watersheds. A (edge) weighted graph is a triplet G = (V, E, w) where V is a finite set, E is a subset of V × V , and w is a map from E into R. The elements of V and E are called vertices and edges (of G), respectively. Let G = (V, E, w) be a weighted graph and let G = (V , E , w) be a graph such that V ⊆ V and

E ⊆ E. We say that G is a subgraph of G. A sequence π = (x 0 , . . . , x n ) of vertices in V is a path (in G ) from x 0 to x n if {x i-1 , x i } is an edge of G for any 1 ≤ i ≤ n. If x 0 = x n
and if there are no repeated edges in π, we say that π is a cycle (in G ). The subgraph G of G is said to be connected if, for any x and x in V , there exists a path from x to x . Moreover, we say that G is a connected component of G if: For instance, Figure 1(a) illustrates a weighted graph with four minima delimited by the dashed lines.

Important remark: in the remainder of this section, G = (V, E, w) denotes a connected weighted graph and n denotes the number of minima of G.

Let G = (V , E , w) be a subgraph of G. A Minimum Spanning Forest (MSF) of G rooted in G is a subgraph G = (V, E , w) of G such that: 1. for every connected component X of G , there is exactly one connected component X of G such that X is a subgraph of X ; 2. every cycle in G is a cycle in G ; and 3.
e∈E w(e) is minimal among all graphs which satisfy conditions (1) and [START_REF] Beucher | The morphological approach to segmentation: the watershed transformation[END_REF].

A partition of V is a set P of disjoint subsets of V such that the union of the elements in P is V . The partition of V induced by a graph G is the partition P such that every element of P is the set of vertices of a connected component of G . A hierarchy of partitions of V is a sequence H = (P 0 , . . . , P n ) of partitions of V such that P n = {V } and such that, for any 0 < i ≤ n, every element of P i is the union of elements of P i-1 .

Any hierarchy of partitions H can be represented as a tree whose vertices correspond to the regions of H and whose edges link nested regions. For instance, Figure 1(b) shows a tree representation of the hierarchy H = (P 0 , P 1 , P 2 , P 3 ), where P 0 = {{a, b}, {c, d}, {e, f }, {g, h}}, P 1 = {{a, b}, {c, d}, {e, f, g, h}}, P 2 = {{a, b, c, d}, {e, f, g, h}} and P 3 = {{a, b, c, d, e, f, g, h}}.

Let S = (M 1 , . . . , M n ) be a sequence of n distinct minima of G such that, for any 0 < i ≤ n, we have M i = (V i , E i , w). The hierarchy of Minimum Spanning Forests of G for S, also known as hierarchical watershed of G for S, is a hierarchy H = (P 1 , . . . , P n ) of partitions of V such that each partition P i is the partition induced by the MSF of G rooted in the graph ( j≥i V j , j≥i E j , w).

A hierarchical watershed of the graph G of Figure 1(a) for the sequence (C, A, B, D) of minima of G is illustrated in Figure 1(b).

Prior knowledge for image processing

Unsupervised data pre-processing methods, such as watershed segmentation and image filtering (e.g. Sobel, Laplacian and morphological filters), are successfully employed in several computer vision tasks, including classification and detection problems. Moreover, when prior knowledge is provided, this can be used to provide further improvements.

In the context of image segmentation, a widespread method to introduce prior knowledge in the results is to consider user-defined markers, which are subsets of image pixels indicating the locations of objects of interest. Such markers guide the segmentation algorithm and assure that the objects of interest are segmented into distinct regions. The notion of markers has been especially explored in watershed segmentations, in which catchment basins are grown from input markers instead of the regional minima of an image (or graph) [START_REF] Beucher | The morphological approach to segmentation: the watershed transformation[END_REF].

The use of markers in a watershed segmentation can go beyond the introduction of new regional minima. The values or spectral signatures of the marked pixels can provide further knowledge about the objects we aim to segment. For instance, in [START_REF] Derivaux | Supervised image segmentation using watershed transform, fuzzy classification and evolutionary computation[END_REF][START_REF] Derivaux | Watershed segmentation of remotely sensed images based on a supervised fuzzy pixel classification[END_REF], the authors use the spectral signature of training samples in the construction of watershed segmentations of multi-spectral images. First, the spectral signature of training pixels is used to train a classifier, which is then applied on the whole image and used to obtain a probability map per class. Then, those maps are combined and used to obtain a single watershed segmentation. Another use of supervised classification for watersheds has been proposed in [START_REF] Lefèvre | Knowledge from markers in watershed segmentation[END_REF], where the watershed segmentation is computed from user-defined markers combined with probability maps computed for each targeted class. More precisely, catchment basins are grown from different markers, and the probability maps, combined with the original data, are used simultaneously in the process. In remote sensing, the later approach has been applied to the detection of buildings [START_REF] Aksoy | Performance evaluation of building detection and digital surface model extraction algorithms: Outcomes of the PRRS 2008 algorithm performance contest[END_REF] and shorelines [START_REF] Lefevre | Weakly supervised image segmentation: application to mapping and monitoring of salt marsh vegetation in the mont-saintmichel bay from high resolution imagery[END_REF] in multi-spectral images. Finally, in [START_REF] De Miranda | Synergistic arc-weight estimation for interactive image segmentation using graphs[END_REF], prior knowledge from markers is employed on several interactive image segmentation methods, including watersheds, in the framework of edge-weighted graphs. Edge weights are defined as a linear combination of the weights obtained from two sources: from the pixel values and from the classification probability maps computed from the markers that are incrementally provided by the users.

More generally, knowledge from markers can be used by other kinds of preprocessing methods beyond watershed segmentation. Namely, spectral signatures of training pixels have been used in [START_REF] Courty | A classwise supervised ordering approach for morphology based hyperspectral image classification[END_REF] to optimize the data pre-processing with alternating sequential filters. Hence, training pixels are used for pre-processing the input data, as well as for the final pixel classification. A related approach is proposed in [START_REF] Velasco-Forero | Supervised ordering in R p : Application to morphological processing of hyperspectral images[END_REF], where training pixels are used to optimize vector orderings for morphological operations applied to hyperspectral images.

In the context of hierarchical segmentation, prior knowledge can play a role in defining which regions should be highlighted at different levels of a hierarchy. In [START_REF] Tarabalka | Marker-based hierarchical segmentation and classification approach for hyperspectral imagery[END_REF], a marker-based hierarchical segmentation is proposed for hyperspectral image classification. Labeled markers are derived from a probability classification map, which is obtained from training samples, as done in [START_REF] Derivaux | Supervised image segmentation using watershed transform, fuzzy classification and evolutionary computation[END_REF][START_REF] Derivaux | Watershed segmentation of remotely sensed images based on a supervised fuzzy pixel classification[END_REF][START_REF] Lefèvre | Knowledge from markers in watershed segmentation[END_REF]. Then, those labeled markers guide the construction of a hierarchical segmentation by preventing regions of different classes to be merged, and by propagating the labeled markers to unlabeled regions. Another related approach, proposed in [START_REF] Fehri | Prior-based hierarchical segmentation highlighting structures of interest[END_REF], uses prior knowledge to keep the regions of interest from being merged early in the hierarchy, i.e., the details in the regions of interest are preserved at high levels of the hierarchy.

Finally, in [START_REF] Lefèvre | Hyperspectral image classification from multiscale description with constrained connectivity and metric learning[END_REF], the authors propose a knowledge-based hierarchical representation for hyperspectral images. In their approach, a dissimilarity measure learned from training pixels is employed in the construction of α-trees.

Attribute profiles

Attribute profile (AP) [START_REF] Dalla Mura | Morphological attribute profiles for the analysis of very high resolution images[END_REF] is a multilevel feature extraction method commonly applied to the analysis of remote sensing images. To convey spatial-spectral features of remote sensing images, APs were initially defined as sequences of filtering operators on the max-and min-trees computed from the original data. Let X : P → Z, P ⊆ Z 2 be a gray-scale image. The calculation of APs on X is achieved by applying a sequence of attribute filters based on a min-tree (i.e. attribute thickening operators {φ A k } K k=1 ) and on a max-tree (i.e. attribute thinning operators {γ A k } K k=1 ) as follows:

(X) = φ A K (X), φ A K-1 (X), . . . , φ A 1 (X), X, γ A 1 (X), . . . , γ A K-1 (X), γ A K (X) , (1)
where φ A k and γ A k are respectively the thickening and thinning operators with respect to the attribute A and to the threshold k, and K is the number of selected thresholds. More precisely, the thickening φ A k (X) of X (resp. thinning γ A k (X) of X) with respect to an attribute A and to a threshold k is obtained as follows: given the min-tree T (resp. max-tree T ) of X, the A attribute values (e.g. area, circularity and contrast) of the nodes of T are computed. If the attribute A is increasing, the nodes whose attribute values are inferior to k are pruned from the tree T ; otherwise other pruning strategies can be adopted [START_REF] Maia | Classification of remote sensing data with morphological attributes profiles: A decade of advances[END_REF]. Finally, the resulting image is reconstructed by projecting the gray levels of the remaining nodes of T into the pixels of X.

