Alasdair Newson

Andrés Almansa

Matthieu Fradet

Yann Gousseau

Patrick Pérez

VIDEO INPAINTING OF COMPLEX SCENES

Keywords: Video inpainting, patch-based inpainting, video textures, moving background AMS subject classifications. 68U10, 65K10, 65C20

We propose an automatic video inpainting algorithm which relies on the optimisation of a global, patch-based functional. Our algorithm is able to deal with a variety of challenging situations which naturally arise in video inpainting, such as the correct reconstruction of dynamic textures, multiple moving objects and moving background. Furthermore, we achieve this in an order of magnitude less execution time with respect to the state-of-the-art. We are also able to achieve good quality results on high definition videos. Finally, we provide specific algorithmic details to make implementation of our algorithm as easy as possible. The resulting algorithm requires no segmentation or manual input other than the definition of the inpainting mask, and can deal with a wider variety of situations than is handled by previous work.

1. Introduction. Advanced image and video editing techniques are increasingly common in the image processing and computer vision world, and are also starting to be used in media entertainment. One common and difficult task closely linked to the world of video editing is image and video "inpainting". Generally speaking, this is the task of replacing the content of an image or video with some other content which is visually pleasing. This subject has been extensively studied in the case of images, to such an extent that commercial image inpainting products destined for the general public are available, such as Photoshop's "Content Aware fill" [1]. However, while some impressive results have been obtained in the case of videos, the subject has been studied far less extensively than image inpainting. This relative lack of research can largely be attributed to high time complexity due to the added temporal dimension. Indeed, it has only very recently become possible to produce good quality inpainting results on high definition videos, and this only in a semi-automatic manner. Nevertheless, high-quality video inpainting has many important and useful applications such as film restoration, professional post-production in cinema and video editing for personal use. For this reason, we believe that an automatic, generic video inpainting algorithm would be extremely useful for both academic and professional communities.

1.1. Prior work. The generic goal of replacing areas of arbitrary shapes and sizes in images by some other content was first presented by Masnou and Morel in [START_REF] Masnou | Level lines based disocclusion[END_REF]. This method used level-lines to disocclude the region to inpaint. The term "inpainting" was first introduced by Bertalmio et al. in [START_REF] Bertalmio | Image inpainting[END_REF]. Subsequently, a vast amount of research was done in the area of image inpainting [START_REF] Bertalmio | Encyclopedia of Computer Vision[END_REF], and to a lesser extent in video inpainting.

Generally speaking, video inpainting algorithms belong to either the "object-based" or "patch-based" category. Object-based algorithms usually segment the video into moving fore-ground objects and background that is either still or displays simple motion. These segmented image sequences are then inpainted using separate algorithms. The background is often inpainted using image inpainting methods such as [START_REF] Criminisi | Object removal by exemplar-based inpainting[END_REF], whereas moving objects are often copied into the occlusion as smoothly as possible. Unfortunately, such methods include restrictive hypotheses on the moving objects' motion, such as strict periodicity. Some object-based methods include [START_REF] Sen-Chin | Efficient object-based video inpainting[END_REF][START_REF] Jia | Video repairing under variable illumination using cyclic motions[END_REF][START_REF] Ling | Virtual contour guided video object inpainting using posture mapping and retrieval[END_REF].

Patch-based methods are based on the intuitive idea of copying and pasting small video "patches" (rectangular cuboids of video information) into the occluded area. These patches are very useful as they provide a practical way of encoding local texture, structure and motion (in the video case).

Patches were first introduced for texture synthesis in images [START_REF] Efros | Texture synthesis by non-parametric sampling[END_REF], and subsequently used with great success in image inpainting [START_REF] Bornard | Missing data correction in still images and image sequences[END_REF][START_REF] Criminisi | Object removal by exemplar-based inpainting[END_REF][START_REF] Drori | Fragment-based image completion[END_REF]. These methods copy and paste patches into the occlusion in a greedy fashion, which means that no global coherence of the solution can be guaranteed. This general approach was extended by Patwardhan et al. to the spatio-temporal case in [START_REF] Patwardhan | Video inpainting of occluding and occluded objects[END_REF]. In [START_REF]Video inpainting under constrained camera motion[END_REF], this approach was further improved so that moving cameras could be dealt with. This is reflected by the fact that a good segmentation of the scene into moving foreground objects and background is needed to produce good quality results. This lack of global coherence can be a drawback, especially for the correct inpainting of moving objects.

Another method leading to a different family of algorithms was presented by Demanet et al. in [START_REF] Demanet | Image inpainting by correspondence maps: a deterministic approach[END_REF]. The key insight here is that inpainting can be viewed as a labelling problem: each occluded pixel can be associated with an unoccluded pixel, and the final labels of the pixels result from a discrete optimisation process. This idea was subsequently followed by a series of image inpainting methods [START_REF]Statistics of patch offsets for image completion[END_REF][START_REF] Komodakis | Image completion using efficient belief propagation via priority scheduling and dynamic pruning[END_REF][START_REF] Liu | Exemplar-based image inpainting using multiscale graph cuts[END_REF][START_REF] Pritch | Shift-map image editing[END_REF] which use optimisation techniques such as graph cuts algorithms [START_REF] Boykov | Fast approximate energy minimization via graph cuts[END_REF], to minimise a global patch-based functional. The vector field representing the correspondences between occluded and unoccluded pixels is referred to as the shift map by Pritch et al. [START_REF] Pritch | Shift-map image editing[END_REF], a term which we shall use in the current work. This idea was extended to the video case by Granados et al. in [START_REF] Granados | How not to be seen: Object removal from videos of crowded scenes[END_REF]. They propose a semi-automatic algorithm which optimises the spatio-temporal shift map. This algorithm presents impressive results on higher resolution images than are previously found in the literature (up to 1120×754 pixels). However, in order to reduce the large search space and high time complexity of the optimisation method, manual tracking of moving occluded objects is required. To the best of our knowledge, the inpainting results of Granados et al. are the most advanced to date, and we shall therefore compare our algorithm with these results.

We also note the work of Herling and Broll [START_REF] Herling | Pixmix: A real-time approach to high-quality diminished reality[END_REF] whose goal is "diminished reality", which considers the inpainting task coupled with a tracking problem. This is the only approach of which we are aware which inpaints videos in a real-time manner. However, the method relies on restrictive hypotheses on the nature of the scene to inpaint and can therefore only deal with tasks such as removing a static object from a rigid platform.

Another family of patch-based video inpainting methods was introduced in the seminal work of Wexler et al. [START_REF] Wexler | Space-time video completion[END_REF]. This paper proposes an iterative method that may be seen as an heuristic to solve a global optimisation problem. This work is widely cited and well-known in the video inpainting domain, mainly because it ensures global coherency in an automatic manner. This method is in fact closely linked to methods such as non-local denoising [START_REF] Buades | A non-local algorithm for image denoising[END_REF]. This link was also noted in the work of Arias et al. [START_REF] Arias | A variational framework for exemplar-based image inpainting[END_REF], which introduced a general non-local patch-based variational framework for inpainting in the image case. In fact, the algorithm of Wexler et al. may be seen as a special case of this framework. We shall refer to this general approach as the non-local patch-based approach. Darabi et al. have presented another variation on the work of Wexler et al. for image inpainting purposes in [START_REF] Darabi | Image melding: combining inconsistent images using patch-based synthesis[END_REF]. In the video inpainting case, the high dimensionality of the problem makes such approaches extremely slow, in particular due to the nearest neighbour search, requiring up to several days for a few seconds of VGA video. This problem was considered in our previous work [START_REF] Newson | Towards fast, generic video inpainting[END_REF], which represented a first step towards achieving high quality video inpainting results even on high resolution videos. Given the flexibility and potential of the non-local patch-based approach, we use it here to form the core of the proposed method. In order to obtain an algorithm which can deal with a wide variety of complex situations, we consider and propose solutions to some of the most important questions which arise in video inpainting.

