Zhe Liu
email: zhe.liu@enpc.fr

Renaud Marlet
email: renaud.marlet@enpc.fr

Virtual Line Descriptor and Semi-Local Matching Method for Reliable Feature Correspondence

Finding reliable correspondences between sets of feature points in two images remains challenging in case of ambiguities or strong transformations. In this paper, we define a photometric descriptor for virtual lines that join neighbouring feature points. We show that it can be used in the second-order term of existing graph matchers to significantly improve their accuracy. We also define a semi-local matching method based on this descriptor. We show that it is robust to strong transformations and more accurate than existing graph matchers for scenes with significant occlusions, including for very low inlier rates. Used as a preprocessor to filter outliers from match candidates, it significantly improves the robustness of RANSAC and reduces camera calibration errors.

Introduction

Finding reliable correspondences between sets of feature points in two images is a key step in many computer vision problems, e.g., image registration, camera calibration and object recognition. To this end, feature detectors such as SIFT [START_REF] Lowe | Distinctive image features from scale-invariant keypoints[END_REF], SURF [START_REF] Bay | Speeded-Up Robust Features (SURF)[END_REF], Harris-affine [START_REF] Mikolajczyk | Scale & affine invariant interest point detectors[END_REF] or MSER [START_REF] Matas | Robust wide-baseline stereo from maximally stable extremal regions[END_REF] robustly identify interest points or areas in images. By design, the detected points or areas are salient enough to likely be also salient in other views of the same scene, under different imaging conditions (viewpoint, lighting, orientation, scale, etc.). Besides, these points or areas can be individually described based on their scale, if any, as well as on an abstraction of their photometric neighborhood, e.g., based on the distribution of local gradients. Such feature descriptors include SIFT [START_REF] Lowe | Distinctive image features from scale-invariant keypoints[END_REF], SURF [START_REF] Bay | Speeded-Up Robust Features (SURF)[END_REF] and MSER shape descriptor [START_REF] Forssén | Shape descriptors for maximally stable extremal regions[END_REF]. Like detectors, these descriptors are designed to be robust, to some extent, to variations such as noise or change of viewpoint, orientation or illumination.

Matching detected features in two images based on the similarity of their descriptor often provides good correspondences (inliers). However, it often includes false matches too (outliers). Eliminating those false matches while preserving true correspondences remains challenging for images with ambiguities or strong transformations. Ambiguity usually arises from repetitive patterns (e.g., facade windows) or lack of texture. In this case, the descriptors are not discriminative enough to safely differentiate feature points. There actually is a balance to find as repeatable descriptors tend to be less distinctive, and vice versa. As for strong transformations, they can sometimes be avoided by carefully controlling imaging conditions. Yet some sharp transformations cannot be escaped, e.g., due to strong occlusions, when a foreground object obstructs very different background areas. To get both a high number of matching inliers and a low outlier ratio, just comparing individual feature descriptors is not enough. Global methods are required, such as RANSAC or graph matching.

Feature matching by RANSAC. For rigid transformations, RANSAC-like methods [START_REF] Fischler | Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography[END_REF] can accurately separate inliers from outliers. They randomly sample subsets of correspondences to build a putative model of the transformation (fundamental/homography matrix) and count the number of matches that are compatible with the model. The largest consensus set defines what is to be considered as inliers, other matches being regarded as outliers.

This works well if the inlier rate ρ is high, not if it is low. The reason is that the number of required sampling iterations is on the order of 1/ρ n , where n is the number of correspondences to draw to define a model. (In general, for the fundamental matrix, n ≥ 7 or 8 [START_REF] Hartley | Multiple View Geometry in Computer Vision[END_REF].) Better drawing strategies such as MLESAC [START_REF] Torr | MLESAC: A new robust estimator with application to estimating image geometry[END_REF] or PROSAC [START_REF] Chum | Matching with PROSAC -progressive sample consensus[END_REF] can greatly reduce the number of models to sample, but they are nonetheless not well suited for inlier rates lower than 50%. Only a few methods such as ORSA [START_REF] Moisan | A probabilistic criterion to detect rigid point matches between two images and estimate the fundamental matrix[END_REF] can treat an inlier rate of 10%. Yet in any case, all RANSAC-like methods inherently suffer from a limitation when estimating the fundamental matrix: they cannot eliminate outliers corresponding to points that have matches near their epipolar line but far from the correct location, which may degrade precision.

Graph matching methods. Graph matching is another tool to determine feature point correspondences, with a global consistency criterion. It applies not only to rigid scenes but also to deformable objects. The idea is to construct a graph where vertices are feature points and edges are pairwise relations. Higher order constraints, involving more than two vertices, can be modeled as hyperedges. Graph matching methods try to establish a vertex correspondence between two graphs, satisfying matching constraints or optimizing a global score. Some can also handle inexact matching, allowing different structures to some extent [START_REF] Conte | Thirty years of graph matching in pattern recognition[END_REF].

For 2 nd -order graph matching, many methods use the relative distance between points as constraint [START_REF] Cho | Reweighted random walks for graph matching[END_REF][START_REF] Leordeanu | A spectral technique for correspondence problems using pairwise constraints[END_REF], possibly in combination with angles [START_REF] Berg | Shape matching and object recognition using low distortion correspondence[END_REF]. Feature orientation and scale are used too, e.g., to define an affine transformation predicting the projection of neighboring points [START_REF] Albarelli | Robust game-theoretic inlier selection for bundle adjustment[END_REF]. Some robust pairwise descriptors combine individual feature descriptors too [START_REF] Gu | Critical nets and beta-stable features for image matching[END_REF].

A better matching accuracy or robustness to noise can be achieved with higher order graph matching [START_REF] Chertok | Efficient high order matching[END_REF][START_REF] Lee | Hyper-graph matching via reweighted random walks[END_REF][START_REF] Ren | High order structural matching using dominant cluster analysis[END_REF]. A common 3 rd -order constraint expresses triangle similarity [START_REF] Lee | Hyper-graph matching via reweighted random walks[END_REF]. 4 th -order constraints typically include consistency w.r.t. a local affine transformation [START_REF] Duchenne | A tensor-based algorithm for high-order graph matching[END_REF][START_REF] Zass | Probabilistic graph and hypergraph matching[END_REF]. Graph matchers supporting edges of even higher order can for instance also express projective-invariant potentials [START_REF] Duchenne | A tensor-based algorithm for high-order graph matching[END_REF]. However, despite recent advances in high-order graph matching, the running time and memory consumption remain an issue, especially for large datasets (images with hundreds or thousands of features): the complexity is at least O(N d ) where N is the number of points and d the order. Besides, although some methods explicitly include a treatment of outliers, e.g., using absorbing nodes [START_REF] Lee | Hyper-graph matching via reweighted random walks[END_REF], the inlier rate is still assumed to be relatively large. For instance, Lee et al. [START_REF] Lee | Hyper-graph matching via reweighted random walks[END_REF] only describe experiments with at least 50% inliers (and at most 60 points), and Duchenne et al. [START_REF] Duchenne | A tensor-based algorithm for high-order graph matching[END_REF] show a severe drop of performance when the inlier rate falls below 30% (with less than 100 points).

Our work. Feature descriptors provide 1 st -order photometric information to estimate correspondence likelihood and identify potential matches. All other information used for matching is generally restricted to geometric information, i.e., relative point location in the image. This is the case for RANSAC methods and for most existing graph matchers. Although some authors mention possible extensions of graph matching potentials to photometric information [START_REF] Duchenne | A tensor-based algorithm for high-order graph matching[END_REF], such uses are scarce and tend to translate into quasi-dense matching [START_REF] Ferrari | Simultaneous object recognition and segmentation by image exploration[END_REF]. We propose here a novel, simple and efficient, 2 nd -order photometric criterion.