Since its appearance, the notion of APs has been extended to other hierarchical representations including tree-of-shapes [START_REF] Dalla Mura | Self-dual attribute profiles for the analysis of remote sensing images[END_REF] and partition trees such as α-tree and ω-tree [START_REF] Bosilj | Attribute profiles from partitioning trees[END_REF]. To obtain a profile from a partition tree instead of a component tree, some adaptations have to be made to the original definition of APs, as discussed in [START_REF] Bosilj | Attribute profiles from partitioning trees[END_REF]. For instance, the nodes of a partition tree are not naturally associated to gray-level values, as it is the case of component trees. The strategy adopted in [START_REF] Bosilj | Attribute profiles from partitioning trees[END_REF] is to represent each node as its level in the tree or as the maximum, minimum, or average gray-level of the leaf nodes (pixels) of this node. For more details about APs' extensions, we invite readers to refer to a recent survey [START_REF] Maia | Classification of remote sensing data with morphological attributes profiles: A decade of advances[END_REF].

Watershed-based attribute profiles

In this article, we extend the notion of AP to hierarchical watershed obtained in the framework of edge-weighted graphs.

As mentioned in section 2.1, hierarchies of partitions, such as hierarchical watersheds, can be equally represented a (partition) tree. Hence, the filtering strategy of Watershed-APs is similar to the strategy described in [START_REF] Bosilj | Attribute profiles from partitioning trees[END_REF] for the αand ω-APs.

As discussed in [START_REF] Bosilj | Attribute profiles from partitioning trees[END_REF], image reconstruction from partition trees is not straightforward as it is from component trees. For node representation, we adopt one of the solutions proposed in [START_REF] Bosilj | Attribute profiles from partitioning trees[END_REF] and already mentioned in section 2.3, in which a node is represented by the average gray-level of the pixels belonging to it. We highlight that, in the case of multiband images, the average grey level computed on each band might lead to spectral values not present in the input image. However, in the context of attribute profiles used for pixel classification, our aim is not image filtering. Hence, the fact that new spectral values (a.k.a. false colors) are created is not a problem as long as they allow us to distinguish between different semantic classes.

Note that hierarchical watersheds are usually constructed from a gradient of the original image, which contains more information about the contours between salient regions than about the spectral signature of those regions. Hence, we consider the original pixel values to obtain the nodes representation instead of the image gradient.

Formally, let X : P → Z be a gray-scale image and let G = (V, E, w) be a weighted graph which represents a gradient of X, i.e., V = P and, for every edge e = {x, y} in E, the weight w(e) represents the dissimilarity between x and y, e.g. w(e) = |X(x) -X(y)|. Let S be a sequence of minima of G ordered according to a given criterion C, and let H be the hierarchical watershed of G for the sequence S. Given the tree representation T of H, a Watershed-AP of X for the criterion C is constructed as a sequence of image reconstructions from filtered versions of T .

As discussed in section 2.2, user-defined markers and prior knowledge can boost the performance of watershed segmentations. In this article, we extend the use of prior knowledge to hierarchical watersheds. More specifically, we aim to enforce regional minima at the regions with high probability of belonging to any given ground-truth class. This is done through a combination of the methods proposed in [START_REF] Derivaux | Supervised image segmentation using watershed transform, fuzzy classification and evolutionary computation[END_REF][START_REF] Derivaux | Watershed segmentation of remotely sensed images based on a supervised fuzzy pixel classification[END_REF][START_REF] Lefèvre | Knowledge from markers in watershed segmentation[END_REF]. Given a dataset I (e.g. a panchromatic or a RGB image) and its training set composed of c classes, we compute its hierarchical watershed using prior knowledge as follows: In the first step of our method, we are aware that: (1) there might be sample pixels of a given class whose spectral values are not represented in the training set, and (2) there might be pixels in the training set with very similar spectral signatures but which belong to distinct classes. In those cases, we expect the classifier to assign low classification probabilities to such pixels. This means that the watershed segmentation at those regions will be mostly guided by the original gray-levels of the image gradient. Then, in the second step of our method, we combine the classification probability maps into a single probability map µ. We expect this combination to provide flat zones of pixels with high probability of belonging to any given class, i.e., subsets of pixels that should be merged early in the resulting hierarchical watershed. In the extreme case where the classifier assigns very high classification probabilities to all pixels of I, we would have a single flat zone and, consequently, a hierarchy with a single segmentation level. However, in that case, we might not need APs to improve the classification results on the image I. In the third step, a weighted graph (V, E, w P ) is obtained from the combined probability map µ. Our choice for computing edge weights as the maximum between the probability values of neighbouring pixels was actually heuristic and was based on a few experiments with the datasets described in the next section. In the steps 4 and 5, a gradient (V, E, w G ) of I is computed and then combined with (V, E, w P ) as a multiplication of edge weights, similarly to [START_REF] Lefèvre | Knowledge from markers in watershed segmentation[END_REF]. We note that the proposed method is related to ones introduced in [START_REF] Fehri | Prior-based hierarchical segmentation highlighting structures of interest[END_REF][START_REF] Lefèvre | Hyperspectral image classification from multiscale description with constrained connectivity and metric learning[END_REF], the main difference being the type of hierarchy under consideration and how the original data is combined with the prior knowledge.

In Figure 2, we show that the proposed method can be effectively used to highlight objects of interest (e.g. cars) in hierarchical watersheds. Given the RGB image I of Figure 2(a) and the set of labeled training samples for the car (in blue) and background (in red) classes of Figure 2(b), we compute the probability map per class and combine them into the map µ of Figure 2(c), in which dark regions are composed of pixels with high probability of belonging to any given class. Then, the weighted graph G P = (V, E, w P ) is computed from µ as described in the third step of our method. Next, we compute the gradient G G = (V, E, w G ) of the Red channel of I such that, for any edge e = (x, y) in E, we have w G (e) = |I(x) -I(y)|. At this step, the Green and Blue channels could have been chosen and other dissimilarity measures could be used as well. Then, the weight maps w P and w G are combined into w P G as described previously. Finally, to compare the hierarchies obtained with and without prior-knowledge, we computed the area-based hierarchical watersheds H and H of G G and G GP , respectively. In Figures 2(d),(e) and (f), we show the reconstructions of the tree representation of H after filtering the nodes with area inferior to 100, 500 and 1000, respectively. Those reconstructions are represented with random colors. Similarly, Figures 2(g),(h) and (i) show the reconstructions performed on the tree representation of H for the same filterings. We observe that, by imposing regional minima at the location of the cars, we make those objects to appear earlier in the hierarchy H when compared to H and, at the same time, to be merged later to their surrounding regions. For instance, the dark blue car on the top left appears as a single region in all reconstructions performed on the tree representation of H , which is not the case of H.

Experimental results

In this section, we evaluate the performance of Watershed-AP (computed with and without prior-knowledge) in the context of land-cover classification of remote sensing images. We first describe the panchromatic and RGB images considered in our study, as well as the experimental settings used for evaluation. Finally, we show that Watershed-AP outperforms AP and its variants including SDAP [START_REF] Dalla Mura | Self-dual attribute profiles for the analysis of remote sensing images[END_REF], α-AP [START_REF] Bosilj | Attribute profiles from partitioning trees[END_REF] and ω-AP [START_REF] Bosilj | Attribute profiles from partitioning trees[END_REF], on both datasets.

(a) (b) (c) (d) (e) (f) (g) (h) (i)

Datasets

We validate our approach on two remote sensing datasets: the panchromatic Reykjavik dataset and a RGB image of Zurich dataset [START_REF] Volpi | Semantic segmentation of urban scenes by learning local class interactions[END_REF]. The Reykjavik dataset is a panchromatic image of size 628 × 700 pixels acquired by the IKONOS Earth imaging satellite with 1-m resolution in Reykjavik, Iceland. This data consists of six thematic classes including residential, soil, shadow, commercial, highway and road. The image was provided with alreadysplit training and test sets (22741 training samples and 98726 test samples). The input image together with its thematic ground truth map are shown in Fig. 3(a).

The Zurich Summer dataset [START_REF] Volpi | Semantic segmentation of urban scenes by learning local class interactions[END_REF] is a collection of 20 NIR+RGB images of various dimensions taken from a QuickBird acquisition of the city of Zurich, Switzerland, in August 2002. The ground-truth provided for each image consists of at most eight thematic classes: roads, buildings, trees, grass, bare soil, water, railways and swimming pools. In our experiments, we only consider the RGB channels of the first image zh1.tif of this dataset. The training set is composed of 1% of the labeled pixels randomly extracted for each class. The input image and its ground-truth are given in Fig. 3(b).

Experimental settings

AP and its variants were computed on the Reykjavik and Zurich datasets with the usual area and moment of inertia (MoI) attributes. The following ten area thresholds and four MoI thresholds were adopted for both datasets: λ area = {25, 100, 500, 1000, 5000, 10000, 20000, 50000, 100000, 150000} and λ moi = {0.2, 0.3, 0.4, 0.5}. For the Zurich dataset, the APs and their extensions are computed independently on each of the RGB bands and are concatenated, leading to Extended Attribute Profiles (EAP).