1.2. Outline and contributions of the paper. The ultimate goal of our work is to produce an automatic and generic video inpainting algorithm which can deal with complex and varied situations. In Section 2, we present the variational framework and the notations which we shall use in this paper. The proposed algorithm and contributions are presented in Section 3. These contributions can be summarised with the following points:

• we greatly accelerate the nearest neighbour search, using an extension of the Patch-Match algorithm [START_REF] Barnes | Patchmatch: a randomized correspondence algorithm for structural image editing[END_REF] to the spatio-temporal case (Section 3.1); • we introduce texture features to the patch distance in order to correctly inpaint video textures (Section 3.3); • we deal with the problem of moving background (Section 3.4) in videos, using a robust affine estimation of the dominant motion in the video; • we describe our initialisation scheme (Section 3.5), which is often left unspecified in previous work; • we give precise details concerning the implementation of the multi-resolution scheme (Section 3.6). As shown in Section 4, the proposed algorithm produces high quality results in an automatic manner, in a wide range of complex video inpainting situations: moving cameras, multiple moving objects, changing background and dynamic video textures. No pre-segmentation of the video is required. One of the most significant advantages of the proposed method is that it deals with all these situations in a single framework, rather than having to resort to separate algorithms for each case, as in [START_REF] Granados | How not to be seen: Object removal from videos of crowded scenes[END_REF] and [START_REF] Granados | Background inpainting for videos with dynamic objects and a free-moving camera[END_REF] for instance. In particular, the problem of reconstructing dynamic video textures has not been previously addressed in other inpainting algorithms. This is a worthwhile advantage since the problem of synthesising video textures, which is usually done with dedicated algorithms, can be achieved in this single, coherent framework. Finally, our algorithm does not need any manual input other than the inpainting mask, and does not rely on foreground/background segmentation, which is the case of many other approaches [START_REF] Sen-Chin | Efficient object-based video inpainting[END_REF][START_REF] Granados | How not to be seen: Object removal from videos of crowded scenes[END_REF][START_REF] Jia | Video repairing under variable illumination using cyclic motions[END_REF][START_REF] Ling | Virtual contour guided video object inpainting using posture mapping and retrieval[END_REF][START_REF]Video inpainting under constrained camera motion[END_REF].

We provide an implementation of our algorithm, which is available at the following address: http://www.telecom-paristech.fr/ ~gousseau/video_inpainting.

Variational framework and notation.

As we have stated in the introduction, our video inpainting algorithm takes a non-local patch-based approach. At the heart of such algorithms lies a global patch-based functional which is to be optimised. This optimisation is carried out using an iterative algorithm inspired by the work of Wexler et al. [START_REF]Space-time completion of video[END_REF]. The central machinery of the algorithm is based on the alternation of two core steps: a search for the nearest neighbours of patches which contain occluded pixels, and a reconstruction step based on the aggregation of the information provided by the nearest neighbours.

This iterative algorithm is embedded in a multi-resolution pyramid scheme, similarly to [START_REF] Drori | Fragment-based image completion[END_REF][START_REF]Space-time completion of video[END_REF]. The multi-resolution scheme is vital for the correct reconstruction of structures and moving objects in large occlusions. The precise details concerning the multi-resolution scheme can be found in Section 3.6.

A summary of the different steps in our algorithm can be seen in Figure 2, and the whole algorithm may be seen in Alg. 4.

Notation. Before describing our algorithm, let us first of all set down some notation. A diagram which summarises this notation can be seen in Figure 1. Let u : Ω → R 3 represent the colour video content, defined over a spatio-temporal volume Ω. In order to simplify the notation, u will correspond both to the information being reconstructed inside the occlusion, and the unoccluded information which will be used for inpainting. We denote a spatiotemporal position in the video as p = (x, y, t) ∈ Ω and by u(p) ∈ R 3 the vector containing the colour values of the video at this position.

Let H be the spatio-temporal occlusion (the "hole" to inpaint) and D the data set (the unoccluded area). Note that H and D correspond to spatio-temporal positions rather than actual video content and they form a partition of Ω, that is Ω = H ∪ D and H ∩ D = ∅.

Let N p be a spatio-temporal neighbourhood of p. This neighbourhood is defined as a rectangular cuboid centred on p. The video patch centered at p is defined as vector W u p = (u(q 1 ) • • • u(q N )) of size 3 × N , where the N pixels in N p , q 1 • • • q N , are ordered in a predefined way.

Let us note D = {p ∈ D : N p ⊂ D} the set of unoccluded pixels whose neighborhood is also unoccluded (video patch W u p is only composed of known color values). We shall only use patches stemming from D to inpaint the occlusion. Also, let H = ∪ p∈H N p represent a dilated version of H.

Given a distance d(•, •) between video patches, a key tool for patch-based inpainting is to define a correspondence map that associates to each pixel p ∈ Ω (notably those in occlusion) another position q ∈ D, such that patches W u p and W u q are as similar as possible. This can be formalized using the so-called shift map φ : Ω → R 3 that captures the shift between a position and its correspondent, that is q = p + φ(p) is the "correspondent" of p. This map must verify that p + φ(p) ∈ D, ∀p (see Figure 1 for an illustration).

Mimizing a non-local patch-based functional. The cost function which we use, following the work of Wexler et al. [START_REF]Space-time completion of video[END_REF], has both u and φ as arguments:

(2.1)

E(u, φ) = p∈H d 2 (W u p , W u p+φ(p) ), with (2.2) d 2 (W u p , W u p+φ(p) ) = 1 N q∈Np u(q) -u(q + φ(p)) 2 2 .
In all that follows, in order to avoid cumbersome notation we shall drop the u from W u p and simply denote patches as W p . We show in Appendix A, that this functional is in fact a special case of the formulation of Arias et al. [START_REF] Arias | A variational framework for exemplar-based image inpainting[END_REF]. As mentioned at the beginning of this Section, this functional is optimised using the following two steps: Matching Given current video u, find in D the nearest neighbour (NN) of each patch W p that has pixels in inpainting domain H, that is, the map φ(p), ∀p ∈ Ω \ D. Reconstruction Given shift map φ, attribute a new value u(p) to each pixel p ∈ H.

These steps are iterated so as to converge to a satisfactory solution. The process may be seen as an alternated minimisation of cost (2.1) over the shift map φ and the video content u. As in many image processing and computer vision problems, this approach is implemented in a multi-resolution framework in order to improve results and avoid local minima.

3. Proposed algorithm. Now that we have given a general outline of our algorithm, we proceed to address some of the key challenges in video inpainting. The first of these concerns the search for the nearest neighbours of patches centred on pixels which need to be inpainted.