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Figure 1: Lines (P i , P j ) and (P i , P j ) are unlikely to be similar unless both matches (P i , P i ) and (P j , P j ) are correct. It is based on the fact that for points P i , P j in image I and P i , P j in image I , it is unlikely to find similar photometric information around lines (P i , P j ) and (P i , P j ) unless both (P i , P i ) and (P j , P j ) are correct matches (see Fig. 1). To express this property, we define a virtual line descriptor (VLD) that captures photometric information between two points. The distance between two such descriptors measures the dissimilarity between the corresponding two virtual lines. It can be used in the 2 nd -order term of graph matchers to improve their accuracy.

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We also define a semi-local matching strategy based on VLD that considerably increases the inlier rate. It can be used as a preprocessor to RANSAC methods to improve the quality of match selection. As it can eliminate false matches near epipolar lines, it greatly improves precision. It also considerably reduces the number of iterations, which improves speed.

Virtual line descriptor (VLD)

The general idea of our descriptor for a virtual line between points P i and P j is to consider a regular covering, with some overlap, of an image strip between P i and P j , and use a SIFTlike descriptor to characterize each element of the covering. The global line descriptor is basically the concatenation of the descriptors of each covering element. It inherits SIFT descriptor's robustness to noise and changes of scale, orientation and illumination.

Geometric consistency. Before describing a line, we actually first check a geometric constraint, extending that of Albarelli et al. [START_REF] Albarelli | Robust game-theoretic inlier selection for bundle adjustment[END_REF]. Given matches m i,i =(P i , P i ) and m j, j = (P j , P j ), and assuming that the local transformation around P i is close to a similitude, we define the point Q j in image I as the expected position of P j (cf. Fig. 2):

Q j = P i + s(P i ) s(P i ) R(a(P i ) -a(P i )) --→ P i P j (1)
where s(P) is the scale of feature point P, a(P) is the main orientation at P, and R(α) is the rotation of angle α. Permuting I and I defines a point Q j in image I as the expected position of P j . The transformation error of (P j , P j ) by (P i , P i ) is measured based on distances d i, j = dist(P i , P j ), t i, j = dist(P i , Q j ), e i, j = dist(P j , Q j ), and likewise in I . The normalized and symmetrized score of geometric consistency for matches m i,i and m j, j is defined as:

χ(m i,i , m j, j )=min(η i,i , j, j , η j, j ,i,i ) where η i,i , j, j = e i, j min(d i, j ,t i, j ) = e i , j min(d i , j ,t i , j ) (2)
Matches m i,i and m j, j are considered as consistent w.r.t. geometry iff χ(m i,i , m j, j ) < χ max . In all our experiments, we use a threshold value χ max = 0.5. This fast prefiltering step eliminates many false matches before photometric comparison while preserving most inliners.

! " ! "#$ "#$ Figure 4: For r ≥ r min , the VLD is computed on the q th image scale on r * -radius disks, where r min ≤ r * = r/2 q/2 < r min √ 2.

# $ # % # & '(' !"#$%&'(!") '$(#!"*"('! +,-) '$(#!"*"('! ) * + , # ,
Line covering. For any two points P i and P j in image I at distance d one from another, we consider U inter-point disks (D u ) of radius r = d U+1 centered on points P i + u U+1 --→ P i P j for u ∈{1,...,U} (see Fig. 3). Each disk is then described at image scale s = max(r/r min , 1) where r min is a minimum description radius. In our experiments, we use U = 10 and r min = 5 pixels, which provides a good balance between discrimination and repeatability. In practice, scales can be discretized and precomputed, to avoid rescaling the image for each new pair of points. As for SIFT [START_REF] Lowe | Distinctive image features from scale-invariant keypoints[END_REF], we construct a pyramid of scaled images. In our experiments, a geometric progression with ratio √ 2 proved enough for repeatability. For any disk radius r in the original image I, we thus use scale s * = 2 q/2 for q natural integer such that 2 q/2 ≤ s < 2 (q+1)/2 , i.e., q = 2 log s log 2 . In the scaled image, the disk radius is r * = r/s * (see Fig. 4). Inter-point gradient histogram. The descriptor for disk D u is a single SIFT-like local gradient histogram [START_REF] Lowe | Distinctive image features from scale-invariant keypoints[END_REF]. (SIFT actually defines a grid of 4 × 4 such histograms.) We use V bins (h u,v ) v∈{1,...,V } to represent the distribution: each pixel in the disk votes in the orientation bin corresponding to its gradient (relatively to the line direction), weighted by the gradient magnitude and by a Gaussian-weighted circular window with σ = 3 2 r * . The line histograms are then normalized so that ∑ U u=1 ∑ V v=1 h u,v = 1. In our experiments, like SIFT, we use V = 8. Inter-point orientation. Inter-point orientation is computed as SIFT too, with some adaptation. We construct an orientation histogram for D u using W bins (O u,w ) w∈{0,...,W -1} . As slightly more variations can be expected on the line between two points than on the feature point themselves, we recommend W > V . In all our experiments we use W = 24 (as opposed to 36 for SIFT). In addition, we treat opposite direction together and actually consider the derived histogram ( O u,w ) w∈{0,...,W -1} defined as O u,w = O u,w -O u,(w+W/2) modW . (Only W/2 bins are actually needed to accumulate gradient information.) This reduction of the number of bins preserves enough discrimination while enhancing robustness. The main orientation w * u is finally defined as the bin of the derived histogram with the highest value. We also define normalizing factors γ u to weight each orientation over the whole line:

w * u = argmax w∈{0,...,W -1} O u,w γ u = O u,w * u ∑ U u=1 O u,w * u ≥ 0 (3)
Each disk D u is represented by the histogram (h u,v ) v∈{1,...,V } , the orientation w * u and the normalizing factor γ u . The size of the overall line descriptor is thus U (V + 2).

High contrast suppression. Lines with high contrast might be unreliable, in particular along image edges. In this case, gradients have strong responses in the orthogonal direction, which leads VLDs to be similar and thus not discriminative. For this reason, we define a line contrast indicator: κ = s * Ud ∑ U u=1 O u,w * u . VLDs with contrast κ above given threshold κ max are considered unreliable and discarded. Experimentally, assuming image intensity in the range 0..255, we use κ max = 30. (Low contrast lines can also be discarded but are very unlikely.)

Distance between two VLDs. Given lines l i, j =(P i , P j ) in I and l i , j =(P i , P j ) in I , we now define the distance between their descriptors. For each inter-point disk D u in I and corresponding inter-point disk D u in I , we compute both the difference of the gradient histograms and the difference of the main orientations. Orientations w * u and w * u are compared modulo W/ 2 (most dissimilar orientation), normalized to 1, and weighted by the average of the orientation normalizing factors γ u and γ u , resulting in a value in [0, 1]. The differences of gradient histograms and main orientations are then linearly combined with factor β ∈ [0, 1]:

τ(l, l )=β U ∑ u=1 V ∑ v=1 |h u,v -h u,v | +(1 -β ) U ∑ u=1 γ u + γ u 2 . min(|w * u -w * u |,W -|w * u -w * u |) W/ 2 (4)
VLD-consistency. The VLD-distance between matches m i,i =(P i , P i ) and m j, j =(P j , P j ) is the VLD-distance between the corresponding lines: τ(m i,i , m j, j )=τ(l i, j , l i , j ) ∈ [0, 1]. It can be used in the pairwise score of a graph matcher, e.g., with a contribution of the form exp(-λτ 2 ). Experimentally, we use β = 0.36 and λ = 100. When an all-or-nothing choice is required, matches m i,i and m j, j are said VLD-consistent iff their virtual lines are valid w.r.t. contrast and τ(m i,i , m j, j ) ≤ τ max . In our experiments, we use τ max = 0.35. The matches are said gVLD-consistent iff they are both geometry-and VLD-consistent.

Parameters. Our descriptor may seem to have many parameters but most of them actually are directly imported from SIFT. Experiments just taught us that SIFT standard values could be weakened here for a lighter but still discriminant and robust descriptor. The only specific VLD parameters are the number of disks (U, with minimum radius r min ), the balance between gradients and orientations (β ) and a possible weight when used for a pairwise score (λ ). Parameters such as β and λ could be learned as in [START_REF] Leordeanu | Unsupervised learning for graph matching[END_REF]. K-VLD below uses additionally consistency thresholds for geometry (χ max ), contrast (κ max ) and photometry (τ max ).