For each dataset I, hierarchical watersheds were computed from two 4connected edge-weighted graphs: from the graph G G = (V, E, w G ) obtained from a gradient of the original data I (without any prior-knowledge from markers), and the second one computed from the combination of the graph G G with the classification probability map obtained from the training set of I, as described in Section 3. For every edge e = (x, y) in E, we define w G (e) as |I(x) -I(y)|.

Supervised pixel classification was performed twice, once for obtaining the classification probability map and then to provide the final land-cover pixel classification. Both were performed using a Random Forest classifier with 100 trees. The number of variables used for training was set to the square root of the feature vectors length. The different approaches are compared using the overall accuracy (OA), average accuracy per class (AA) and κ coefficient, as done in [START_REF] Dalla Mura | Morphological attribute profiles for the analysis of very high resolution images[END_REF]. For each tested method, we report the average and standard deviation of the classification scores over ten runs.

As defined in section 2.1, hierarchical watersheds can be computed for any given ordering on the minima of a weighted graph. In our experiments, such orderings are obtained from extinction values [START_REF] Grimaud | New measure of contrast: the dynamics[END_REF][START_REF] Vachier | Extinction value: a new measurement of persistence[END_REF] based on the area, dynamics and volume attributes.

Results and discussion

Tables 1-4 present the classification results of the Reykjavik and Zurich datasets. We compare the performance of the following methods: AP-maxT and AP-minT obtained by filtering the max-and min-tree, respectively; AP [START_REF] Dalla Mura | Morphological attribute profiles for the analysis of very high resolution images[END_REF], obtained as a concatenation of AP-maxT and AP-minT; SDAP [START_REF] Dalla Mura | Self-dual attribute profiles for the analysis of remote sensing images[END_REF]; α-AP and ω-AP [START_REF] Bosilj | Attribute profiles from partitioning trees[END_REF]; and the Watershed-AP computed with and without prior knowledge. To simplify the notations, Watershed-AP computed without and with prior knowledge are denoted respectively as A-WS-AP and A-PWS-AP, where A is the attribute used in the construction of the hierarchical watersheds, namely Area, Dynamics (Dyn) and Volume (Vol).

For the Reykjavik dataset, Watershed-AP constructed with the area and volume attributes outperform all other methods. As shown in Table 1, the best classification result, obtained with Area-WS-AP, outperforms AP by 1.39%, 3.08% and 1.76% in terms or OA, AA and κ, respectively. Moreover, considering only the APs obtained from partition trees (α-AP, ω-AP and Watershed-AP), the Watershed-AP outperform both α-and ω-APs by more than 10% with respect to OA, AA and κ. In terms of classification results per class (see Table 2), the highest scores are achieved by Watershed-AP and AP, except for the shadow class, for which none of those methods where able to outperform the classification based only on panchromatic pixel values.

Let us now analyse the influence of the use of prior knowledge in the performance of Watershed-AP. For the Watershed-AP computed with the dynamics attribute, the use of prior knowledge led to an improvement of 6.97%, 9.43% and 8.79% in terms of OA, AA and κ, respectively, and better accuracy scores for all six semantic classes. Whereas, the same did not happen for the Watershed-AP computed with area and volume, for which the overall classification results decreased by more than 1%.

On the Zurich dataset, the Watershed-APs outperform all other methods (see Table 3). The best method, Vol-PWS-AP, outperforms SDAP by 3.11%, 3.16% and 4.21% in terms of OA, AA and κ, respectively. Similar to the Reykjavik dataset, the Watershed-AP yielded the best results per class, except for the 'swimming pools' class (see Table 4). Moreover, on this dataset, the use of prior knowledge in the construction of hierarchical watersheds led to small improvements for all three Watershed APs. More precisely, we observed an improvement of up to 0.25% in terms of OA, AA and κ for the Watershed-APs constructed with the area, dynamics, and volume attributes.

Conclusion

We proposed the Watershed-AP as an extension of AP to hierarchical watersheds computed from (edge) weighted graphs. Besides, we investigate the relevance of using semantic prior knowledge in the construction of such hierarchies. We validated our approach on the pixel classification of two remote sensing images, which showed the potential of hierarchical watersheds in this field. On both datasets, Watershed-APs, computed with and without prior-knowledge, presented the highest evaluation scores when compared to the standard APs. As future work, we will to explore the versatility of hierarchical watersheds by considering other methods to obtain the gradient of remote sensing images, as well as different ways of including prior knowledge in the computation of those hierarchies. We are also interested in theoretical properties of the Watershed-AP, namely its link with other related methods such as Feature Profiles [START_REF] Pham | Feature profiles from attribute filtering for classification of remote sensing images[END_REF] and Extinction Profiles [START_REF] Ghamisi | Extinction profiles for the classification of remote sensing data[END_REF].

Fig. 1 .

 1 Fig. 1. (a): A weighted graph G = (V, E, w). (b): A tree representation of the hierarchical watershed H of G for the sequence (C, A, B, D) of minima of G.

1 .

 1 Train a classifier using the training set of I and compute per-pixel classification probabilities (p 1 , . . . , p c ) for all pixels of I; 2. Combine the classification probabilities into a single map µ : X → [0, 1] such that, for any pixel x in X, µ(x) = 1 -p 1 (x) 2 + . . . p c (x) 2 3. Compute a 4-or 8-connected weighted graph G P = (V, E, w P ) from µ such that, for any edge e = (x, y) in E, we have w P (e) = max(µ(x), µ(y)); 4. Compute a 4-or 8-connected weighted graph G G = (V, E, w G ) which represents a gradient of I. For instance, for any edge e = (x, y) in E, we may have w G (e) = |I(x) -I(y)| or w G (e) = (I(x) -I(y)) 2 ; 5. Combine the weight maps w P and w G into a map w GP such that, for any edge e in E, we have w GP = w P (e) × w G (e); and 6. Finally, compute the hierarchical watershed of G GP = (V, E, w GP ) for a given sequence of minima of G.

Fig. 2 .

 2 Fig. 2. (a): original image I. (b) training samples in red (background) and blue (cars). (c) classification probability map µ. (d), (e) and (f): image reconstructions obtained from a hierarchical watershed of I, by filtering the nodes with area inferior to 1000, 5000 and 10000, respectively. (g), (h) and (i): image reconstructions obtained from a hierarchical watershed of the combination of I and µ, by filtering the nodes with area inferior to 1000, 5000 and 10000, respectively.

Fig. 3 .

 3 Fig. 3. Two data sets used in our experimental study. (a) The 628 × 700 Reykjavik data. Left to right: panchromatic image, ground truth and training set including six thematic classes: residential, shadow, highway, soil, commercial, road. (b) The 610 × 340 Zurich data. Left to right: RGB image and ground truth including seven thematic classes: roads, buildings, trees, grass, bare soil, railways, swiming pools.

Table 1 .

 1 Classification result of Reykjavik dataset obtained by different methods using the default 4-connectivity and 1-byte quantization.

	Method	Dimension		Classification result
			OA (%)	AA (%)	κ × 100
	Panchromatic	1	63.86 ± 0.01	53.35 ± 0.05	52.70 ± 0.00
	AP-maxT [8]	16	76.01 ± 0.23	69.15 ± 0.10	68.89 ± 0.29
	AP-minT [8]	16	72.90 ± 0.48	64.01 ± 0.56	64.97 ± 0.61
	AP [8]	30	84.45 ± 0.43	78.22 ± 0.37	80.11 ± 0.53
	SDAP [7]	16	81.51 ± 0.37	74.83 ± 0.34	76.45 ± 0.45
	α-AP [3]	16	74.99 ± 0.14	66.48 ± 0.22	67.90 ± 0.19
	ω-AP [3]	16	75.06 ± 0.22	66.69 ± 0.38	67.99 ± 0.30
	Area-WS-AP	16	85.84 ± 0.33	81.30 ± 0.5	81.87 ± 0.41
	Dyn-WS-AP	16	75.96 ± 0.13	66.86 ± 0.18	69.20 ± 0.14
	Vol-WS-AP	16	85.62 ± 0.20	80.13 ± 0.31	81.54 ± 0.27
	Area-PWS-AP	16	84.63 ± 0.09	79.05 ± 0.12	80.21 ± 0.07
	Dyn-PWS-AP	16	82.93 ± 0.36	76.29 ± 0.55	77.99 ± 0.48
	Vol-PWS-AP	16	83.86 ± 1.22	79.15 ± 1.25	79.41 ± 1.51

Table 2 .

 2 Classification result of Reykjavik dataset obtained by different methods using the default 4-connectivity and 1-byte quantization.