3.1. Approximate Nearest Neighbour (ANN) search. When considering the high complexity of the NN search step, it quickly becomes apparent that searching for exact nearest neighbours would take far too long. Therefore, an approximate nearest neighbour (ANN) search is carried out. Wexler et al. proposed the k-d tree based approach of Arya and Mount [START_REF] Arya | Approximate nearest neighbor queries in fixed dimensions[END_REF] for this step, but this approach remains quite slow. For example, one ANN search step takes about an hour for a video containing 120 × 340 × 100 pixels, with about 422, 000 missing pixels, which represents a relatively small occlusion (the equivalent of a 65 × 65 pixel box in each frame). We shall address this problem here, in particular by using an extension of the PatchMatch algorithm [START_REF] Barnes | Patchmatch: a randomized correspondence algorithm for structural image editing[END_REF] to the spatio-temporal case. We note that the PatchMatch algorithm has also been used in conjunction with a 2D version of Wexler's algorithm for image inpainting, in the Content-Aware Fill tool of Photoshop [1], and by Darabi et al. [START_REF] Darabi | Image melding: combining inconsistent images using patch-based synthesis[END_REF].

Barnes et al.'s PatchMatch is a conceptually simple algorithm based on the hypothesis that, in the case of image patches, the shift map defined by the spatial offsets between ANNs is piece-wise constant. This is essentially because the image elements which the ANNs connect are often on rigid objects of a certain size. In essence, the algorithm looks randomly for ANNs and tries to "spread" those which are good. We extend this principle to the spatio-temporal setting. Our spatio-temporal extension of the PatchMatch algorithm consists of three steps: (i) initialisation,(ii) propagation and (iii) random search.

Let us recall that H is a dilated version of H. Initialisation consists of randomly associating an ANN to each patch W p , p ∈ H, which gives an initial ANN shift map, φ. In fact, apart from the first iteration, we already have a good initialisation: the shift map φ from the previous iteration. Therefore, except during the initialisation step (see Section 3.5), we use this previous shift map in our algorithm instead of initialising randomly.

The propagation step encourages shifts in φ which lead to good ANNs to be spread throughout φ. In this step, all positions in the video volume are scanned lexicographically. For a given patch W p at location p = (x, y, t), the algorithm considers the following three candidates : W p+φ(x-1,y,t) , W p+φ(x,y-1,t) and W p+φ(x,y,t-1) . If one of these three patches has a smaller patch distance with respect to W p than W p+φ(p) , then φ(p) is replaced with the new, better shift. The scanning order is reversed for the next iteration of the propagation, and the algorithm tests W p+φ(x+1,y,t) , W (p+φ(x,y+1,t) and W p+φ(x,y,t+1) . In the two different scanning orderings, the important point is obviously to use the patches which have already been processed in the current propagation step.

The third step, the random search, consists in looking randomly for better ANNs of each W p in an increasingly small area around p + φ(p), starting with a maximum search distance. At iteration k, the random candidates are centred at the following positions:

(3.1) q = p + φ(p) + ⌊r max ρ k δ k ⌋,
where r max is the maximum search radius around p + φ(p), δ k is a 3-dimensional vector drawn from the uniform distribution over unit cube [-1, 1] × [-1, 1] × [-1, 1] and ρ ∈ (0, 1) is the reduction factor of the search window size. In the original PatchMatch, ρ is set to 0.5. This random search avoids the algorithm getting stuck in local minima. The maximum search parameter r max is set to the maximum dimension of the video, at the current resolution level.

The propagation and random search steps are iterated several times to converge to a good solution. In our work, we set this number of iterations to 10. For further details concerning the PatchMatch algorithm in the 2D case, see [START_REF] Barnes | Patchmatch: a randomized correspondence algorithm for structural image editing[END_REF]. Our spatio-temporal extension is summarized in Algorithm 1.

We note here that other ANN search methods for image patches exist which outperform PatchMatch [START_REF] He | Computing nearest-neighbor fields via propagation-assisted kd-trees[END_REF][START_REF] Korman | Coherency sensitive hashing[END_REF][START_REF] Olonetsky | Treecann -k-d tree coherence approximate nearest neighbor algorithm[END_REF]. However, in practice, PatchMatch appeared to be a good option because of its conceptual simplicity and nonetheless very good performance. Furthermore, to take the example of the "TreeCANN" method of Olonetsky and Avidan [START_REF] Olonetsky | Treecann -k-d tree coherence approximate nearest neighbor algorithm[END_REF], the reported reduction in execution time is largely based on a very good ANN shift map initialisation followed by a small number of propagation steps. In our case, we already have a good initialisation (from the previous iteration), which makes the usefulness of such approaches questionable. However, further acceleration is certainly something which could be developed in the future.

= p -(1, 0, 0), b = p -(0, 1, 0), c = p -(0, 0, 1); q = arg min r∈{p,a,b,c} d(W u p , W u p+φ(r) ); if p + φ(q) ∈ D then φ(p) ← φ(q); end else /* Propagation on odd iteration */ for p = p | H| to p 1 do a = p + (1, 0, 0), b = p + (0, 1, 0), c = p + (0, 0, 1); q = arg min r∈{p,a,b,c} d(W u p , W u p+φ(r) ); if p + φ(q) ∈ D then φ(p) ← φ(q); end end for p = p 1 to p | H| do /* Random search */ q = p + φ(p) + ⌊r max ρ k RandUniform([-1, 1] 3 ) ⌋; if d(W u p , W u p+φ(q) ) < d(W u p , W u p+φ(p)
) and p + φ(q) ∈ D then φ(p) ← φ(q); end end the following manner:

(3.2) u(p) = q∈Np s q p u(p + φ(q)) q∈Np s q p , ∀p ∈ H, with (3.3) s q p = exp - d 2 (W q , W q+φ(q) ) 2σ 2 p .
Wexler et al. proposed the use of an additional weighting term to give more weight to the information near the occlusion border. We dispense with this term, since in our scheme it is somewhat replaced by our method of initialising the solution which will be detailed in Section 3.5. Parameter σ p is defined as the 75th percentile of all distances {d(W q , W q+φ(q) ), q ∈ N p } as in [START_REF]Space-time completion of video[END_REF].

Observe that in order to minimise (2.1), the natural approach would be to do the reconstruction with the non-weighted scheme (s q p = 1 in Equation 3.2) that stems from ∂E ∂u(p) = 0. However, the weighted scheme above tends to accelerate the convergence of the algorithm, meaning that we produce good results faster.

Original frame : "Crossing ladies" [START_REF]Space-time completion of video[END_REF] Weighted average

Mean shift Best patch We observe that the proposed reconstruction using only the best patch at the end of the algorithm produces similar results to the use of the mean shift algorithm, avoiding blur induced by weighted patch averaging, while being less computationally expensive. Please note that the blurring effect is best viewed in the pdf version of the paper.

An important observation is that, in the case of regions with high frequency details, the use of this mean reconstruction (weighted or unweighted) often leads to blurry results, even if the correct patches have been identified. This phenomenon was also noted in [START_REF]Space-time completion of video[END_REF]. Although we shall propose in Section 3.3 a method to correctly identify textured patches in the matching steps, this does not deal with the reconstruction of the video. We thus need to address this problem, at least in the final stage of the approach: throughout the algorithm, we use the unweighted mean given in Equation 3.2 and, at the end of the algorithm, when the solution has converged at the finest pyramid level, we simply inpaint the occlusion using the best patch among the contributing ones. This corresponds to setting σ p to 0 in (3.2-3.3) or, seen in another light, may be viewed as a very crude annealing procedure. Final reconstruction at position p ∈ H reads:

(3.4) u (final) (p) = u(p + φ (final) (q * )), with q * = arg min q∈Np d(W (final-1) q , W q+φ (final) (q) ).
Another solution to this problem based on the mean shift algorithm was proposed by Wexler et al. in [START_REF]Space-time completion of video[END_REF], but such an approach increases the complexity and execution time of the algorithm. Figure 3 shows that very similar results to those in [START_REF]Space-time completion of video[END_REF] may be obtained with our much simpler approach.