K-VLD: a K-connected VLD-based matching method

VLD can be directly used as a pairwise constraint in 2 nd or higher-order graph matching methods. Yet, existing graph matching methods do not scale well to large numbers of matches and, as shown in the experiment section, may perform poorly when the foreground creates background occlusions. Besides, some of them are not well suited for large outlier elimination. In this section, we introduce K-VLD, a novel matching method that overcomes these limitations. It is semi-local in the sense that the score of a match depends on its consistency with neighboring matches. The consistency is both geometric and photometric, using our VLD criterion. The basic idea is that, given a potential match (P i , P i ), if there are in the neighborhood of P i and P i at least K other matches (P j k , P j k ) k∈{1,...,K} that are gVLDconsistent with (P i , P i ), then (P i , P i ) is likely to be a correct match. The method can be seen as a simplified 2 nd -order graph matcher specialized for image features. It provides a binary assessment for each match (correct or not) as well as a consistency score for further filtering.

Neighborhoods. K-connected gVLD-consistency has to apply only within a neighborhood of the points. We adapt the size of the neighborhoods to the density ρ of feature points. Neighborhoods are defined as disks centered on P i and P i , with respective radius B and B . Given a set M of potential matches between I and I , with minimum inlier rate ρ min , then the average number of correct matches K B in a B-neighborhood is:

K B = π B 2 area(I) ρ min |M| B K = K area(I) πρ min |M| (5)
As B should be chosen such that K B ≥ K, we get a definition for the minimum radius B K of the neighborhood (see Eq. 5). Moreover, for stability reason, we exclude neighboring points P j that are too close to P i , within B min pixels. Wrapping up, we say that a match (P j , P j ) is a neighbor of (P i , P i ) iff P j is in the (B min , B K )-annulus centered on P i in I, or P j is in the (B min , B K )-annulus centered on P i in I . (The definition is symmetrical.) In the unlikely case that we discover a posteriori that ρ < ρ min , B K has to be expanded accordingly and the algorithm has to be rerun. In all our experiments, we set ρ min = 3% and B min = 10 pixels. Then, given a set of matches M and a match m ∈ M, we define the following neighborhoods:

• N M (m)={m ∈ M | m and m are neighbors}

• N M,geom (m)={m ∈ M | m and m are geometry-consistent neighbors}⊂N M (m) • N M,gvld (m)={m ∈ M | m and m are gVLD-consistent neighbors}⊂N M,geom (m)
Problem statement. Experimentally, requiring that good matches have at least K gVLD-consistent neighbors eliminates many outliers, but some may still remain, especially in ambiguous scenes. We found that adding an extra constraint on the proportion of geometryconsistent neighbors helped in removing many of these remaining outliers. More formally, given a set of potential matches M, we look for a subset M ⊂Msuch that, for all m ∈ M,

|N M,gvld (m)|≥K and |N M,geom (m)| |N M (m)| ≥ ω min or ∑ m ∈N M (m) χ(m, m ) |N M (m)| ≤ χmax (6)
We are actually interested in a set M * with maximum cardinality satisfying this condition. The absence of ambiguous matches in M can also be imposed (see below).

Algorithm. For efficiency reasons, we actually only look for sets M of large cardinality satisfying equation [START_REF] Chum | Matching with PROSAC -progressive sample consensus[END_REF]. Our algorithm starts with M = M and repeatedly performs:

1. remove all m ∈ M such that N M,gvld (m) < K 2. remove all m ∈ M such that |N M,geom (m)| |N M (m)| < ω min and ∑ m ∈N M (m) χ(m, m ) |N M (m)| > χmax
until no match is removed. Upon termination, which always occurs as |M| strictly decreases (after around 3 iterations, at most 5 in practice), either M = / 0 or M satisfies condition [START_REF] Chum | Matching with PROSAC -progressive sample consensus[END_REF]. In practice, this yields a large set of matches for M, almost never empty. gVLD-consistency is enforced first because it is the strongest condition; the value of M is thus best estimated when performing step 2. In all our experiments, we use K = 3, ω min = 30% and χmax = 1.2.

Dealing with ambiguity. Ambiguous matches, i.e., matches that share a point, correspond to a special kind of outliers. Treating them as ordinary matches when eliminating outliers does not guarantee an ambiguity-free set of final matches. For this reason, a heuristic elimination is often performed to keep at most one match per point, generally by keeping only the one with the best score. But sometimes there is no easy choice, e.g., with ambiguous points on an epipolar line. In that case we can use VLD information to improve disambiguation. For this, we sort the matches in M so that matches m with highest number of gVLDconsistent neighbors are preferred or, if equal, highest average of VLD score among these gVLD-consistent neighbors, i.e., T (m)=∑ m ∈N M,gvld τ(m, m )/|N M,gvld (m)|. More formally, matches are sorted (less likely matches first) according to the order relation:

m M m iff |N M,gvld (m)| < |N M,gvld (m )| or |N M,gvld (m)| = |N M,gvld (m )| and T (m) ≥ T (m ) (7)
And we add a third step to the algorithm:

3. for each m ∈ M in M -sort order, remove m if ∃m ∈ M\{m} s.t. m conflicts with m
It is to be performed only once as all conflicts get removed and none can later be created.

The sorting can also be used in step 1 and 2 of the algorithm for picking m ∈ M, updating M as matches are removed. However, it does lead to much better results experimentally.

Optimizations and heuristics. Given a match m, we need to count the number of gVLD-consistent neighbors m , which requires computing τ(m, m ). To avoid recomputation, we keep theses values in a cache. Besides, to speedup the algorithm, we do not have to enumerate all gVLD-consistent neighbors. It is enough to stop after N max K neighbors are found, as m is then extremely unlikely to be later removed (e.g., N max = 20 for K = 3).

Evaluation

We experimented with existing matching methods, we augmented some of them with VLD, and we compared with K-VLD. We evaluated matching accuracy in various imaging conditions. We also tested K-VLD as a prefilter to RANSAC-based calibration.

All extracted features are SIFT keypoints, as implemented in VLFeat [START_REF] Vedaldi | VLFeat: An open and portable library of computer vision algorithms[END_REF]. We selected a range of state-of-art methods presenting the rich variety of graph matching methods. We use the authors' code for probabilistic hypergraph matching (HGM) [START_REF] Zass | Probabilistic graph and hypergraph matching[END_REF], hypergraph matching via reweighted random walks (RRWHM) [START_REF] Lee | Hyper-graph matching via reweighted random walks[END_REF] and tensor matching (TM) [START_REF] Duchenne | A tensor-based algorithm for high-order graph matching[END_REF], and our own implementation of spectral matching (SM) [START_REF] Leordeanu | A spectral technique for correspondence problems using pairwise constraints[END_REF], which computes the same matching results as integer projected fixed point (IPFP) [START_REF] Leordeanu | Unsupervised learning for graph matching[END_REF], and game-theoretic matching (GTM) [START_REF] Albarelli | Robust game-theoretic inlier selection for bundle adjustment[END_REF]. VLD is incorporated to SM (2 nd -order method) and HGM. For calibration, we used the IPOL implementation of ORSA [START_REF] Moisan | A probabilistic criterion to detect rigid point matches between two images and estimate the fundamental matrix[END_REF][START_REF] Moisan | Automatic homographic registration of a pair of images, with a contrario elimination of outliers[END_REF], which is a parameterless, state-of-art RANSAC variant.