	Method	Dim.			Classification results per class
			Residential	Soil	Shadow	Commercial	Highway	Road
	Panchromatic	1	6.02 ± 0.36	73.81 ± 0.00	90.94 ± 0.01	77.86 ± 0.09	16.60 ± 0.00	54.87 ± 0.01
	AP-maxT [8]	16	44.55 ± 0.42	79.64 ± 1.29	88.28 ± 0.01	88.41 ± 1.51	51.15 ± 0.15	62.88 ± 0.18
	AP-minT [8]	16	30.79 ± 0.68	85.55 ± 0.22	81.04 ± 0.73	81.89 ± 0.16	39.93 ± 6.37	64.89 ± 3.42
	AP [8]	30	62.11 ± 0.57	91.29 ± 1.52	76.58 ± 1.61	92.79 ± 0.36	74.45 ± 0.32	72.08 ± 0.27
	SDAP [7]	16	56.73 ± 1.11	89.57 ± 1.17	78.50 ± 0.89	90.45 ± 0.12	62.77 ± 0.24	70.94 ± 1.20
	α-AP [3]	16	45.68 ± 1.33	88.45 ± 0.39	81.40 ± 0.08	82.11 ± 0.05	69.34 ± 0.37	31.93 ± 0.58
	ω-AP [3]	16	47.15 ± 1.24	88.60 ± 0.38	81.39 ± 0.06	82.02 ± 0.16	69.20 ± 0.31	31.75 ± 1.66
	Area-WS-AP	16	79.23 ± 1.39	94.98 ± 0.28	87.38 ± 0.66	89.33 ± 0.08	87.99 ± 1.52	48.88 ± 1.65
	Dyn-WS-AP	16	47.16 ± 1.08	89.74 ± 0.15	80.26 ± 0.16	84.47 ± 0.48	71.68 ± 0.19	27.85 ± 0.41
	Vol-WS-AP	16	74.41 ± 0.82	95.98 ± 0.19	84.00 ± 0.70	89.85 ± 0.04	87.62 ± 1.33	48.90 ± 2.07
	Area-PWS-AP	16	71.05 ± 0.11	91.08 ± 0.08	80.67 ± 0.50	91.90 ± 0.13	88.09 ± 0.00	51.54 ± 0.34
	Dyn-PWS-AP	16	65.70 ± 1.17	94.87 ± 0.23	80.81 ± 0.67	88.55 ± 0.22	77.99 ± 0.19	49.84 ± 3.51
	Vol-PWS-AP	16	75.05 ± 0.16	89.91 ± 3.24	79.50 ± 0.45	90.27 ± 0.16	84.89 ± 3.37	55.29 ± 5.31

Table 3 .

 3 Classification result of Zurich dataset obtained by different methods using the default 4-connectivity and 1-byte quantization.

	Method	Dimension		Classification result
			OA (%)	AA (%)	κ × 100
	RGB	3	80.21 ± 0.03	69.32 ± 0.07	73.18 ± 0.04
	AP-maxT [8]	48	88.44 ± 0.02	87.13 ± 0.04	84.37 ± 0.03
	AP-minT [8]	48	87.90 ± 0.04	81.56 ± 0.07	83.69 ± 0.05
	AP [8]	90	92.83 ± 0.04	92.43 ± 0.10	90.34 ± 0.05
	SDAP [7]	48	93.78 ± 0.04	92.17 ± 0.10	91.61 ± 0.05
	α-AP [3]	48	92.55 ± 0.05	86.10 ± 0.15	89.96 ± 0.07
	ω-AP [3]	48	92.48 ± 0.03	85.79 ± 0.14	89.86 ± 0.04
	Area-WS-AP	48	96.61 ± 0.03	95.30 ± 0.03	95.44 ± 0.04
	Dyn-WS-AP	48	93.88 ± 0.03	88.70 ± 0.12	91.76 ± 0.04
	Vol-WS-AP	48	96.79 ± 0.02	95.09 ± 0.11	95.69 ± 0.03
	Area-PWS-AP	48	96.73 ± 0.04	95.46 ± 0.08	95.60 ± 0.05
	Dyn-PWS-AP	48	94.07 ± 0.04	88.60 ± 0.16	92.01 ± 0.06
	Vol-PWS-AP	48	96.89 ± 0.05	95.33 ± 0.17	95.82 ± 0.06

Table 4 .

 4 Classification result of Zurich dataset obtained by different methods using the default 4-connectivity and 1-byte quantization.

	Method	Dim.			Classification result per class
		Roads	Buildings	Trees	Grass	Bare Soil	Railways	Swimming Pools
	RGB

Acknowledgements

This work was partially supported by the ANR Multiscale project under the reference ANR-18-CE23-0022. The authors would like to thank Prof. Jon Atli Benediktsson for making available the Reykjavik image.


Etienne Pessard
email: etienne.pessard@ensam.eu

Manon Lavialle

Pascal Laheurte

Paul Didier

Myriam Brochu

High-cycle fatigue behavior of a laser powder bed fusion additive manufactured Ti-6Al-4V titanium: effect of pores and tested volume size

Keywords: high cycle fatigue, additive manufacturing, Ti-6Al-4V, titanium, porosity, size effect, powder bed fusion, Kitagawa-Takahashi diagram

This work is focused on the effect of natural defect on the fatigue resistance of a laser powder bed fusion additively manufactured Ti-6Al-4V titanium. To reveal the fatigue strength variability and its sensitivity to the defect size, push-pull fatigue tests have been undertaken on specimens with different sizes of highly loaded volume of material. In order to easily vary the size of the highly loaded volume, specimens containing different numbers of surface hemispherical shape holes of 600 µm in diameter have been tested. This method also allowed to test small volume which triggered crack initiation from microstructural features.

The fatigue damage mechanisms observed and the average natural defect size measured on the failure surfaces depend on the size of the highly stressed region. A higher fatigue strength is observed for smaller stressed volumes and defect free regions. To reduce the impact lack-offusion on fatigue and increase the probability of triggering crack initiation from a microstructural feature, the specimens were built in the horizontal direction. For specimens where fatigue cracks initiated at natural discontinuities, the results reported in a Kitagawa-Takahashi diagram revealed a critical defect size ( √ area) in the range of 30 µm. In addition, a probabilistic approach based on the weakest link theory is proposed. The model describes a probabilistic Kitagawa-Takahashi diagram accounting for the size of both the highly stressed volume and the natural defect.

Introduction

In LPBF (Laser Powder Bed Fusion) components, there is a consensus that the presence of defects is the factor that limits the material's fatigue strength [START_REF] Bian | Mechanical properties and microstructural features of direct laserdeposited ti-6al-4v[END_REF][START_REF] Rafi | A comparison of the tensile, fatigue, and fracture behavior of ti-6al-4v and 15-5 PH stainless steel parts made by selective laser melting[END_REF][START_REF] Facchini | Ductility of a ti-6al-4v alloy produced by selective laser melting of prealloyed powders[END_REF][START_REF] Le | Interpretation of the fatigue anisotropy of additively manufactured ta6v alloys via a fracture mechanics approach[END_REF]. It is recognized by several authors that the variability in pore size, shape and location is pronounced, even under closely controlled manufacturing conditions [START_REF] Le | Interpretation of the fatigue anisotropy of additively manufactured ta6v alloys via a fracture mechanics approach[END_REF][START_REF] Günther | Fatigue life of additively manufactured ti-6al-4v in the very high cycle fatigue regime[END_REF][START_REF] Greitemeier | Fatigue performance of additive manufactured TiAl6v4 using electron and laser beam melting[END_REF]. In addition, the microstructure resulting from layer manufacturing may also show other heterogeneities such as in the grain size and shape as discussed in the literature review by Kok et al. [START_REF] Kok | Anisotropy and heterogeneity of microstructure and mechanical properties in metal additive manufacturing: A critical review[END_REF].

The Ti-6Al-4V titanium alloy studied in this work is one of the most common alloys manufactured by LPBF with applications in many industries, in particular the aeronautic and the medical domains. The monotonic mechanical properties of the Ti-6Al-4V alloy processed by LPBF are close to those of its hot rolled counterpart [START_REF] Bian | Mechanical properties and microstructural features of direct laserdeposited ti-6al-4v[END_REF][START_REF] Rafi | A comparison of the tensile, fatigue, and fracture behavior of ti-6al-4v and 15-5 PH stainless steel parts made by selective laser melting[END_REF][START_REF] Facchini | Ductility of a ti-6al-4v alloy produced by selective laser melting of prealloyed powders[END_REF][START_REF] Le | Interpretation of the fatigue anisotropy of additively manufactured ta6v alloys via a fracture mechanics approach[END_REF]. Nevertheless, the presence of pores, typical of LPBF manufacturing, reduces the average fatigue strength and increases its dispersion [START_REF] Le | Interpretation of the fatigue anisotropy of additively manufactured ta6v alloys via a fracture mechanics approach[END_REF][START_REF] Günther | Fatigue life of additively manufactured ti-6al-4v in the very high cycle fatigue regime[END_REF][START_REF] Greitemeier | Fatigue performance of additive manufactured TiAl6v4 using electron and laser beam melting[END_REF].