3.3. Video texture pyramid. In order for any patch-based inpainting algorithm to work, it is necessary that the patch distance identify "correct" patches. This is not the case in Approximate nearest neighbours without texture features

Approximate nearest neighbours with texture features several situations. Firstly, as noticed by Liu and Caselles in [START_REF] Liu | Exemplar-based image inpainting using multiscale graph cuts[END_REF], the use of multi-resolution pyramids can make patch comparisons ambiguous, especially in the case of textures, in images and videos. Secondly, it turns out that the commonly used ℓ 2 patch distance is ill-adapted to comparing textured patches. Thirdly, PatchMatch itself can contribute to the identification of incorrect patches. These reasons are explored and explained more extensively in Appendix B. A visual illustration of the problem may be seen in Figure 4. We note here that similar ambiguities were also identified by Bugeau et al. in [START_REF] Bugeau | A comprehensive framework for image inpainting[END_REF], but their interpretation of the phenomenon was somewhat different.

We propose a solution to this problem in the form of a multi-resolution pyramid which reflects the textural nature of the video. We shall refer to this as the texture feature pyramid. The information in this texture feature pyramid is added to the patch distance in order to identify the correct patches.

In order to identify textures, we shall consider a simple, gradient-based texture attribute. Following Liu and Caselles [START_REF] Liu | Exemplar-based image inpainting using multiscale graph cuts[END_REF], we consider the absolute value of the image derivatives, averaged over a certain spatial neighbourhood ν. Obviously, many other attributes could be considered, however they seemed too involved for our purposes.

More formally, we introduce the two-dimensional texture feature T = (T x , T y ), computed at each pixel p ∈ Ω:

(3.5) T (p) = 1 card(ν) q∈ν (|I x (q)|, |I y (q)|),
where I x (q) (resp. I y (q)) is the derivative of the image intensity (grey-level) in the x (resp.

y) direction at the pixel q. The squared patch distance is now defined as

(3.6) d 2 (W p , W q ) = 1 N r∈Np u(r) -u(r -p + q)) 2 2 + λ T (r) -T (r -p + q) 2 2 ,
where λ is a weighting scalar. The feature pyramid is then set up by subsampling the texture features of the fullresolution input video u. We note here that each level is obtained by subsampling the information contained at the finest pyramid resolution, and not by calculating T ℓ based on the subsampled video at the level ℓ:

(3.7) ∀(x, y, t) ∈ Ω ℓ , T ℓ (x, y, t) = T (2 ℓ x, 2 ℓ y, t), ℓ = 1 • • • L.
This is an important point, since the required textural information does not exist at coarser levels. Features are not filtered before subsampling, since they have already been averaged over the neighbourhood ν. In the experiments done in this paper, this neighbourhood is set, by default, to the area to which a coarsest-level pixel corresponds, which is a square of size 2 L-1 , as is done in [START_REF] Liu | Exemplar-based image inpainting using multiscale graph cuts[END_REF]. However, in a more general setting, the size of this area should be independent of the number of levels, so care should be taken in the case where few pyramid levels are used.

A notable difference with respect to the work of Liu and Caselles [START_REF] Liu | Exemplar-based image inpainting using multiscale graph cuts[END_REF] is the fact that we use the texture features at all pyramid levels. Liu and Caselles do not do this, since they perform graph cut based optimisation at the coarsest level, and at the finer levels only consider small relative shifts with respect to the coarse solution.

A final choice which must be made when using the texture features is how they are themselves reconstructed. In shift maps based algorithms, this is not a problem, since by definition an occluded pixel takes on the characteristics of its correspondent in D (colour, texture features or anything else).

In our case, we inpaint the texture features using the same reconstruction scheme as is used for colour information (see Eq. 3.2):

(3.8) T (p) = q∈Np s q p T (p + φ(q)) q∈Np s q p , ∀p ∈ H.
Conceptually, the use of these features is quite simple, and easily fits into our inpainting framework. To summarise, these features may be seen as simple texture descriptors which help the algorithm avoid making mistakes when choosing the area to use for inpainting.

The methodology which we have proposed for dealing with dynamic video textures is important for the following reasons. Firstly, to the best of our knowledge, this is the first inpainting approach which proposes a global optimisation and which can deal correctly with textures in images and videos, without restricting the search space (contrary to [START_REF] Criminisi | Object removal by exemplar-based inpainting[END_REF][START_REF] Darabi | Image melding: combining inconsistent images using patch-based synthesis[END_REF][START_REF] Granados | How not to be seen: Object removal from videos of crowded scenes[END_REF][START_REF] Liu | Exemplar-based image inpainting using multiscale graph cuts[END_REF][START_REF] Pritch | Shift-map image editing[END_REF]). Secondly, while the problem of recreating video textures is a research subject in its own right and algorithms have been developed for their synthesis [START_REF] Doretto | Dynamic textures[END_REF][START_REF] Kwatra | Graphcut textures: image and video synthesis using graph cuts[END_REF][START_REF] Schödl | Video textures[END_REF], ours is the first algorithm to achieve this within an inpainting framework. Finally we note that algorithms such Original frame : "Jumping man" [START_REF] Granados | Background inpainting for videos with dynamic objects and a free-moving camera[END_REF] Inpainting without realignement Inpainting with realignement as that of Granados et al. [START_REF] Granados | Background inpainting for videos with dynamic objects and a free-moving camera[END_REF], which are specifically dedicated to background reconstruction, cannot deal with background video textures (such as waves), since they suppose that the background is rigid. This hypothesis is clearly not true for video textures. An example of the impact of the texture features in video inpainting may be seen in Figure 7.

3.4.

Inpainting with mobile background. We now turn to another common case of video inpainting, that of mobile backgrounds. This is the case, for example, when hand-held cameras are used to capture the input video.

There are several possible solutions to this problem. Patwardhan et al. [START_REF]Video inpainting under constrained camera motion[END_REF] segment the video into moving foreground and background (which may also display motion) using motion estimation with block matching. Once the moving foreground is inpainted, the background is realigned with respect to a motion estimated with block matching, and the background is filled by copying and pasting background pixels. In this case, the background should be perfectly realigned.

Granados et al. [START_REF] Granados | Background inpainting for videos with dynamic objects and a free-moving camera[END_REF] propose a homography-based algorithm for this task. They estimate a set of homographies between each frame and choose which homography should be used for each occluded pixel belonging to the background.

Both of these algorithms require that the background and foreground be segmented, which we wish to avoid here. Furthermore, they have the quite strict limitation that pixels are simply copied from their realigned positions, meaning that the realignment must be extremely accurate. Here, we propose a solution which allows us to use our patch-based variational framework for both tasks (foreground and background inpainting) simultaneously, without any segmentation of the video into foreground and background.

The fundamental hypothesis behind patch-based methods in images or videos is that content is redundant and repetitive. This is easy to see in images, and may appear to be the case in videos. However, the temporal dimension is added in video patches, meaning that a sequence of image patches should be repeated throughout the video. This is not the case when a video displays random motion (as with a mobile camera): even if the required content Algorithm 2: Pre-processing step for realigning the input video.