Changing imaging conditions. We use Mikolajczyk's dataset, that evaluates feature detectors and descriptors under different image transformations, including change of viewpoint and illumination, zoom, blur and rotation [START_REF] Mikolajczyk | A performance evaluation of local descriptors[END_REF]. It is composed of 8 sequences of 6 images with increasing variation. For each sequence, we successively match image 1 with the others. We extract the best 400 SIFT matches (i.e., with lowest descriptor distance) as candidates for each image pair. 400 features was about the limit that methods TM and RRWHM could handle on a 24 GB computer, running in 200 s; K-VLD runs in 1s, with a performance quasi linear in |M|. For each method, we extract the N best matches according to the method, where N is the number of ground truth inliers. Matches with less than 5-pixel transformation error are considered as inliers, and accuracy is the proportion of inliers among the N returned matches [START_REF] Lee | Hyper-graph matching via reweighted random walks[END_REF]. This dataset features image transformations that can be described by a single homography; as lines are preserved, VLDs are expected to be relatively stable. In fact, results in Fig. 5 show that K-VLD often outperforms other methods. Besides, VLD significantly improves existing methods, especially for scenes with viewpoint or scale changes. Strong occlusions. To evaluate the case of occlusions, we use the Dětenice fountain dataset [START_REF]Dětenice fountain dataset[END_REF], from which we took a sequence of 43 images. The occluding foreground (a statue) creates strong variations in the background. A ground truth calibration is first constructed by selecting 50 inliers by hand. As above, we extract the best 400 SIFT matches for each pair of successive images. We then measure the actual inlier rate and the accuracy. Results are shown in Fig. 6, where strong local variations have been smoothed and resampled in plotted graphs for readability. K-VLD creates clusters of consistent matches despite occlusions, outperforming other methods most of the time. VLD improves SM moderately (5-10% more inliers) and HGM only slightly, as it already has an excellent performance. Ambiguity and RANSAC prefiltering. To evaluate the benefits of K-VLD as a preprocessing filter for RANSAC-based calibration, we use Strecha's castle dataset [START_REF] Strecha | On benchmarking camera calibration and multi-view stereo for high resolution imagery[END_REF]. It is composed of a looping sequence of 19 images of a courtyard and provides ground truth for both internal and external camera calibration. The scene is highly ambiguous due to many repeated windows, which degrades registration, as illustrated on Fig. 7. Around 5,000 to 7,000 SIFT points are extracted in each image. Potential matches are generated for each pair of consecutive images and for different values of the Lowe rejection threshold [START_REF] Lowe | Distinctive image features from scale-invariant keypoints[END_REF], i.e., the distance ratio of closest to second closest keypoint (often 0.8 in the literature). We also test the case of symmetric matches only, i.e., points P whose match P in the second image has P as match in the first image. This yields 300 to 3,000 matches per pair. We measure the average angle error over each pair and the error standard deviation. As the last image can be compared with the first one, we also measure the accumulated angle error independently of the ground truth by multiplying all the rotation matrices and measuring the angle difference with identity. We compare two methods for estimating the fundamental matrix: using ORSA alone, or prefiltering the matches with K-VLD before ORSA. Results are shown in Fig. 8. The use of K-VLD as a match prefilter greatly improves stability (deviation) and precision.

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Conclusion

For 2 nd -order graph matching, distinctive pairwise constraints are crucial, just as distinctiveness is crucial for ordinary, 1 st -order feature matching. As our experiments show, our virtual line descriptor VLD provides such distinctiveness, offering a better accuracy to 2 nd -order graph matchers and preserving the scalability (time and space) to a large number of points contrary to high-order graph matchers. Besides, compared to other graph matching methods, our K-VLD matcher provides among the best accuracy, especially when the inlier rate drops, even down to a few percents (less than 5 or 10%) and despite strong occlusions (thanks to its semi-local nature) or strong viewpoint or scale changes. Moreover, used as a preprocessor to RANSAC, K-VLD eliminates most outliers, including near epipolar lines and despite possible ambiguities, which greatly improves calibration precision.

These results were achieved with unchanged parameter values after experimenting a few options, but a more systematic study à la SIFT [START_REF] Lowe | Distinctive image features from scale-invariant keypoints[END_REF] on a larger benchmark would be valuable. Interestingly, K-VLD only uses 2 nd -order constraints; 1 st -order terms could also be added.

Figure 2 :

 2 Figure 2: Distances used for the computation of the transformation error η i,i , j, j .

Figure 3 :

 3 Figure 3: Top: disk covering of line (P i , P j ). Bottom: 8-bin histogram of gradient orientation for disk D u , and main orientation w * u .

Figure 5 :

 5 Figure 5: Matching accuracy measured on Mikolajczyk's dataset.

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Figure 6 :

 6 Figure 6: Dětenice fountain: K-VLD clusters (1 image pair) & average accuracy on all pairs.

Figure 7 :Figure 8 :

 78 Figure 7: Left: inliers by ORSA. Middle: false matches near epipolar lines not eliminated by ORSA but rejected by K-VLD. Right: inliers by K-VLD + ORSA. (Symmetric matches; Lowe criterion threshold = 0.8; only 1/4 matches shown, thus matches may show or hide.)


Bernard Viguier
email: bernard.viguier@ensiacet.fr

Fabienne Touratier

Eric Andrieu

High-temperature creep of single-crystal nickel-based superalloy: microstructural changes and effects of thermal cycling

Keywords: Ni-based superalloys, dislocation mechanics, precipitation, microstructure change anisothermal, creep, superalloy, MC2, climb, dislocations, rafting

High-temperature creep of single-

Introduction

Single-crystal nickel-based superalloys are extensively used in the aeronautic industry, particularly in the hottest parts of turbo-reactors for aircraft and helicopters.

The optimized microstructure consists of an array of γ' cubic precipitates aligned along the <100> cubic directions and coherently embedded in an austenitic matrix γ.

Creep at high temperature leads to changes in this microstructure where cubes coarsen to form platelets in a direction perpendicular to the [001] tensile axis depending on the sign of the misfit: this is rafting [START_REF] Nabarro | The physics of creep[END_REF]. The creep behavior of superalloys and the associated morphological changes have been extensively studied over the last 30 years but almost exclusively by means of isothermal creep tests [START_REF] Feller-Kniepmeier | [END_REF][3][4][5]. Real service conditions in aircraft or helicopter engines, including certification tests, imply much 2/25 more severe thermo-mechanical loading of the superalloys, far from isothermal and where temperatures can reach up to 1200°C during emergency regimes. Nevertheless, very few studies have dealt with non isothermal creep [6][START_REF] Raffaitin | Influence du cyclage thermique sur les comportements en oxydation/corrosion et en fluage de systèmes MCrAlY/superalliage à base de nickel[END_REF][START_REF] Cormier | [END_REF][9][10]. Raffaitin et al. [6,[START_REF] Raffaitin | Influence du cyclage thermique sur les comportements en oxydation/corrosion et en fluage de systèmes MCrAlY/superalliage à base de nickel[END_REF] studied the behavior of the superalloy MC2 with anisothermal creep at very high temperatures. During thermal cycling creep tests under 1150°C and 80MPa, with cycles composed of heating, high-temperature dwell and cooling steps, they reported a creep behavior drastically different from that observed under isothermal loading, reducing creep life 3-or 4-fold and increasing the secondary creep rate 10-fold.

Besides, a focus on the strain corresponding to each cycle underlines that it is not linear but shows, at the beginning of the high-temperature dwell, a transient period followed by a linear phase. This acceleration recalling a primary creep stage involves the acceleration of the global creep rate; however, its origin has not yet been fully elucidated.

Superalloy high-temperature isothermal creep is being increasingly studied and results show the presence of a<010> prismatic edge dislocations aligned along <110> directions and moving by climb within the rafts [11][12][13][14][15][16][17][18][19][20]. It has been observed that the majority of these systems are a[100] or a[010] dislocations, so they experience no force at all from the [001] applied stress (their Schmid factor for glide and climb is equal to 0). One way to explain climbing of such systems into the rafts was proposed by Epishin et al. [16]. Their model consists in a vacancy exchange between two different dislocation systems, moving by pure climb or by a combined glide and climb process: the first is localized at the γ/γ' interface, takes part in the interfacial network, and is composed of a[001] dislocations activated by the applied stress, while the second is composed of a[100] and a[010] dislocations, fed by the vacancies from the first system. This mechanism was also investigated by Mompiou and Caillard to explain quasi-crystal deformation [21,22]. The climbing of a [100] type dislocations does not produce a creep strain along the load axis directly, but it is an essential sequence of the global deformation mechanism occurring during the secondary stage of nickel-based superalloy creep. Indeed when the superalloy microstructure is fully rafted, the deformation can be limited to the γ corridors otherwise dislocations would accumulate at the γ/γ' interface and strain would be rapidly stopped. Steady-state deformation requires a recovery process for the annihilation of the accumulated dislocations, which implies that the γ' rafts must be crossed. The climb of [010] dislocations, even if it does not produce any deformation along the [001] axis, can be considered as the limiting step of the global creep mechanism.