From the literature, all fatigue specimens made of Ti-6Al-4V without a HIP (Hot Isostatic Pressure) treatment, encountered crack initiation from process related discontinuities, typically, gas pores or lack of fusion pores [START_REF] Günther | Fatigue life of additively manufactured ti-6al-4v in the very high cycle fatigue regime[END_REF][START_REF] Greitemeier | Fatigue performance of additive manufactured TiAl6v4 using electron and laser beam melting[END_REF][START_REF] Fatemi | Multiaxial fatigue behavior of wrought and additive manufactured ti-6al-4v including surface finish effect[END_REF][START_REF] Fatemi | Torsional fatigue behavior of wrought and additive manufactured ti-6al-4v by powder bed fusion including surface finish effect[END_REF][START_REF] Wycisk | Effects of defects in laser additive manufactured ti-6al-4v on fatigue properties[END_REF]. On the other hand, for HIP-LPBF-Ti-6Al-4V, the literature data agree that the material fatigue strength is closed to its hot rolled counterpart [START_REF] Greitemeier | Fatigue performance of additive manufactured TiAl6v4 using electron and laser beam melting[END_REF][START_REF] Wycisk | Fatigue performance of laser additive manufactured ti-6al-4v in very high cycle fatigue regime up to 10 9 cycles[END_REF][START_REF] Chastand | Comparative study of fatigue properties of ti-6al-4v specimens built by electron beam melting (ebm) and selective laser melting (slm)[END_REF][START_REF] Vayssette | Numerical modelling of surface roughness effect on the fatigue behavior of ti-6al-4v obtained by additive manufacturing[END_REF].

HIPping also efficiently reduces the scatter in the fatigue strength that characterizes LPBF-Ti-6Al-4V [START_REF] Chastand | Comparative study of fatigue properties of ti-6al-4v specimens built by electron beam melting (ebm) and selective laser melting (slm)[END_REF][START_REF] Vayssette | Numerical modelling of surface roughness effect on the fatigue behavior of ti-6al-4v obtained by additive manufacturing[END_REF]. In HIP specimens, prior-β grain boundaries and/or adjacent α colonies have been found to play a crucial role in fatigue crack initiation [START_REF] Cao | A review of the fatigue properties of additively manufactured ti-6al-4v[END_REF]. To the author's knowledge, the fatigue resistance of un-HIPped defect free LPBF-Ti-6Al-4V samples has yet to be characterized.

It is generally accepted that the fatigue strength of metals is sensitive to the tested volume.

The so-called size effect was first studied in the 1960s by Pogoretskii who observed that fatigue strength decreased with an increase in specimen length [START_REF] Pogoretskii | Effect of test piece length on the fatigue strength of steel in air[END_REF]. The size effect is most often revealed when testing materials containing discontinuities like inclusions in laminated steels [START_REF] Beretta | Defect tolerant design of automotive components[END_REF][START_REF] Beretta | Sif and threshold for small cracks at small notches under torsion[END_REF] or pores in cast alloys [START_REF] Ai | Probabilistic modeling of fatigue life distribution and size effect of components with random defects[END_REF][START_REF] Khoukhi | Experimental investigation of the size effect in high cycle fatigue: Role of the defect population in cast aluminium alloys[END_REF]. It has has also been observed in materials having for weakest link microstructural features such as Al 7050 alloys [START_REF] Abroug | Hcf of aa7050 alloy containing surface defects: Study of the statistical size effect[END_REF]. Because of the size effect, the experimentally determined fatigue strength can be influenced by the size and geometry of the specimens used. In the case of AM-materials the minimum and maximum pore sizes observed at the fatigue crack initiation site are limited by the tested volume. In the work by Hu et al. [START_REF] Hu | A new approach to correlate the defect population with the fatigue life of selective laser melted ti-6al-4v alloy[END_REF]on a LPBF-Ti-6Al-4V, the lack of fusion pore sizes observed at the initiation sites are comprised between 30 et 72 µm, even if 21 specimen were tested. This small range illustrates the difficulty in characterising the defect size sensitivity in fatigue using only one specimen geometry. The solution to this problem would therefore be to use very small specimens to obtain failure from small defects or even without a defect and to use very large specimens to characterize the sensitivity to the largest defects.

In order to simplify the experimental problems caused by using different specimens geometries, different numbers of hemispherical surface holes have been machined into the specimens to modify the size of the highly stressed volume. A similar strategy, has been successful used by Abroug et al. [START_REF] Abroug | Hcf of aa7050 alloy containing surface defects: Study of the statistical size effect[END_REF] on a 7050 aluminium alloy.

The goal of this work is to study the effect of the tested volume size on the fatigue strength at 2 × 10 6 cycles of LPBF-Ti-6Al-4V. Three specific objectives are treated: 1) to determine the fatigue strength sensitivity to the natural defect size, 2) to characterize the "defect-free" material fatigue strength for cracks initiating from micostructural features not classified as defects, 3) to characterize the size effect for "defect-free" samples and to compare the behaviour with the results gathered in 1).

In order to fulfil these objectives, a vast experimental fatigue campaign under push-pull loading conditions (load ratio of R=-1) has been undertaken. In the first part of this paper, the manufacturing process used to obtained specimens with different highly stressed volumes is presented. The material, its microstructure and the pore population, characterized by optical microscopy, are then presented. In the second section, the fatigue strengths determined at 2 × 10 6 cycles for four specimen configurations with different Highly Stressed Volumes (noted HSV in this paper). Finally, a probabilistic approach that leads to a Kitagawa-Takahashi diagram [START_REF] Kitagawa | Applicability of fracture mechanics to very small cracks[END_REF] is proposed and discussed.

Experimental methodology

Specimen manufacturing

Cylindrical bars were manufactured using a SLM280 HL machine with a 278×278×325 mm 3 chamber, with a regulated argon atmosphere containing less than 0.1% in volume of oxygen. The specimens were built in the horizontal direction to reduce the impact of the lack-of-fusion on fatigue and increase the probability of triggering crack initiation from a microstructural feature as shown in Fig. 1, on a plateform preheated at 200 • C.

The powder layer thickness, set to 30 µm, was controlled by the vertical displacement of the building platform. A laser with a spot diameter of about 70 µm traveled in a continuous scan mode at a scanning speed of V=775 mm/s. A cross hatching strategy was used where the scanning direction is switched by 67 • between successive layers. The hatch (distance between two laser tracks) was set to 100 µm and the laser power to 175 W.

All bars were removed from the build plate and then annealed at 850 • C for 2 hours under vacuum and cooled down within the furnace to obtain an alpha + beta microstructure. Fatigue specimens with the dimensions given in Fig. 2 were machined from the heat treated bars. The reduced section of the specimens were manually polished with SiC abrasive paper to provide a mirror-like surface with a roughness of Ra< 0.1 µm. Hemispherical artificial defects with a diameter of 600 µm were then machined on the surface of some specimens by micro-drilling using a carbide tool. After drilling, the defects were scanned with a 3D BRUKER profilometer to Build Direction (Z) SLM built geometry measure their depth. An average depth of 300 µm±5 µm was obtained for 20 measurements. The notches have been distributed, in four lines of the specimen (situated at 0, 45, 90 and 135 • ) around the specimen contour to facilitate micro drilling. The hole-position on the sample axis has been chosen to avoid interaction between defects. On a given line the distance between defects is equal to 3mm. The distance between 2 drilling planes is 0.75mm to avoid positioning two defects in the same cross section.

A final heat treatment was carried out at 650 • C for 4 hours under vacuum followed by ambient cooling to relax any potential residual stresses introduced by micro-drilling. A sample containing 32 artificial defect is shown in Fig. 6 a).

Material, microstructure and natural pores

The Ti-6Al-4V alloy used in this study is grade 23 ELI dedicated to aeronautical applications.

Its standradized chemical composition as per ASTM F3001-14 in weight fraction is given in Table 1.

Element Ti

Al V C Fe H N O wt% Bal. 5.5-6.0 3.5-4.5 ≤ 0.08 ≤ 0.25 ≤ 0.012 ≤ 0.05 ≤ 0.13

Table 1: Standardized chemical compositions of the Ti-6Al-4V alloy as per ASTM F3001-14 [START_REF]Standard Specification for Additive Manufacturing Titanium-6 Aluminum-4 Vanadium ELI (Extra Low Interstitial) with Powder Bed FusionCited By[END_REF].

The Ti-6Al-4V powder, obtained by inert gas atomization, was supplied by TLS (Technik GmbH&Co Speziapulver-Germany). The particle size distribution was measured using a dynamic image analysis instrument "CAMSIZER XT" that provides the following results: D10=36 µm, D50=46 µm, D90=53 µm. This indicates that 10% in volume of the powder particles have a diameter smaller than 36 µm and 10% larger than 53 µm.

To characterize the microstructure and the pore size distribution, eleven cross-sections were extracted from one drilled and heat treated specimen and prepared for metallographic observations. The microstructure of the polished and etched samples was characterised using an optical microscope. A columnar microstructure showing grains elongated parralell to the building direction was observed as shown in Fig. 4 a). Typical lack of fusion and gas pores were also observed as shown on Fig. 4 c) and 4 d) respectively.