Data: Input video u Result: Aligned video and affine warps N f = number of frames in video;

N m = ⌊ N f 2 ⌋; for n = 1 to N f -1 do θ n,n+1 ← EstimateAffineMotion(u n , u n+1 ); end for n = 1 to N f do if n < N m then θ n,Nm = θ Nm-1,Nm • • • • • θ n,n+1 ; if n > N m then θ n,Nm = θ -1 Nm-1,Nm • • • • • θ -1 n,n+1 ; u n ← AffineWrap(u n , θ n,Nm )
; end appears at some point in the sequence, there is no guarantee that the required spatio-temporal patches will repeat themselves with the same motion. Empirically, we have observed that this is a significant problem even in the case of motions with small amplitude.

To counter this problem, we estimate a dominant, affine motion between each pair of successive frames, and use this to realign each frame with respect to one reference frame. In our work, we chose the reference frame to be the middle frame of the sequence (this should be adapted for larger sequences). We use the work of Odobez and Bouthemy [START_REF] Odobez | Robust multiresolution estimation of parametric motion models[END_REF] to realign the frames. The colour values of the pixels in the realigned frames are obtained using linear interpolation. The occlusion H is obviously also realigned. Once the frames are realigned with the reference frame (Alg. 2), we inpaint the video as usual. Finally, when inpainting is finished, we perform the inverse affine transformation on the images and paste the solution into the original occluded area. Figure 5 compares the results with and without this preprocessing step, on a prototypical example. Without it, it is not possible to find coherent patches which respect the border conditions.

Initialisation of the solution.

The iterative procedure at the heart of our algorithm relies on an initial inpainting solution. The initialisation step is very often left unspecified in work on video inpainting. As we shall see in this Section, it plays a vital role, and we therefore explain our chosen initialisation method in detail.

We inpaint at the coarsest level using an "onion peel" approach, that is to say we inpaint one layer of the occlusion at a time, each layer being one pixel thick.

More formally, let H ′ ⊂ H be the current occlusion, and ∂H ′ ⊂ H ′ the current layer to inpaint. We define the unoccluded neighbourhood N ′ p of a pixel p, with respect to the current occlusion H ′ as:

(3.9) N ′ p = {q ∈ N p , q / ∈ H ′ }.
Some choices are needed to implement this initialisation method. First of all, we only compare the unoccluded pixels during a patch comparison. The distance between two patches . Impact of initialisation schemes on inpainting results. Note that the only scheme which successfully reconstructs the cardboard tube is the "onion peel" approach where first filling at the coarsest resolution is conducted in a greedy fashion.

W p and W p+φ(p) is therefore redefined as:

(3.10) d 2 (W p , W p+φ(p) ) = 1 |N ′ p | q∈N ′ p u(q) -u(q + φ(p)) 2 2 + λ T (q) -T (q + φ(p)) 2 2 .
We also need to choose which neighbouring patches to use for reconstruction. Some will be quite unreliable, as only a small part of the patches are compared. In our implementation, we only use the ANNs of patches whose centres are located outside the current occlusion layer. Formally, we reconstruct the pixels in the current layer by using the following formula, modified from Equation 3.2:

(3.11) u p = q∈N ′ p s q p u(p + φ(q)) q∈N ′ p s q p .
The same reconstruction is applied to the texture features. A pseudo-code for the initialisation procedure may be seen in Algorithm 3.

Algorithm 3: Inpainting initialisation

Data: Coarse level inputs u L , T L , φ L , H L Result: Coarse initial filling u L , T L and map φ L B ← 3 × 3 × 3 structuring element; H ′ ← H L ; while H ′ = ∅ do ∂H ′ ← H ′ \Erosion(H ′ , B); φ L ← ANNsearch(u L , T L , φ L , ∂H ′ ) ; // Alg.1, with partial distance (3.10) u L ← Reconstruction(u L , φ L , ∂H ′ ) ; // Eq.3.11 T L ← Reconstruction(T L , φ L , ∂H ′ ) ; // Eq.3.11 H ′ ← Erosion(H ′ , B); end
Figure 6 shows some evidence to support a careful initialisation scheme. Three different initialisations have been tested: random initialisation, zero-Laplacian interpolation and onion peel. Random initialisation is achieved by initialising the occlusion with pixels chosen randomly from the image. Zero Laplacian (harmonic) interpolation is the solution of the Laplace equation ∆u = 0 with Dirichlet boundary conditions stemming from D. It may be seen (Figure 6) that the first two initialisations are unable to join the two parts of the cardboard tube together, and that the subsequent iterations do not improve the situation. In contrast, the proposed initialisation produces a satisfactory result.

In order to make our method as easy as possible to reimplement, we now present some further algorithmic details, which are in fact very important for achieving good results.

3.6. Implementation of the multi-resolution scheme. Our first remarks concern the implementation of the multi-resolution pyramids. Wexler et al. and Granados et al. both note that temporal subsampling can be detrimental to inpainting results. This is due to the difficulty of representing motion at coarser levels. For this reason we do not subsample in the temporal direction, as in [START_REF] Granados | How not to be seen: Object removal from videos of crowded scenes[END_REF]. The only case where we need to temporally subsample is when the objects spend a long time behind the occlusion, (this was only done in the "Jumping girl" sequence). This is quite a hard problem to solve since it becomes increasingly difficult to decide what motion an occluded object should have when the occlusion time grows longer, unless there is strictly periodic motion. We leave this as an open question, which could be investigated in further work.

A crucial choice when using multi-resolution schemes is the number of pyramid levels to use. Most other methods leave this parameter unspecified, or fix the size of the image/video at the coarsest scale, and determine the resulting number of levels [START_REF] Granados | How not to be seen: Object removal from videos of crowded scenes[END_REF][START_REF] Liu | Exemplar-based image inpainting using multiscale graph cuts[END_REF][START_REF] Pritch | Shift-map image editing[END_REF]. In fact, when one considers the problem in more detail, it becomes apparent that the number of levels should be set so that the occlusion size is not too large in comparison to the patch size. This intuition is supported by experiments in very simple image inpainting situations, which showed that the occlusion size should be somewhat less than twice the patch size. In our experiments, we follow this general rule of thumb.

Original frame : "Waves"

Inpainting without features Inpainting with features Another question which is of interest is how to pass from one pyramid level to another. Wexler et al. presented quite an intricate scheme in [START_REF]Space-time completion of video[END_REF] to do this, whereas Granados et al. propose a simple upsampling of the shift map. This is conceptually simpler than the approach of Wexler et al. and, after experimentation, we chose this option as well. Therefore, the shift map φ is upsampled using nearest neighbours interpolation, and both the higher resolution video and the higher resolution texture features are reconstructed using Equation 3.2. One final note on this point is that we use the upsampled version of φ as an initialisation for the PatchMatch algorithm at each level apart from the coarsest (this differs to our previous work in [START_REF] Newson | Towards fast, generic video inpainting[END_REF]).

Various implementation details.

We also require a threshold which will stop the iterations of the ANN search and reconstruction steps. In our work, we use the average colour difference in each channel per pixel between iterations as a stopping criterion. If this falls below a certain threshold, we stop the iteration at the current level. We set this threshold to 0.1. In order to avoid iterating for too long, so we also impose a maximum number of 20 iterations at any given pyramid level.