The present study aims at improving our understanding of the fundamental mechanisms occurring during high-temperature creep of single-crystal superalloys.

We focused our study on two scale levels. First the evolution of the γ/γ' microstructure is studied during the creep tests with an emphasis on the changes occurring during the imposed thermal cycle. Then the active dislocation systems are characterized in order to elucidate the creep rate-controlling mechanism. Finally, the high-temperature mechanisms will be discussed with regard to these results and the consequences of thermal cycling conditions upon dislocation mechanism will be examined.

Experimental details

The material for this study was the single crystal nickel-based superalloy MC2 (nominal composition in wt. % Ni-7.2Cr-8Co-2Mo-7W-5.5Al-1.3Ti-5.6Ta) provided by Turbomeca in the form of a single crystal plate subjected to the standard solutioning treatment: 3h at 1300°C. Flat thin-wall tensile creep specimens (3x20 mm² gage length, 1mm thick) were electro-discharge machined with the long axis of the specimen along the [001] direction while the flat face was parallel to the (100) plane. After cutting, the specimens were heat treated for 6h at 1080°C followed by air cooling and 20h at 870°C then final air cooling. All the faces of the creep specimens were ground with SiC paper, down to grade 4000 for the two large faces and grade 1000 for the edges, in order to remove the electro-discharge machine damaged zone.

It was checked that the initial microstructure consisted of cubic γ' precipitates (with a 440 nm mean edge length), well aligned along the <001> directions in the γ austenitic matrix. Creep tests were performed at 950°C/200MPa and at 1150°C/80MPa along the [001] direction under laboratory air on a creep frame loaded with a dead weight (i.e. constant load conditions) equipped with a radiation furnace allowing fast temperature changes. Isothermal and thermal cycling creep was studied, in both cases the specimen was first heated to the high-temperature value (either 950 or 1150°C) and then loaded by setting the required weight. The deformation was recorded using a contact-less laser extensometer as described in detail in [23]. For thermal cycling creep experiments, the temperature was monitored as shown in Figure 1: the cycle duration was 60 minutes with a 5 minute heating stage up to 950 or 1150°C, a dwell time of 30 minutes and a 25 minutes cooling stage (see below for the temperature profile during the cooling step) down to about 45°C. The load was kept constant during the thermal cycling.

Special care was taken in the way specimens were cooled for the interrupted tests. Indeed, this is of prime importance in order to get information on the evolution of the γ/γ' microstructure and specially the γ' dissolution phenomenon. First, we shall consider the cooling rate that the sample experiences during the normal thermal cycle. This is shown as the "slow" rate curve in Figure 2, it corresponds to a controlled cooling rate of 3.75°C/s down to 600°C and then an even slower cooling rate due to the furnace and load train thermal inertia. Some creep experiments were interrupted at the end of such cooling steps for microstructure observations which are representative of the low temperature state during the thermal cycle (point B in Figure1). Secondly, some specimens were cooled as fast as possible, by suddenly switching off the radiation furnace and opening the specimen chamber, so that the specimen is cooled directly by laboratory air flow. This is illustrated by the "quench" curve in Figure 2, and corresponds to a cooling rate of 33°C/s down to 800°C (the specimen cooled from 1150 to 850°C in 9 s). This cooling procedure was applied at the end and at the beginning of the high-temperature dwell of the thermal cycles (points A and C respectively in Figure 1). Finally some specimens were also cooled by switching off the radiation furnace and letting the specimen cool down in the furnace, this gave an intermediate cooling rate, labelled "fast" in Figure 2 and which was mainly used for isothermal creep tests. After the creep tests, the specimens were mechanically ground, first with SiC paper, then with diamond paste down to 1µm and finally etched with an electrolytic solution (13% H 3 PO 4 , 45% H 2 SO 4 and 42% HNO 3 under 3V) allowing γ matrix dissolution. Scanning electron microscopy (SEM) was performed with a JEOL JSM 6700 F equipped with a field emission gun operating at 15 kV using secondary electron mode. The observations were made on the flat surface of the creep specimen.

In all SEM images reported the tensile axis [001] is vertical in the image plane -(100) plane. Quantitative data on the γ' raft morphology were obtained using image analysis of the SEM micrographs by Image J software [24]. The surface fraction of the rafts was determined by manually redrawing the contour of rafts on seven images for each experiment. Note that only rafts were considered, not the secondary precipitates neither in spherical nor in cuboid shape. The thickness of the rafts and the periodicity of the structure were measured along intercept lines drawn parallel to the [001] direction, more details on such measurements are given in [START_REF] Touratier | Etude des mécanismes de déformation et d'endommagement du superalliage base nickel MC2 en fluage aux très hautes températures[END_REF].

Thin foils for transmission electron microscopy (TEM) were sawn from thin wall creep samples as squares 2.1 mm in length and 200 µm in thickness. These slices were then mechanically polished down to about 50 µm while tilted in order to achieve an orientation of the slice close to (211). The foils were then dimpled to about 20 µm and ion milled using a Gatan PIPS ® system. Dislocation observations for further indexation were achieved in two-beam diffraction contrast on a Jeol 2010 operating at 200 kV at the TEMSCAN Service of University Paul Sabatier in Toulouse.

Results

Mechanical behaviour

The results of creep tests performed at 950°C/200 MPa and 1150°C/80 MPa are shown in Figure 3 for isothermal and thermal cycling conditions. With thermal cycling, the curves show only the deformation accumulated during the hightemperature dwell, following the procedure described by Raffaitin et al. [6]. As reported in Figure 3, the thermal cycling strongly affects the creep resistance at 1150°C, with a significant increase of the secondary creep rate and reduction of the creep life duration. The average secondary creep rate increases from 7.5 10 -8 s -1 for isothermal conditions to 4 10 -7 s -1 in the present anisothermal creep conditions. As already noted [6] the strain produced during each high-temperature dwell is not a linear function of the time but instead presents a slope change analogous to primarysecondary creep stages as illustrated in Figure 4.a. The occurrence of this primary Similar behaviour and the same occurrence of primary creep after a thermal change was also reported by Cormier et al [9]. For comparison, isothermal and thermal cycling creep experiments were conducted at 950°C, under 200 MPa. Due to the very long creep life in those conditions the tests were not conducted to rupture but stopped during the secondary stage. The curves shown in Figure 3 indicate that, for 950°C, the thermal cycling did not affect the secondary creep rate. Indeed the same average secondary creep rate (10 -8 s -1 ) was measured for isothermal and anisothermal conditions. Furthermore close examination of the deformation curve of the thermal cycling creep test shows that for each high-temperature dwell, the strain is a linear function of time (constant creep rate), see Figure 4.b, unlike the primary and secondary stages observed for thermal cycling creep at 1150°C. This observation supports the link between the global creep acceleration due to thermal cycling at 1150°C and the occurrence of a primary creep stage for each dwell.