The pore size distribution, shown in Fig. 5 was characterized using the square root of the pore area ( √ area). Pores with a √ area greater than 7 µm were counted on the 2D images observed by an optical microscope. A total of 2109 pores were characterized on a surface covering 425 mm 2 which resulted in an average defect density of 5 pores/mm 2 .

Fatigue testing

In order to characterize the scale effect, specimens containing different number of surface holes (1, 8 and 32) were tested (see Table 2). In addition, 4 specimens without hemispherical surface holes, were also tested to characterize the fatigue strength for large highly stressed volumes.

The fatigue tests were carried out on a dynamic drive Rumul Testronic machine. An axial cyclic stress was applied normal to the building direction with a load ratio of R = F min /F max = -1.

The test frequency was approximately 105 Hz. The step technique proposed by Maxwell and

Nicholas [START_REF]Fatigue and fracture mechanics[END_REF] developed to rapidly characterize the fatigue strength at a given number of cycles In order to take into account the potential damage caused by the preliminary fatigue cycles, the fatigue strength at 2 × 10 6 cycles was adjusted using equation 1 proposed by Nicholas et al. [START_REF]Fatigue and fracture mechanics[END_REF].

σ f,a = (σ a,n -σ a,n-1 ) * N f 2 * 10 6 + σ a,n-1 (1)
where σ f,a is the adjusted fatigue strength, σ a,n is the stress amplitude at wich failure occurred, σ a,n-1 is the stress amplitude of the previous block load and N f the number of cycles at failure for σ a,n . This method assumes that the damage induced by the previous loading blocks (n-i with i>1) is negligible. This assumption is applied to all the results presented in this paper as no drop in frequency was observed for the load blocks n-i where i>1. Unless indicated, all samples survived at least one full block of 2 × 10 6 cycles.

A fatigue strength correction was also made for specimens with a failure plane located outside the reduced section of the specimen. The corrected nominal fatigue strength was given by the axial force divided by cross-sectional area of the failure plane, which results in slightly lower stress amplitudes than the nominal stress (generally 0.92 to 1.0 times the nominal stress amplitude calculated in the gauge section).

Experimental results and pore effect analysis

Fatigue results

The results of the fatigue tests are reported in Table 2 and Fig. 7. When comparing the average nominal stress amplitudes that caused the failure of specimens without artificial defects (i.e.

specimens D0-i) to the specimens with artificial defects (i.e. specimens D1-i, D8-i, D32-i), two observations stand out. The specimens without artificial defects did not resist significantly higher nominal stress amplitudes than the specimens with one artificial defect. Some specimens (D1-7

and D8-6) with an artificial defect failed due to a nominal stress amplitude that was greater than the maximum nominal strength of smooth samples (D0-1, D0-2, and D0-3). This is counter intuitive given that the presence of an artificial defect locally raises the stress by a factor of 2, as will be shown in section 3.3. Nevertheless, as the number of artificial defects increases, the average nominal stress at failure converges toward a lower value. This trend will be explained by studying the synergy between the artificial defects and natural defects.

Fatigue damage mechanisms

The fractographic observation of all failed specimens revealed the presence of two categories of features at the fatigue crack initiation sites: 1) process related discontinuities (e.g. lack of fusion pores) and 2) microstructural features. This is reported in the 8 th column of Table 2.

As part of the 18 specimens for which a crack initiated at a natural defect, 17 were caused by a lack of fusion pore (LoF) located at the surface or less than 20 µm from the surface. As illustrated in Fig. 8, LoF defects have been observed at the crack initiation sites of 13 specimens with artificial defects and 4 specimens without artificial defects. perpendicular to the applied stress) [START_REF] Murakami | Effect of mean stress on the fatigue strength of high-strength steels containing small defects or nonmetallic inclusions[END_REF]. These values are reported in the last column of Table 2. cracks and have a detrimental effect on the fatigue strength. An observation of the overall fracture surface shows that the LoF pores are aligned between melt pools. This kind of failure surface, not much discussed in the literature, is characteristic of delamination which is occasionally reported in L-PBF materials. Andreau et al. [START_REF] Andreau | A competition between the contour and hatching zones on the high cycle fatigue behaviour of a 316l stainless steel: Analyzed using x-ray computed tomography[END_REF] observed delamination on a 316L L-PBF stainless steel in samples where pores covered more than 10% of the cross section. Similarly, the failure surface of the delaminated sample presented in Fig. 9, also showed a high surface fraction of pores (in black on the failure surface).

For 7 out of 25 specimens, no pores were observed at the crack initiation sites as shown in Fig. 10. In these cases, it was assumed that cracks initiated from the microstructure, most probably from α laths or colonies as report in the literature [START_REF] Cao | A review of the fatigue properties of additively manufactured ti-6al-4v[END_REF][START_REF] Xu | The influence of building direction on the fatigue crack propagation behavior of ti6al4v alloy produced by selective laser melting[END_REF][START_REF] Ellyson | Characterization of bending vibration fatigue of slm fabricated ti-6al-4v[END_REF] for similar alloys.

Impact of artifical defects on fatigue strength

Finite element simulation of the elastic stresses around an artificial defect were conducted using the following material elastic constants: E = 110 GPa and ν =0. notched cylindrical specimens (K t = 2.72) under push-pull loads. Therefore, it is assumed that the local stress at the root of the artificial defect can be estimated by:

σ f,a,local = K t σ a,nominal (2)
On the plane of maximal normal stress, the K t remains quasi constant around the defect with a decrease of only 4.2% when comparing stress value in the depth and at the surface. Equation 2was used to calculated σ f,a,local reported in Table 2 for all of the samples identified as "ADmicrostructure".

For specimen with a natural defects found in the vicinity of an artificial defect (AD-natural defect), the local stress at the tip of the natural defect is reported. The results of finite element analysis was used to quantify the stress concentration factor (K t,d * ) for specific positions underneath the artificial defect, d * , as shown in Fig. 11 and Fig. 12. The natural defect is therefore not explicitly modelled but the stress gradient in the vincity of the hemispherical hole is take into account.

For each specimen, the local stress was quantified at the tip of the initiation defect using equa-tion 3: The local fatigue strength σ f,a,local is reported in the 7 th column of Table 2. These values are calculated at the tip of the artificial defect for all specimens with the crack initiation type "ADmicrostructure" and at the tip of the natural defect for the specimens with the "AD-natural defect" initiation type.

σ f,a,local = K t,d * σ f,a,nominal (3)
In Fig. 13, the local fatigue strength of all sample are plotted as a function of square root area √ area of the defect observed at the crack initiation site. This diagram is presented on a log-log scale in order to appreciate the power law relationship that typically describes the influence of large defects on the fatigue strength as proposed by Kitagawa-Takahashi in 1976 [START_REF] Kitagawa | Applicability of fracture mechanics to very small cracks[END_REF]. For crack initiation at microstructural features (i.e. defect-free), the fatigue strength is plotted at √ area = 1 µm.

In presence of natural defects, the data obtained from specimen with and without hemispherical surface hole presented on Fig. 13 are not significantly different. This indicates that the proposed tress correction for presence of natural defect in the vicinity of a hole is fair.

Fig. 13 demonstrates the detrimental effect of defects on the fatigue strength. For all specimens containing a natural defect with √ area > 19 µm , the fatigue strength can be estimated by a power law relationship of the defect size ( √ area) with an exponent of -0.5 (σ f,a,local = B( √ area) 0.5 ).

Such a power law relationship well describes the linear elastic fracture mechanics principles where the fatigue strength can be expressed by:

σ a = ∆K th F π √ area (4)
where F is a dimensionless factor which equal to 0.65 for an hemispherical surface defected considered as an pre-existing crack in a body of infinite size. For load ratios smaller or equal to 0 (R≤0) it is generally recognized that the positive portion of the load is the sole contributor to long crack propagation in metals, as proposed in ASTM E647 [START_REF]Standard Test Method for Measurement of Fatigue Crack Growth Rates[END_REF]. Therefore, in equation 4, the expression of delta ∆K = K amp = K max has been used and ∆σ can be replaced by σ max . µm.

In the next section an attempt to correlate the dispersion in the measured fatigue strength to the size of the Highly Stressed Volume (HSV) is presented. In addition the scatter in ∆K th will also be accounted for in a probabilistic Kitagawa-Takahashi diagram.

Probabilistic Kitagawa-Takahashi diagram

Analysis of the volume effect

To quantify the size of the HSV the approach proposed by Sonsino et al. [START_REF] Sonsino | Transferability of material data for the example of a randomly loaded forged truck stub axle[END_REF] is used. These authors proposed that fatigue damage on component experiencing a stress gradient occurs essentially in the HSV. The authors defined the HSV to be the volume within which the Von-Mises stress amplitude is greater than 90% of the maximum stress amplitude in the component. Here, this principle is used to quantified the HSV in samples containing a stress gradient caused by the presence of the artificial defects. For the specimens described in this paper, V90% equals 2.15, 17.2 and 68.9 ×10 -3 mm 3 respectively for samples containing 1, 8 and 32 hemispherical holes.