The patch size parameters were set to 5 × 5 × 5 in all of our experiments. We set the texture feature parameter λ to 50. Concerning the spatio-temporal PatchMatch, we use ten iterations of propagation/random search during the ANN search algorithm and set the window size reduction factor β to 0.5 (as in the paper of Barnes et al. [START_REF] Barnes | Patchmatch: a randomized correspondence algorithm for structural image editing[END_REF]).

The complete algorithm is summarized in Alg. 4.

Experimental results.

The goal of our work is to achieve high quality inpainting results in varied, complex video inpainting situations, with reduced execution time. Therefore, we shall evaluate our results in terms of visual quality and execution time.

We compare our work to that of Wexler et al. [START_REF]Space-time completion of video[END_REF] and to the most recent video inpainting method of Granados et al. [START_REF] Granados | How not to be seen: Object removal from videos of crowded scenes[END_REF]. All of the videos in this paper (and more) can be viewed and downloaded along with occlusion masks at http://www.telecom-paristech.fr/ ~gousseau/video_inpainting An implementation of our method is also available at this address.

Visual evaluations.

First of all, we have tested our algorithm on the videos proposed by Wexler et al. [START_REF]Space-time completion of video[END_REF] and Granados et al [START_REF] Granados | How not to be seen: Object removal from videos of crowded scenes[END_REF]. The visual results of our algorithm may be Original frame : "Girl" [START_REF] Granados | Background inpainting for videos with dynamic objects and a free-moving camera[END_REF] Result from [START_REF] Granados | Background inpainting for videos with dynamic objects and a free-moving camera[END_REF] Our result Figure 8. A comparison of our inpainting result with the that of the background inpainting algorithm of Granados et al. [START_REF] Granados | Background inpainting for videos with dynamic objects and a free-moving camera[END_REF]. In such cases with moving background, we are able to achieve high quality results (as do Granados et al.), but we do this in one, unified algorithm. This illustrates the capacity of our algorithm to perform well in a wide range of inpainting situations.

"Beach Umbrella" "Crossing Ladies" "Jumping girl"

Original frames

Inpainting result from [START_REF]Space-time completion of video[END_REF] Our inpainting result seen in Figures 9 and10. We note that the inpainting results of the previous authors on these examples are visually almost perfect, so very little qualitative improvement can be made. It may be seen that our results are of similarly high quality to the those of the previous algorithms. In particular we are able to deal with situations where several moving objects must be correctly recreated, without requiring manual segmentation as in [START_REF] Granados | How not to be seen: Object removal from videos of crowded scenes[END_REF]. We also achieve these results in at least an order of magnitude less time than the previous algorithms. We Original frame : "Duo" Original frame : "Museum"

Inpainting result from [START_REF] Granados | How not to be seen: Object removal from videos of crowded scenes[END_REF] Our inpainting result We achieve similar results to those of [START_REF] Granados | How not to be seen: Object removal from videos of crowded scenes[END_REF] in an order of magnitude less time, without user intervention. The occlusion masks are highlighted in green.

Algorithm 4: Proposed video inpainting algorithm.

Data: Input video u over Ω, occlusion H, resolution number L Result: Inpainted video

(u, θ) ←AlignVideo(u) ; // Alg.2 {u ℓ } L ℓ=1 ←ImagePyramid(u); {T ℓ } L ℓ=1 ←TextureFeaturePyramid(u); // Eqs.3.5-3.7 {H ℓ } L ℓ=1 ←OcclusionPyramid(H); φ L ←Random; (u L , T L , φ L ) ←Initialisation(u L , T L , φ L , H L ); // Alg.3 for ℓ = L to 1 do k = 0, e = 1; while e > 0.1 and k < 20 do v = u ℓ ; φ ℓ ←ANNsearch(u ℓ , T ℓ , φ ℓ , H ℓ ) ; // Alg.1 u ℓ ←Reconstruction(u ℓ , φ ℓ , H ℓ ) ; // Eqs.3.2-3.3 T ℓ ←Reconstruction(T ℓ , φ ℓ , H ℓ ); e = 1 3|H ℓ | u ℓ H ℓ -v H ℓ 2 ; k ← k + 1; end if ℓ = 1 then u ←FinalReconstruction(u 1 , φ 1 , H) ; // Eq.3.4 else φ ℓ-1 ←UpSample(φ ℓ , 2) ; // Sec.3.6 u ℓ-1 ←Reconstruction(u ℓ-1 , φ ℓ-1 , H ℓ-1 ); T ℓ-1 ←Reconstruction(T ℓ-1 , φ ℓ-1 , H ℓ-1 ); end end u ←UnwarpVideo(u, θ)
note that it is not feasible to apply the method of Wexler et al. to the examples of [START_REF] Granados | How not to be seen: Object removal from videos of crowded scenes[END_REF], whose resolution is too large (up to 1120×754×200 pixels).

Next, we provide experimental evidence to show the ability of our algorithm to deal with various situations which appear frequently in real videos, some of which are not dealt with by previous methods. Figure 7 shows an example of the utility of using texture features in the inpainting process: without them, the inpainting result is quite clearly unsatisfactory. We have not directly compared these results with previous work. However it is quite clear that the method of [START_REF] Granados | How not to be seen: Object removal from videos of crowded scenes[END_REF] cannot deal with such situations. This method supposes that the background is static, and in the case of dynamic textures, it is not possible to restrict the search space as proposed in the same method for moving objects. Furthermore, the background inpainting algorithm of Granados et al. [START_REF] Granados | Background inpainting for videos with dynamic objects and a free-moving camera[END_REF] supposes that moving background undergoes a homographic transformation, which is clearly not the case for video textures. By relying on a plain colour distance between patches, the algorithm of Wexler et al. is likely to produce results similar to the one which may be seen in Figure 7 (middle image). Finally, to take another algorithm of the literature, the method of Patwardhan et al. [START_REF]Video inpainting under constrained camera motion[END_REF] would encounter the same problems as that of [START_REF] Granados | Background inpainting for videos with dynamic objects and a free-moving camera[END_REF], since they copy-and-paste pixels directly after compensating for a locally estimated motion. More examples of videos containing dynamic textures can be seen at http://www.telecom-paristech.fr/~gousseau/video_inpainting.

Our algorithm's capacity to deal with moving background is illustrated by Figure 8. We do this in the same unified framework used for all other examples in this paper, whereas a specific algorithm is needed by Granados et al. [START_REF] Granados | Background inpainting for videos with dynamic objects and a free-moving camera[END_REF] to achieve this. Thus, we see that the same core algorithm (iterative ANN search and reconstruction) can be used in order to deal with a series of inpainting tasks and situations. Furthermore, we note that no foreground/background segmentation was needed for our algorithm to produce satisfactory results. Finally, we note that such situations are not managed using the algorithm of [START_REF]Space-time completion of video[END_REF]. Again, examples containing moving backgrounds can be viewed at the referenced website.

The generic nature of the proposed approach represents a significant advantage over previous methods, and allows us to deal with many different situations without having to resort to manual intervention or the creation of specific algorithms.

Execution times.

One of the goals of our work was to accelerate the video inpainting task, since this was previously the greatest barrier to development in this area. Therefore, we compare our execution times to those of [START_REF]Space-time completion of video[END_REF] and [START_REF] Granados | How not to be seen: Object removal from videos of crowded scenes[END_REF] 1 Partial and total execution times on different examples. The partial inpainting times represent the time taken for the ANN search for all occluded patches at the full resolution. Note that for the "museum" example, Granados's algorithm is parallelised over the different occluded objects and the background, whereas ours is not.