Microstructural characterisation

γ/γ' microstructure evolution

The microstructure of the crept specimens was investigated firstly at the γ/γ' scale (SEM observations). Figure 5 shows the microstructure of specimens tested at 950°C under isothermal and thermal cycling conditions, and interrupted after 117 and 147 hours respectively (time at high temperature). After creep interruption, the isothermal sample was cooled using the "fast" procedure and the thermal cycling test was interrupted at the low-temperature state of the cycle (corresponding to point B in Figure 1). The microstructures showed a very clear rafted state as commonly observed in the secondary creep stage [START_REF] Nabarro | The physics of creep[END_REF]. Quantitative image analyses reveal a surface fraction of the γ' phase of 75 and 73.5 % and raft thickness of 420 ± 20 and 405 ± 25 nm for and experienced about 1% creep strain. For comparison the isothermal specimen, shown in Figure 6(a), was also stopped after 1% creep strain, which in this case corresponds to 14 hours spent at 1150°C. The general and common feature of the microstructures observed in Figure 6 is the presence of large γ' precipitates -the rafts -associated with smaller ones -secondary precipitates -within the γ corridors. It is also apparently evident that the surface fraction of the rafts is much smaller than after creep at 950°C. These observations indicate the role of γ' dissolution during creep at 1150°C, indeed for such temperatures the equilibrium volume fraction of γ' decreased significantly [START_REF] Cormier | [END_REF]. A much more precise comparison of these microstructures was obtained by quantitatively measuring the raft characteristics, in terms of raft surface to be submitted to Phil Mag. 9/25 fraction (f sR ), raft thickness (t) and periodicity of the γ/γ' sequence (λ). The results of these measurements, listed in Table 1, clearly reveal two main features: firstly a difference between the isothermal and the anisothermal microstructures, and secondly, an evolution of the microstructure during one thermal cycle. The most important microstructural difference between isothermal and anisothermal creep concerns the surface fraction of the rafts which was significantly lower for the anisothermal creep (f sR = 40 -47%) than for the isothermal (f sR = 56%), despite the fact that anisothermal specimens spent less time at 1150°C than isothermal specimens. The surface fraction of rafts also changed during one thermal cycle, since it rose from about 40% for the high-temperature states (40% for point A and 42% for C) to 47% for the low-temperature point B. This variation of raft surface fraction already indicates that the thermal cycling enhanced the dissolution of γ' rafts, and that during cycling, rafts continuously dissolved and reprecipitated. It also appears that the raft thickness depends on the creep conditions. Indeed, after isothermal creep the average thickness of the rafts was t = 410 nm, the same value was measured after thermal cycling was interrupted at point C (beginning of the high-temperature dwell during cycling creep) whereas the thickness was slightly lower for the lowtemperature point (t = 400 nm for point B) and significantly lower at the end of the dwell (t = 356 nm for point A). The parameter λ also varied during the thermal cycle: λ = 908 nm at the end of the dwell -point A -it dropped to 690 nm for the lowtemperature point -B -and increased back to 805 nm at the beginning of the following dwell -point C. This parameter is correlated to the number of rafts present in the microstructure, and its variations thus show that rafts are successively created and dissolved during the thermal cycle. Besides the variation of raft structure, secondary γ' precipitates are visible in the micrographs of Figure 6 in the γ matrix channels. These small precipitates are due to γ' phase dissolution during the high-temperature dwell and its re-precipitation during the cooling phase as is currently reported in the literature [START_REF] Grosdidier | [END_REF][27][28]. These secondary precipitates exhibit different morphologies and sizes, mainly depending on the cooling rate as shown in Figure 7. For rapid cooling, such as "quench" and "fast" procedures, the secondary precipitation consists in the homogeneous distribution of spherical precipitates with a diameter of around 30 nm, as exemplified in Figure 7(a).

For the "slow" cooling rate, corresponding to point B in the thermal cycle, secondary precipitation forms cuboid precipitates with a mean edge length of 100 nm. Unlike the spheres, these cubes are not homogeneously distributed in the γ channels but are concentrated in the middle of the channel and surrounded by the γ phase containing fine spherical precipitates with 23 nm mean diameter Figure 7(b). Examination of the microstructure at a larger scale in the B state also reveals that some of the cuboid secondary precipitates coalesced to form new rafts, in agreement with the decrease of the period λ which was measured between states A and B.

If we now summarise the overall change of the γ' precipitates during a cycle of an anisothermal creep test at 1150°C, it can be noticed that during the cooling phase (A to B) the thickness of the rafts increased, the period of the structure decreased considerably and as a consequence, the surface fraction of the rafts increased. During the heating phase (B to C), the thickness of the rafts remained constant, the period of the structure increased but remained below the value of the state A and the surface fraction of the rafts decreased slightly. Finally, on the hightemperature dwell (C to A), the thickness of the rafts decreased, the period of the rafts [29,30]. Thus, we focus our study on the dislocation microstructure within the rafts. A view of the dislocation landscape after creep under thermal cycling conditions at 950°C/200 MPa is shown in Figure 8. It is seen that γ channels contain only a few dislocations (with a ½[110] type Burgers vector) related to interfacial dislocations. In contrast, quite numerous, isolated dislocations can be seen in the γ' rafts. They either go right across the rafts (dislocation 1 in Figure 8) or they form loops connected to one side of the raft and which bow into the γ' raft (dislocation 2 in Figure 8). Similar configurations were observed in specimens crept under isothermal conditions. A more detailed analysis of different dislocation segments was performed on a specimen crept under isothermal conditions at 1150°C/80 MPa. Figure 9 shows images of the same area obtained with two different diffraction vectors. The whole analysis required the use of up to seven different diffraction vectors; the direction of different dislocation segments was determined using trace analysis on the foil stereographic projection.

Note that this analysis was made more complex by the fact that for dislocations with Burgers vectors of the type a[010] the g.b = 0 criterion is not sufficient and quite a 12/25 strong residual contrast occurs when g.b ^ξ ξ ξ ξ ≠ 0 as shown previously in the literature [15 , 31]. The results of such analysis are shown in Table 2, showing that the dislocations present within the γ' raft have a[010] Burgers vector. Their line segment is often parallel to <101> directions, so they are edge or 45° mixed dislocations.

Moreover, spatial configuration of dislocations, such as the segments labelled 4, 5 and 6 in Figure 9 show that they do not lie in their glide plane as showed in the scheme in

Discussion

The creep behaviour of MC2 single crystal superalloy was studied at 950 and 1150°C under isothermal and thermal cycling conditions. We showed that thermal cycling has a tremendous effect on creep properties at 1150°C, with an acceleration of creep strain and reduction of life-time, while the creep curves on each high-temperature dwell exhibit a succession of primary and secondary creep stages. In contrast, the creep rate does not seem to be affected by thermal cycling at 950°C and the individual creep steps during high-temperature dwell show a monotonous increase of strain with time (steady -state type creep). Two microstructural phenomena revealed in the present study may play a significant role on creep behaviour at such high temperatures. These are i) the evolution of γ' precipitates with dissolution-reprecipitation and ii) the climb of b = a<010> dislocations within the γ' rafts.

Dissolution / reprecipitation of γ γ γ γ' phase

The changes that occur in the γ/γ' microstructure during high-temperature creep is of prime importance. The rafting phenomenon that transforms the initial cuboid precipitate morphology into large plates has been studied for a many years [START_REF] Nabarro | The physics of creep[END_REF]3]. For moderate temperatures, as is the case in the present study for creep at 950°C, this rafting occurs at constant γ' volume fraction as compared to the initial state. Only very severe plastic deformation leads to destabilisation of the rafts during the tertiary stage of the creep curve [32,33]. However, for higher temperatures as is the case in the present study of creep at 1150°C, dissolution of the γ' phase may occur [START_REF] Cormier | [END_REF]34]. to be submitted to Phil Mag.

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The first evidence of γ' dissolution is given by the values of the raft surface fraction. Indeed in the initial microstructure, the surface fraction of γ' cuboids is about 70 %. This surface fraction does not change significantly after creep at 950°C either under isothermal or cycling conditions. In contrast, creep at 1150°C involves a decrease of the surface fraction down to 56 % for isothermal conditions and down to about 40 % for thermal cycling creep. A direct consequence of γ' dissolution is to create a supersaturated γ matrix that will lead to the formation of fine γ' secondary precipitates upon cooling [35], such as those visible within the γ channels in the micrographs in Figure 6. It is striking to observe the influence of the cooling rate on secondary precipitate morphology as shown in Figure 7. Indeed the only difference between samples A and B is the cooling rate: "quench" for A and "slow" for B, according to the notation used in Figure 2. This precipitation has huge consequences on the way the microstructure changes, since precipitation during the slow cooling steps of the thermal cycles involves the creation of new small rafts as discussed below.