For specimen without artificial defect, the HSV (V90%) is theoretically the volume of the reduced section, which equals 770 mm 3 . Nevertheless, experiments performed by Andreau et al.

[26] on a 316L stainless steel showed that cracks initiate at the surface even in the presence of internal defects that are 4 to 10 times larger than the surface defects. Based on this observation, it is proposed that fatigue damage mainly occurs within a surface layer or skin with a thickness of 20 µm, leading to a V90% of 4.4 mm 3 . This thickness has been chosen to be equal to the α colony size, which is known to play a crucial role in crack initiation [START_REF] Cao | A review of the fatigue properties of additively manufactured ti-6al-4v[END_REF][START_REF] Ellyson | Characterization of bending vibration fatigue of wbd fabricated ti-6al-4v[END_REF]. Neikter et al. [START_REF] Neikter | Microstructural characterization and comparison of ti-6al-4v manufactured with different additive manufacturing processes[END_REF] proposed a typical colony size of 20 µm in their extensive characterization work on LPBF-Ti-6Al-4V.

In Fig. 14, the evolution of the local fatigue strength σ f,a,local as a function of the highly stressed volume is presented for all specimens. For specimens with crack initiation at a pore (♦), the fatigue strength is decreasing with the HSV. This can be explained by an increasing probability of encountering a large pore within larger HSV. The pore size reported in Table 3 is in agreement with this statement. This size effect has been reported by several authors who account for it using extreme values defect distribution [START_REF] Le | Interpretation of the fatigue anisotropy of additively manufactured ta6v alloys via a fracture mechanics approach[END_REF][START_REF] Murakami | Effects of small defects and nonmetallic inclusions[END_REF][START_REF] Makkonen | Size effect in fatigue based on the extreme value distribution of defects[END_REF].

For specimen with no observable defects at the crack initiation site (•), the average fatigue strength is even more sensitive to the HSV size. This size effect can be explain by microstructural heterogeneities. A larger HSV increases the probability of having a large colony of alpha laths favourably oriented for slip. For the smallest HSV, the fatigue strength reached values as high as 663 MPa for specimen D1-7. These results illustrate that the local fatigue strength of the LPBF material can be very high, even higher than those usually obtained for a conventional specimen geometry with a HIP treatment (HIP-LPBF-Ti-6Al-4V -around 510 MPa [START_REF] Vayssette | Numerical modelling of surface roughness effect on the fatigue behavior of ti-6al-4v obtained by additive manufacturing[END_REF]) and closed to those of hot rolled Ti-6Al-4V material (around 675 MPa [START_REF] Vayssette | Numerical modelling of surface roughness effect on the fatigue behavior of ti-6al-4v obtained by additive manufacturing[END_REF])

A probabilistic approach to model the volume effect

In this section, a probabilistic Kitagawa-Takahashi diagram that accounts for the effect of the HSV on the fatigue strength and the experimental dispersion, measured in terms of ∆K th is pro- symbols for failures that initiated at a natural defect (diamonds) and microstructure (circles). For all specimens, when a crack initiated from a natural defect at a smooth surface (not in an artificial defect), the result is reported in the category of the large HSV (4,4 mm 3 ). The size of spheres is proportionate to the size of the tested HSV.

Failure probability for defect free fatigue crack initiation

To model the effect of the HSV on the fatigue strength when the microstructure controls the fatigue strength, a crack initiation threshold is defined at the mesoscopic scale (at the grain scale or the alpha colony scale) using a Weibull distribution. In the following, for convenience, the maximum principal stress criterion (σ a for an axial load) is chosen, but other more complex thresholds could be used (i.e. equivalent stress, strain, energy, etc.). Hence, the condition for crack initiation is defined by:

σ a σ th (5)
where σ th is the intrinsic fatigue stress necessary to trigger crack initiation in a colony. In order to take into account the inherent stochastic nature of the fatigue resistance at the microstructural scale a two parameter Weibull distribution [START_REF] Weibull | A Statistical Theory of the Strength of Materials, Generalstabens litografiska anstalts förlag[END_REF][START_REF] Weibull | A statistical distribution function of wide applicability[END_REF] is used to describe the intrinsic fatigue strength.

The probability density function proposed to described this distribution is given in Eq.6.

f 01 (σ th ) = m 1 σ th0 σ th σ th0 m 1 -1 exp - σ th σ th0 m 1 (6)
where σ th0 is the scale parameter and m 1 is the shape parameter (or the Weibull exponent) used to reflect the scatter associated with the local intrinsic fatigue strength within the microstructure.

Such a distribution can represent, for example, the effect of colony orientation on the initiation of a fatigue crack. The probability of triggering crack initiation can be expressed as the probability of having the applied mesoscopic stress amplitude reaching the local fatigue strength as in equations 7 and 8.

P F 01 = (σ th < σ a ) = σ eq 0 f 01 (σ th )dσ a (7)
P F 01 = 1 -exp - σ a σ th0 m 1 (8)
In order to account for the size of the HSV in the probability function, the weakest link theory proposed by Freudenthal [38] is used. In this framework, the probability of survival (1 -P F 1 ) is defined as the product of the survival probabilities of each elementary volume V 0 in the volume

V Ω . 1 -P F 1 = Π V Ω 1 -P F 01 (M) (9)
resulting in:

P F 1 = 1 -exp - 1 V 0 V Ω σ a σ th0 m 1 dV (10)
Within the experimental campaign presented in Table 3, four sample types with a specific volume sizes ,V Ω ,were tested. The failure probability of each sample type is described by equation 11.

P F 1 = 1 -exp - V Ω V 0 σ a σ th0 m 1 (11)
From Eq. 11, the relation between the fatigue strengths noted σ a,1 and σ a,2 of two different loading volume sizes V Ω1 and V Ω2 can be written as:

σ a,2 σ a,1 = V Ω1 V Ω2 1/m 1 (12)
The parameter m 1 can therefore be obtained experimentally by fitting a power equation to the fatigue strength results as a function of the tested volume V Ω as illustrated in Fig. 14. This leads to a m 1 value of 10.98 for a failure probability of 0.5.

Then, knowing m 1 and the average fatigue strength for an HSV, the gamma function is used to calculate the Weibull scale factor (see Eq.13).

σ th0 = σ a Γ (1 + 1/m 1 ) (13)
where Γ(t) = ∞ 0 x t-1 e -x is the gamma function [START_REF] Abramowitz | Handbook of Mathematical Functions[END_REF]. Assuming V 0 is equal to the volume of a cube with a side length of 20 µm can be used to model a α colony, this lead to a Weibull scale factor of the theoretical local intrinsic fatigue strength σ th of 1.73x10 5 MPa.

Failure probability due to propagation from natural defects

As shown previously, the evolution of the fatigue strength as a function of the natural defect size (Fig. 13) is well described using the LEFM approach defined by Eq.4 when √ area > 30 µm.

To estimate the failure probability caused by the propagation of a crack from a natural defect, a linear elastic fracture mechanic approach is therefore proposed. As defined in Eq.14, according to LEFM and the Murakami approach [START_REF] Murakami | Effects of small defects and nonmetallic inclusions[END_REF], a defect of size √ area, considered as a pre-existing crack will not propagate under cyclic loads if the applied stress intensity range, ∆K I , is less than the material crack propagation threshold, ∆K th .

∆K I = 0.65 σ a π √ area ≤ ∆K th (14)
As for the intrinsic local fatigue resistance, the stochastic character of the long crack propagation threshold is modelled using a Weibull distribution [START_REF] Weibull | A Statistical Theory of the Strength of Materials, Generalstabens litografiska anstalts förlag[END_REF][START_REF] Weibull | A statistical distribution function of wide applicability[END_REF] for which the probability density function is given by:

f 02 (∆K th ) = m 2 ∆K th0 ∆K th ∆K th0 m 2 -1 exp - ∆K th ∆K th0 m 2 (15)
In this case, the shape factor or the Weibull exponent, m 2 , governs the scatter and ∆K th0 is the scale factor of the distribution.

The probability of a crack propagating, corresponds to the probability that ∆K is above the material ∆K th as expressed in Eq.16:

P F 2 = 1 -exp - ∆K I ∆K th0 m 2 (16)
Assuming that all specimens with a natural defect at the initiation site broke when the stress intensity factor range reached the crack propagation threshold, the cumulative probability of the crack propagation threshold can be ploted (as in Fig. 15). In this figure the two points with the lowest defect size locate on the Kitagawa-Takahashi plateau were not considered. Note that two data points in Fig. 15 do not seem to follow the Weibull trend. These two data with a particular high crack propagation threshold were not taken into account to fit the Weibull distribution. The crack propagation threshold data presented in Fig. 15

Combined failure probability

Finally, it is proposed that the survival probability of a volume V Ω , failing by one or the other damage scenarios, is equal to the product of the two survival probabilities described by Eq.11 and

Eq.16. Essentially, the weakest link hypothesis [38], is employed once again, which assumes that the two damage mechanisms are independent, leading to equation 17:

1 -P F = (1 -P F 1 )(1 -P F 2 ) (17)
The combined probability of failure for a tested volume size V Ω can thus be written as in equation 18 describing a bimodal Weibull distribution.:

P F = 1 -exp            -             V Ω V 0 σ a σ th0 m 1 +             F σ a π √ area ∆K th0             m 2                        (18)
The volume size effect on the microstructural fatigue strength is explicitly taken into account via the terms V Ω V 0 . In addition the defect size is directly introduced as an input parameter. The influence of the tested volume size on the probability of finding a large defect is not modelled.