Comparisons with Wexler's algorithm should be obtained carefully, since several crucial parameters are not specified. In particular, the ANN search scheme used by Wexler et al. requires a parameter, ε, which determines the accuracy of the ANNs. More formally, if W p is a source patch, W q is the exact NN of W p and W r is an ANN of W p , then the work of [START_REF] Arya | Approximate nearest neighbor queries in fixed dimensions[END_REF] guarantees that d(W p , W r ) ≤ (1 + ε)d(W p , W q ). This parameter has a large influence on the computational times. For our comparisons, we set this parameter to 10, which produced ANNs with a similar average error per patch component as our spatio-temporal PatchMatch. Another parameter which is left unspecified by Wexler et al. is the number of iterations of ANN search/reconstruction steps per pyramid level. This has a very large influence on the total execution time. Therefore, instead of comparing total execution times we simply compare the ANN search times, as this step represents the majority of the computational load. We obtain a speedup of 20-50 times over the method of [START_REF] Arya | Approximate nearest neighbor queries in fixed dimensions[END_REF]. We also include our total execution times, to give a general idea of the time taken with respect to the video size. These results show a speedup of around an order of magnitude with respect to the semi-automatic methods of Granados et al [START_REF] Granados | How not to be seen: Object removal from videos of crowded scenes[END_REF]. In Table 1, we have also added our execution times without the use of texture features to illustrate the additional computational load which this adds.

These computation times show that our algorithm is clearly faster than the approaches of [START_REF]Space-time completion of video[END_REF] and [START_REF] Granados | How not to be seen: Object removal from videos of crowded scenes[END_REF]. This advantage is significant because not only is the algorithm more practical to use, but it is also much easier to experiment and therefore make progress in the domain of video inpainting.

5. Further work. Several points could be improved upon in the present paper. Firstly, the case where a moving object is occluded for long periods remains very difficult and is not dealt with in a unified manner here. The one solution to this problem (temporal subsampling) does not perform well when complex motion is present. Therefore, other solutions could be interesting to explore. Secondly, we have observed that using a multi-resolution texture feature pyramid produces very interesting results. Therefore, we could perhaps enrich the patch space with other features, such as spatio-temporal gradients. Finally, it is acknowledged that videos of high resolutions still take quite a long time to process (up to several hours). Further acceleration could be achieved by dimensionality-reducing transformations of the patch space.

Conclusion.

In this paper, we have proposed a non-local patch-based approach to video inpainting which produces good quality results in a wide range of situations, and on high definition videos, in a completely automatic manner. Our extension of the PatchMatch ANN search scheme to the spatio-temporal case reduces the time complexity of the algorithm, so that high definition videos can be processed. We have also introduced a texture feature pyramid which ensures that dynamic video textures are correctly inpainted. The case of mobile cameras and moving background is dealt with by using a global, affine estimation of the dominant motion in each frame.

The resulting algorithm performs well in a variety of situations, and does not require any manual input or segmentation. In particular, the specific problem of inpainting textures in videos has been addressed, leading to much more realistic results than other inpainting algorithms. Video inpainting has yet not been extensively used, in a large part due to prohibitive execution times and/or necessary manual input. We have directly addressed this problem in the present work. We hope that this algorithm will make video inpainting more accessible to a wider community, and help it to become a more common tool in various other domains, such as video post-production, restoration and personal video enhancement.

7.

Acknowledgments. The authors would like to express their thanks to Miguel Granados for his kind help, for making his video content publicly available and for answering several questions concerning his work. The authors also thank Pablo Arias, Vicent Caselles and Gabriele Facciolo for numerous insightful discussions and for their help in understanding their inpainting method.

Appendix A. On the link between the non-local patch-based and shift map-based formulations.

In the inpainting literature, two of the main approaches which optimise a global objective functional include non-local patch-based methods [START_REF] Arias | A variational framework for exemplar-based image inpainting[END_REF][START_REF]Space-time completion of video[END_REF], which are closely linked to Non-Local Means denoising [START_REF] Buades | A non-local algorithm for image denoising[END_REF], and shift map-based formulations such as [START_REF] Granados | How not to be seen: Object removal from videos of crowded scenes[END_REF][START_REF] Liu | Exemplar-based image inpainting using multiscale graph cuts[END_REF][START_REF] Pritch | Shift-map image editing[END_REF]. We will show here that the two formulations are closely linked, and in particular that the shift map formulation is a specific case of the very general formulation of Arias et al. [START_REF] Arias | A variational framework for exemplar-based image inpainting[END_REF][START_REF] Arias | Analysis of a variational framework for exemplar-based inpainting[END_REF].

As far as possible we keep the notation introduced previously. In particular p is a position in H, q is a position in N p and r is a position in D. In its most general version, the variational formulation of Arias et al. is not based on a one-to-one shift map, but rather on a positive weight function w : Ω × Ω → R + that is constrained by r w(p, r) = 1 to be a probability distribution for each point p. Inpainting is cast as the minimisation of the following energy functional (that we rewrite here in its discretised version):

(A.1) E arias (u, w) = p∈H   r∈ D w(p, r)d 2 (W p , W r ) + γ r∈ D w(p, r) log w(p, r)   ,
with γ a positive parameter. The first term is a "soft assignment" version of (2.1), while the second term is a regulariser that favors large entropy weight maps.

Arias et al. propose the following patch distance:

(A.2) d 2 (W p , W r ) = q∈Np g a (p -q)ϕ[u(q) -u(r + (p -q))],
where g a is the centered Gaussian with standard deviation a (also called the intra-patch weight function), and ϕ is a squared norm. This is a very general and flexible formulation. Arias et al. optimise this function using an alternate minimisation over u and w, and derive solutions for various patch distances. Let us choose the ℓ 2 distance for d(W p , W r ), as is the case in many inpainting formulations (and in particular the one which we use in our work). In this case, the minimisation scheme leads to the following expressions:

w(p, r) = 1 Z p exp - d 2 (W p , W r ) γ , (A.3) u(p) = q∈Np g a (p -q) r∈ D w(q, r)u(r + (p -q)) . (A.4)
The parameter γ controls the selectivity of the weighting function w. Let us consider the case where each weight function w(p, .) is equal to a single Dirac centered at a single match p + φ(p). If, in addition, we consider that the intra-patch weighting is uniform, in other words a = ∞, the cost function E arias reduces to:

(A.5) E arias (u, φ) = p∈H q∈Np u(q) -u(q + φ(p)) 2 2 ,
which is the formulation of Wexler et al. [START_REF]Space-time completion of video[END_REF]. Rewriting (A.4) in the particular case just described, yields the optimal inpainted image

Table 2

Probability of producing a random 2D or 3D patch that is closer to a random reference patch than to a constant one with same mean value. Values are obtained through numerical simulations averaged over ten runs for each experiment. Components of random patches are i.i.d. according to the centred normal law with a grey level variance of 25.

of detail, leading to difficulties in correctly identifying textures. In fact, we found that this difficulty may occur at all the pyramid levels in images and videos. Roughly speaking, we observed that textured patches are quite likely to be matched with smooth ones. The following simple computations quantify this phenomenon.