From the microstructural study and the morphological data (Table 1) extracted from interrupted thermal creep tests A, B and C at 1150°C/80MPa, we can propose the following microstructural alterations during one thermal cycle. From A to B, this is the slow rate cooling step, the period λ of the rafted structure significantly decreased (from 910 to 690 nm) which means that new rafts appeared in the matrix corridors.

The new rafts were created during cooling by the coalescence of the cubic secondary precipitates of the γ' phase, as observed in Figure 7(b). It is notable that the new rafts are created very rapidly, since the time necessary to decrease the temperature from 1150 to 900°C is less than 90 s. From B to C, when the temperature was increased from room temperature to 1150°C, the period of the structure increased (but λ is This dissolution of the thinner rafts resulted in the cut off of the size distribution of raft thickness for the smaller values which implied an increase of the mean value of the distribution. Indeed during this heating step, the mean thickness of the rafts increased slightly from 400 nm (point B at room temperature) to 410 nm (point C at 1150°C). It is even possible that during this step the thicker rafts were moderately eroded but to a less extend than the small ones. This highlights the stability of the larger rafts probably related to a very well established interfacial dislocation network.

Indeed the interfacial dislocation network does not appear to evolve along the thermal cycle [36]. During the temperature dwell of the thermal cycle (from C to A), we can note that the thickness of the rafts decreased and that the period of the structure increased which means that the small rafts formed during the cooling step were totally dissolved and that the larger ones began to dissolve slowly under the synergetic effect of temperature and plastic strain.

During thermal cycling, not only very small spherical γ' precipitates, but also cuboid ones and even small rafts are regularly created and dissolved. According to the numbers given in Table 1, the precipitation kinetics were more rapid than the dissolution kinetics, since the surface fraction (f sR ) and the number of rafts (through the variation of λ) was higher at the point C than A. The low-temperature incursion during creep creates new rafts, but on the other hand thermal cycling enhances global γ' dissolution (overall raft surface fraction was always smaller in thermal cycling conditions versus isothermal) and thus accelerates the creep damage [37]. This to be submitted to Phil Mag.

16/25 apparent contradiction may be rationalised firstly by considering that imposing thermal cycling from the beginning of the creep hinders the stabilisation of numerous rafts and secondly by the synergetic effect between the microstructure evolution and creep strain. Indeed in superalloys the role of plastic strain, and the associated dislocation motion, on the transformation of the γ' microstructure has been repeatedly demonstrated either for formation of rafts [38] or for their destabilisation [32].

Climb of b = a<010> dislocations

We reported in section 3.2.2 the presence of numerous dislocations within γ' rafts with a[010] Burgers vectors. This type of dislocation was the most frequent type observed after creep at the highest temperatures tested. Such a[010] dislocation has been mentioned in earlier reports, in single-phase Ni 3 Al compounds [39,[START_REF] Baluc | Contribution à l'étude des défauts et de la plasticité d'un composé intermetallique ordonné : Ni3Al[END_REF] and in the γ' phase of the superalloy CMSX 2 [11,12]. More recently they have been repeatedly observed to cut γ' rafts after high-temperature low-stress creep experiments in various superalloys: CMSX 6 [13], SRR 99 [14],CMSX 4 [29], CMSX 10 [16], TMS-138 [20]. In the last studies in which such dislocations were reported, in superalloys NASAIR-100 [19] and LEK94 [18], the authors recognised that this γ' cutting by climbing a[100] dislocations is a generic recovery mechanism for hightemperature low-stress creep of superalloys. In line with these findings, we have shown that a similar mechanism also operates in MC2 superalloy, the activity of a[010] dislocations being enhanced by the testing temperature as reported in Table 3.

A common feature to most of these observations, from the literature and from the present study, is the fact that a[010] dislocations move through a climb mechanism. It has been reported that the a[100] dislocations may dissociate forming either to be submitted to Phil Mag. 17/25 superpartial dislocations [29] or Shockley type partial dislocations involving stacking faults [START_REF] Kohler | [END_REF] similarly to the configurations studied in other L1 2 compounds [42]. Such dissociation seems to stabilize the line orientation along <101> directions, as observed in the present study, but does not strongly impact the motion mode which is still mainly climb. Even though to our knowledge, the sign of the climbing dislocation has never been determined, it is postulated that a[010] dislocations climb by absorbing vacancies which results in shrinking the specimen (a climbing process associated to the emission of vacancies would result in thickening the specimen in a direction perpendicular to the applied tensile stress, which is not realistic).

The driving force for the activation of a[010] dislocations has been questioned in the literature. Indeed when creep load is imposed along the [001] direction, the a[010] dislocations do not experience any resolved force (neither glide nor climb) from the applied stress. It is thus believed that their activation is related to a kind of secondary process that recovers the activation of a primary system which is actually driven by the applied stress and provides creep strain along [001]. Such recovery processes include i) the annihilation of γ/γ' interface dislocations when the a[010]

dislocations have crossed the raft [19] and ii) the absorption of vacancies emitted by the climb of a primary system in the interface. Epishin and Link [16] were the first to show that such a mechanism is involved in the high-temperature creep strain of CMSX-4 and CMSX-10 superalloys. In their description, the creep strain along This mechanism involves two dislocation systems: system 1 accounts for the longitudinal strain along the load direction and creates vacancies, and system 2 absorbs vacancies and produces shrinkage of the specimen normal to the load axis.

Dislocation system 1 can ensure a steady-state macroscopic strain only in the case where the vacancies it creates are eliminated in some way. If not, the osmotic force due to the oversaturation of vacancies would block this deformation. The elimination of vacancies requires two main points: i) the existence of sinks for the vacancies, and ii) that the flux of vacancies to these sinks is fast enough compared to the strain rate.

These two points may be rate limiting and the creep rate of rafted single-crystal superalloys could be controlled by the dislocations in γ' raft, either through their limited climb velocity [29] or limited density [30]. We recently calculated [44] the flux of vacancies generated by their diffusion between the γ/γ' interface (where an oversaturation is due to the activity of system 1 dislocations) and the centre of γ' rafts assumed to be at thermal equilibrium. The diffusion data show that for T = 950°C and 1150°C, this flux is more than large enough to ensure the transportation of vacancies from system 1 to system 2 and thus can ensure condition ii) as stated above. The thermal diffusion to pre-existing pores, which has been proposed as a alternative vacancy sink [16], was shown to be much more difficult since the distance to be travelled by the vacancies was in this case much larger than the raft thickness [44]. As speculated in the literature [16,30], we may assume that the density of sinks, that is the density of system 2, climbing a[100] dislocations, may be the key point. This hypothesis is supported by our thermal cycling results. Indeed, during the thermal cycles the γ' phase repeatedly precipitates and dissolves. In particular, during the cooling step we show that small rafts are formed, their rafted shape indicates that they relaxed coherency stress, so they must be surrounded by a dislocation network for the compensation of γ/γ' misfit. Such a coarsened secondary precipitate, formed during anisothermal creep experiment -called micro-raft by the authors -has been indeed observed to be surrounded by dislocations, see Figure 12 in [10]. Upon heating, these rafts dissolve, we believe that the dislocations involved in the network are then released and pushed apart by the applied stress (stress is maintained during the whole cycle). By providing such fresh dislocations, the cooling step during thermal cycling allows an acceleration of the creep strain at the beginning of the high temperature dwell in the form of the primary stage, as observed in Figure 4 and in [6]. Once these dislocations are exhausted, the creep rate is lowered, leading to the secondary stage in the creep curve portion reported in Figure 4. Further support of the present conclusion on the predominant role of dislocation density in the γ' raft to be used as vacancy sinks is brought by the observation of samples crept at 950°C. In these conditions, the γ' rafts are not dissolved, so no fresh dislocations are provided during creep, and a close examination of the crept microstructure reveals the presence of numerous loops inside the rafts, indicated by white arrows in Fig. 11. The loops result from the coalescence of oversaturated vacancies, that could not be eliminated by climbing a[100] dislocations.