Eq.18, plotted as a function of the defect size, √ area, results in a probabilistic Kitagawa-Takahashi diagram as shown in Fig. 16. The probabilistic approach correctly predict the fatigue strength trends experimentally observed. It also predict a decreasing critical defect size with a decreasing HSV.

As shown in Fig. 16, care should be taken when using a deterministic Kitagawa-Takahashi diagram because it often neglects the volume or size effect on the fatigue plateau for small defects sizes. Particular care must be taken when the dimensioning curves are derived from tests carried out on small test specimens. The proposed approach is a first step to consider, at the same time, the fatigue strength sensitivity to the loaded volume size in presence of two different damage mechanisms (initiation with and without natural defects).

The comparison of the fatigue strength without natural defects to those obtained from Ti-6AL-4V LPBF HIP and hot-rolled materials [START_REF] Vayssette | Numerical modelling of surface roughness effect on the fatigue behavior of ti-6al-4v obtained by additive manufacturing[END_REF] illustrates the importance of better controlling the Ti-6AL-4V LPBF microstructre. According to the results obtained in this study, removing the natural defects (LoF, gaseous pore, etc.) would only have a limited impact on improving the fatigue resistance of Ti-6AL-4V LPBF. In the absence of a natural defect, the high sensitivity of the fatigue strength to the volume size illustrates significant microstructure variability which results in high scatter and a limited average value. In the absence of natural defects, the optimization of LPBF processes in order to guarantee a more homogeneous microstructures and less sensitive to the scale effect in fatigue is a challenge to be taken up to guarantee better resistance of Ti-6Al-4V-LPBF components in fatigue.

Conclusion

This paper presents the results of a fatigue test campaign performed on Ti-6Al-4V manufac- to litterture data [START_REF] Vayssette | Numerical modelling of surface roughness effect on the fatigue behavior of ti-6al-4v obtained by additive manufacturing[END_REF].

size of the Highly Stressed Volume (HSV). For specimens with small highly stressed volumes it was possible to characterize the material fatigue strength without pores.

The following conclusions can be drawn from the result analysis:

• In the absence of pores in the HSV, the material fatigue strength ranges from 342 to 663

MPa and is very sensitive to the HSV size.

• The critical defect size affecting the fatigue strength of the LPBF Ti-6Al-4V is approximately 30 µm for push-pull loads and fatigue life of 10 6 cycles.

• When a crack initiates from a pore, the fatigue strength decreases with the increasing size of the HSV. This can be explained by the direct relationship between the probability of finding a large defect and the size of the tested volume.

• The fatigue sensitivity to the defect size is correctly predicted using a linear elastic fracture mechanic approach, in crack opening mode I and the Murakami defect size parameter √ area.

• A probabilistic approach, based on the size effect is proposed to predict the occurence of crack initiation from a microstructure feature or from a natural defects.

Fig. 1 :

 1 Fig. 1: Ti-6AL-4V fatigue sample in the SLM 280HL machine.

Fig. 2 :

 2 Fig. 2: Cylindrical fatigue sample dimensions with the as-built geometry, (dimensions are in mm).

Fig. 3 :

 3 Fig. 3: Hemispherical holes surface spatial distribution on the specimenmerged (dimensions are in mm).

Fig. 4 :

 4 Fig. 4: Optical microscopy microstructure observation at different magnification showing a) the columnar microstructure, c) gas pore, b) and d) typical lack of fusion defects.

2 Fig. 5 :

 25 Fig. 5: Cumulative size distribution of the defects for √ area ≥ 7µm

Fig. 6 :

 6 Fig.6: a) Picture of the specimen containing 32 artificial defects, b) at higher magnification a crack that initiated from an artificial defect is surrounded in red.

148Fig. 7 :

 7 Fig. 7: Average and range of the Nominal Stress Amplitude at Failure in function of the artificial defects number.

34 .Fig. 8 :

 348 Fig. 8: Fracture surface of samples a-b) D1-3 where the lack of fusion that caused crack initiation is not in the vicinity of an artificial defect and c-d) D32-5 showing crack initiation from lack of fusion pore, located at the root of an artificial defect.

Fig. 9 :Fig. 10 :

 910 Fig. 9: Fracture surface of sample D32-8 showing a layer delamination.

Fig. 11 :

 11 Fig. 11: a) Finite Element simulation in elasticity of a elementary volume containing a 600 µm diameter hemispherical defect, the arbitrary intensity stress load in tension is 1MPa, maximum principal stress σ I field are given by the color code, b) stress distribution in depth from the free surface of the hemispherical surface hole.

Fig. 13

 13 Fig.13can also be use to estimate the material's resistance to fatigue in the absence of a crack initiating discontinuity. The results presented by open circles were obtained for specimens with no observable pores in the vicinity of the crack initiation region. These results (AD-microstructure) ranged from 342 to 663 MPa. It can also be observed that the presence of defects smaller than 30 µm at the crack initiation sites resulted in fatigue strengths of 341, 383, 384 and 389 MPa, which is the same range as the defect-free samples. It suggests that the critical defect size, which is detrimental to the fatigue strength of the studied LPBF-Ti-6Al-4V is approximately √ area =30

Fig. 12 :Fig. 13 :

 1213 Fig. 12: Fracture surface of samples a) D32-4 and b) D32-2 illustrates the distances d* .

Fig. 14 :

 14 Fig. 14: Evolution of the local fatigue strength in function of the size of HSV. Results are presented using distinct

2 [ 42 ]Fig. 15 :

 24215 Fig. 15: ∆ Kth cumulative probabilty and Weibull function identified.

3 HSV=4.4 mm 3 Fig. 16 :

 3316 Fig. 16: Kitagawa-Takahashi diagram, corresponding to Eq.18, for differents HSV and failure probabilities compared

Table 2 :

 2 Fatigue test results fo a target fatigue life of 2x10 6 cycles

		Specimen	Defects Number at Failure (MPa) Average (MPa) Number (10 6 cycles) (MPa) Nominal Stress Amplitude Step N f σ f,a,local	Initiation Type	√ area (µm)
		D0-1	0	210		1	0.92	199 2	natural defect	116
		D0-2 D0-3	0 0	250 270	275	1 5	0.20 1.73	232 2 248 1	natural defect natural defect	150 73
		D0-4	0	370		10	1.63	341 1	natural defect	26
		D1-1	1	170		2	0.72	147 1	natural defect	169
		D1-2	1	250		3	0.50	357	AD-natural defect	35
		D1-3	1	270		4	1.07	249 1	natural defect	74
		D1-4	1	270	270	4	0.22	252	natural defect	55
		D1-5	1	290		9	0.57	551	AD-microstructure	/
		D1-6	1	290		5	0.69	554	AD-microstructure	/
		D1-7	1	350		11	0.17	663	AD-microstructure	/
		D8-1	8	210		4	0.79	290	AD-natural defect	41
		D8-2	8	230		3	0.2	307	AD-natural defect	31
		D8-3 D8-4	8 8	250 250	257	4 4	0.31 0.19	384 264	AD-natural defect AD-natural defect	25 54
		D8-5	8	270		7	1.15	522	AD-microstructure	/
		D8-6	8	330		10	0.06	621	AD-microstructure	/
		D32-1	32	130		1	1.05	127 2	AD-natural defect	225
		D32-2	32	150		1	0.56	161 2	AD-natural defect	127
		D32-3	32	230		6	0.36	389	AD-natural defect	13
		D32-4	32	250	201	7	0.33	383	AD-natural defect	19
		D32-5	32	270		8	0.76	310	AD-natural defect	81
		D32-6	32	190		4	0.10	342	AD-microstructure	/
		D32-7	32	190		3	0.44	349	AD-microstructure	/
		D32-8	32	110	110	1	0.51	98 2	Layer delamination	/
	146
	147	in terms of the Murakami parameter	√ area (square root of the defect area projected onto a plane

1 : Corrected stress, fracture plane outside the gauge length. natural defect initiation type : σ f,a,local = σ f,a,nominal 2 : Fracture during the first step. AD-microstructure initiation type : σ f,a,local = K t σ f,a,nominal AD: crack initiation from an artificial defect. AD-natural defect initiation type : σ f,a,local = K t,d * σ f,a,nominal All process related defects, observed at the crack initiation sites were measured and reported

Table 3 :

 3 Average pore size observed at crack initiation sites and percentage of failure without pore for the differents

	HSV tested
	Number of Artificial Defect	1	8	32	0
	HSV (10 -2 mm 3 )	0.215 1.72 6.89 440
	average pore size (µm)	35	39	92	95
	failure without pores (%)	42	33	25	0
	posed.