Conclusions

The creep behaviour of the MC2 single-crystal superalloy was studied at very high temperature, 950°/200 MPa and 1150°C/80 MPa under both isothermal and thermal to be submitted to Phil Mag.

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Tables Table 1: Quantitative characteristics of the rafted structure after creep at 1150°C/80 MPa in both isothermal and thermal cycling conditions. The surface fraction of the rafts (f sR ), their mean thickness (t) and the periodicity of the structure normal to the raft plane (λ) are given.

Table 2: Diffraction vectors ---g ---and visibility of different dislocation segments shown in Figure 9, allowing their Burgers vector -b -determination. For each segment the line direction -ξ --ξ --ξ --ξand the character ----ϕ ---were determined by stereographic analysis. V: Visible, I: Invisible and RC: Residual Contrast. [ ]

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  cycle is believed to be responsible for the global creep acceleration.

  thermally cycled specimens respectively. Those values correspond to a classical rafted structure in MC2 alloy under tensile creep at temperatures lower than 1000°C. They indicate that γ' dissolution does not occur during exposure to 950°C whether in isothermal or cycling conditions. In contrast, the microstructures observed on specimens crept at 1150°C under isothermal figure 6(a) and thermal cycling tests figures 6(b to d) interrupted at different points of the thermal cycle are quite different. The thermal cycling creep tests were interrupted at the end of the 12 th -beginning of 13 th cycle, as schematised in Figure 1. The first specimen, shown in Figure 6(b), corresponds to the interruption of the test at the end of the 12 th with a "quench" cooling; it is representative of point A in Figure 1. The second specimen, Figure 6(c), also stopped at the end the 12 th cycle, was cooled with the "slow" procedure and thus represents low-temperature point -B -of the thermal cycle. Finally the microstructure seen in Figure 6(d) is from a sample that was "quenched" from the beginning of the 13 th thermal cycle and represents point C. These three specimens spent the same time (6 hours) at 1150°C

  be submitted to Phil Mag. 11/25 structure increased and so, the surface fraction decreased. This will be discussed below in terms of dissolution and reprecipitation of the γ' rafts and its consequences on the creep behaviour during thermal cycling creep tests.3.2.2 Dislocation microstructuresDislocation microstructures were observed in the specimens crept in the different conditions. Since we are working with rafted structures, recovery mechanisms of the γ dislocations are required to allow macroscopic deformation of the superalloy. It has been repeatedly proposed that such recovery may occur by dislocations crossing the γ'

Figure 9 .

 9 Figure 9.c. Such a spatial configuration could only be obtained by an intensive climb

  C than it is at point A) which means that only one part of the rafts -the smallest -formed during the cooling phase dissolved during reheating. This dissolution of the smaller fresh rafts is supported by the complete dissolution of cuboid secondary precipitates which is visible when comparing Figures6(c) and 6(d).

  [001] is due to the transverse glide-climb of a/2<011> dislocations, thus producing vacancies. These excess vacancies can be absorbed by pores and by a<100> dislocations which implies their climb in γ' rafts. In turn, the crossing of rafts by a[100] dislocations annihilates interface dislocation, recovering the deformation mechanism in the γ channels. Such a mechanism, involving two dislocation systems has been demonstrated to operate during compression testing of AlPdMn quasicrystals[21,43]. The TEM observations reported here show that a similar mechanism must operate in the MC2 superalloy single crystal during hightemperature creep along the [001] axis.

  It was shown that the thermal cycling strongly affects creep behaviour at 1150°C while no effect was observed at 950°C, this was related to the partial dissolution of the γ' phase for the highest temperatures. Indeed for 1150°C testing, the thermal cycling during the creep test induces repeated precipitation and dissolution of γ' particles, as secondary precipitates but also as small rafts. This continuous modification of the precipitation microstructure during thermal cycles alters the raft morphology by reducing the overall surface ratio of the rafts and accelerating their degradation as compared to isothermal conditions. The dislocation microstructures have been observed after interrupted isothermal and thermal cycling creep tests. The role of a[100] dislocations crossing the γ' rafts by a climbing process was revealed. This process acts as a recovery mechanism in regards to the dislocations responsible for the strain along the loading direction. This necessary recovery process could control the creep rate, however it seems that diffusion rate or dislocation climb velocity is fast enough and may not be rate limiting while the density of a[100] seems to be the crucial parameter. Thermal cycling creep, through the dissolution of γ' rafts formed during the low temperature incursion provides new dislocations which may explain the observed acceleration of the creep rate.

Figure 1 :

 1 Figure 1: Specimen temperature during thermal cycling creep experiments. Interrupted tests were stopped at three different points: i) at the end of the hightemperature dwell -point A; ii) at the low-temperature -point B; iii) at the beginning of the high temperature dwell -point C.

Figure 2 :

 2 Figure 2: Specimen temperature showing the three different cooling rates (slow, fast and quench) used for interrupted tests.

Figure 3 :

 3 Figure 3: Comparison of the creep curves obtained on isothermal and thermal cycling conditions for 950°C/200 MPa and 1150°C/80 MPa.

Figure 4 :

 4 Figure 4: Details of the creep strain during one cycle (only the high-temperature dwell is shown) showing the primary-secondary stages for creep performed at 1150°C/80 MPa (a) and linear dependence of strain versus time for 950°C/200 MPa (b).

Figure 5 :

 5 Figure 5: Microstructures after creep at 950°C/200MPa for isothermal (a) and thermal cycling (b) conditions.

Figure 6 :

 6 Figure 6: Microstructures after creep at 1150°C/80 MPa for isothermal (a) and thermal cycling conditions. For the latter, tests were interrupted at different points of the thermal cycle, as indicated in Figure 1: point A, end of the 12 th high temperature dwell (b); point B, low temperature point of the 12 th cycle (c) and point C, beginning of the 13 th high temperature dwell (d).

Figure 7 :

 7 Figure 7: Details of the γ' secondary precipitation during creep at 1150°C the images correspond to cycling creep specimen interrupted at the end of the high-temperature dwell -point A -(a) and at the low temperature point -B-(b). Note that both samples only differ by the cooling rate from 1150°C to room temperature: "quench" for A and "slow" for B, as in the curves in Figure2.

Figure 8 :

 8 Figure 8: Dislocation microstructures in a specimen crept at 950°C / 200 MPa under thermal cycling conditions. Note dislocations (labelled 1 and 2) crossing or bowing into the γ' rafts and a/2<110> type dislocations in the γ corridors (labelled with x).

Figure 9 :

 9 Figure 9: Example of diffraction contrast images used to identify the Burgers vector of the dislocations observed in the γ' rafts. The corresponding specimen was crept at 1150°C/80 MPa under isothermal conditions to rupture. The dislocations are imaged in BF two beam conditions with different diffraction vectors in (a) and (b). The

  scheme in (c) illustrates the geometrical configuration of the dislocation segments 5,4,6 (thick black line) together with the cubic unit directions <100> (thin grey lines and cubes). The Burgers vector b= a[010] of the dislocation is also indicated.

Figure 10 :

 10 Figure 10: Overview of the dislocation microstructures in a specimen crept at 1150°C/80 MPa under thermal cycling conditions. The creep test was stopped at the low-temperature step of the thermal cycle (point B in Figure 2).

Figure 11 :

 11 Figure 11: Detail of the dislocation microstructure in a specimen crept at 950°C/200 MPa under thermal cycling conditions -interrupted at low-temperature point of the thermal cycle -showing the presence of dislocation loops within the γ' rafts.

Table 3 :

 3 Estimate of dislocation densities inside the γ' rafts separating dislocations with b = a[010] and other types of dislocations, depending on the creep conditions.

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Acknowlegments

The authors would like to thank the company Safran-Turbomeca for providing the materials studied and for financial support.