B.1. Comparing patches with the classical ℓ 2 distance. The first reason concerns the patch distance. Let us consider a white noise patch, W , which is a vector of i.i.d. random variables W 1 • • • W N , where N is the number of components in the patch (number of pixels for grey-level patches), and the distribution of all W i 's is f W . Let µ and σ 2 be, respectively, the average and variance of f W . Let us consider another random patch V following same distribution, and the constant patch Z, composed of

Z i = µ, i = 1 • • • N .
In this simple situation, we see that

E[ W -V 2 2 ] = 2E[ W -Z 2 2 ]
. Therefore, on average, the sum-of-squared-differences (SSD) between two patches of the same distribution is twice as great as the SSD between a randomly distributed patch and a constant patch of value µ.

The previous remark is only valid on average between three patches W , V and Z. In reality, we have many random patches V to choose from, and it is sufficient that one of these be better than Z for the least patch distance to identify a "textured" patch. Therefore, a more interesting question is the following. Given a white noise patch W , what is the probability that the patch V will be better than the constant patch Z. This is slightly more involved, and we shall limit ourselves to the case where W and V consist of i.i.d. pixels with a normal distribution.

The SSD patch distance between W and V follows a chi-square distribution χ 2 (0, 2σ 2 ), and that between W and Z follows χ 2 (0, σ 2 ). With this, we may numerically compute the probability of a random patch being better than a constant one. Since the chi-squared law is tabulated, it is much the same thing to use numerical simulations.

In Table 2, we show the corresponding numerical values for both 2D (image) and 3D (video) patches. It may be seen that for a patch of size 9 × 9, there is very little chance of finding a better patch than the constant patch. In the video case, we see that in the case of 5 × 5 × 5 patches, there is a 2 × 10 -7 probability of creating a better patch randomly. This corresponds to needing an area of 170 × 170 × 170 pixels in a video in order to produce on average one better random patch. While this is possible, especially in higher-definition videos, it remains unlikely for many situations.

The question naturally arises of why the problem of comparing textures has not been more discussed in the patch-based inpainting literature. Indeed, to the best of our knowledge, only Bugeau et al. [START_REF] Bugeau | A comprehensive framework for image inpainting[END_REF] and Liu and Caselles [START_REF] Liu | Exemplar-based image inpainting using multiscale graph cuts[END_REF] have clearly identified this problem in the case of image inpainting. This is due to the fact that most other inpainting algorithms restrict the ANN search space to a local neighbourhood around the occlusion. Unfortunately, this restriction principle does not hold in video inpainting since the information can be found anywhere in the video volume, in particular when a complex movement must be reconstructed. B.2. ANN search with PatchMatch. We have indicated that the ℓ 2 grey-level/colour patch distance is problematic for inpainting in the presence of textures. Additionally, this problem is exacerbated by the use of PatchMatch. Indeed, the values of φ which lead to textures are not piecewise constant, and are therefore not propagated through φ during the PatchMatch algorithm. On the other hand, smooth patches represent on average a good compromise as ANNs and are the shifts which lead to them are piecewise constant and therefore well propagated throughout φ. Another problem which may lead to smooth patches being used is the weighted average reconstruction scheme. This can lead to blurry results which in turn means that smooth patches are identified.

One solution to these problems is the use of our texture feature pyramid ( §3.3). This pyramid is inpainted simultaneously with the colour video pyramid, and thus helps to guide the algorithm in the choice of which patches to use for inpainting.

Figure 11 shows an interesting situation: we wish to inpaint a region which contains white noise. This toy example serves as an illustration of the appeal of our texture features. Indeed, it is quite clear that without them, there is no chance of inpainting the occlusion in a manner which would seem "natural" to a human observer, whereas with them it is possible, in effect, to "inpaint noise".

Figure 1 .

 1 Figure 1. Illustration of the notation used for proposed video inpainting algorithm.

Figure 2 .

 2 Figure 2. Diagram of the proposed video inpainting algorithm.

3. 2 .

 2 Video reconstruction. Concerning the reconstruction step, we use a a weighted mean based approach, inspired by the work of Wexler et al., in which each pixel is reconstructed in Algorithm 1: ANN search with 3D PatchMatch Data: Current inpainting configuration u, φ, H Result: ANN shift map φ for k = 1 to 10 do if k is even then /* Propagation on even iteration */ for p = p 1 to p | H| (pixels in H lexicographically ordered) do a

Figure 3 .

 3 Figure 3. Comparison of different final reconstruction methods.We observe that the proposed reconstruction using only the best patch at the end of the algorithm produces similar results to the use of the mean shift algorithm, avoiding blur induced by weighted patch averaging, while being less computationally expensive. Please note that the blurring effect is best viewed in the pdf version of the paper.

Figure 4 .

 4 Figure 4. Illustration of the necessity of texture features for inpainting. Without the texture features, the correct texture may not be found.

Figure 5 .

 5 Figure 5. A comparison of inpainting results with and without affine realignment. Notice the incoherent reconstruction on the steps, due to random camera motion which makes spatio-temporal patches difficult to compare. These random motions are corrected with affine motion estimation and inpainting is performed in the realigned domain.

Figure 6

 6 Figure 6. Impact of initialisation schemes on inpainting results. Note that the only scheme which successfully reconstructs the cardboard tube is the "onion peel" approach where first filling at the coarsest resolution is conducted in a greedy fashion.

Figure 7 .

 7 Figure 7. Usefulness of the proposed texture features. Without the features, the algorithm fails to recreate correctly the waves, which is a typical example of complex video texture.

Figure 9 .

 9 Figure 9. Comparison with Wexler et al. We achieve results of similar visual quality compared to those in [39], with a reduction of the ANN search time by a factor of up to 50 times.

Figure 10 .

 10 Figure 10. Comparison with Granados et al. We achieve similar results to those of[START_REF] Granados | How not to be seen: Object removal from videos of crowded scenes[END_REF] in an order of magnitude less time, without user intervention. The occlusion masks are highlighted in green.

8 ×

 8 10 -2 10 -2 6 × 10 -3 4.1 × 10 -4 5.5 × 10 -6 3 × 10 -7 2 × 10 -7

Figure 11 .

 11 Figure 11. A toy example of the utility of the textures features. With them, we are able to distinguish between white noise (right) and the constant area (left), and thus recreate the noise.

  in Table 1.

	Algorithm	ANN execution times for all occluded pixels at full resolution.
		Beach Umbrella Crossing Ladies Jumping Girl	Duo	Museum
		264 × 68 × 98	170 × 80 × 87	300 × 100 × 239	960 × 704 × 154	1120 × 754 × 200
	Wexler (kdTrees)	985 s	942 s	7877 s	-	-
	Ours (3D PatchMatch)	50 s	28 s	155 s	29 min	44 min
	Algorithm		Total execution time
	Granados	11 hours	-	-	-	90.0 hours
	Ours	24 mins	15 mins	40 mins	5.35 hours	6.2 hours
	Ours w/o texture	14 mins	12 mins	35 mins	4.07 hours	4.0 hours
			Table

as the (aggregated) average of the examples indicated by the NNs of the patches which contain p. Suppose that each pixel is reconstructed using the NN of the patch centred on it: (A. [START_REF] Bertalmio | Image inpainting[END_REF] u(p) = u(p + φ(p)), ∀p ∈ H, as is the case in the shift map-based formulations. Then the functional becomes 1 :

which effectively depends only upon φ.

If we look at the shift map formulation proposed by Pritch et al., we find the following cost function over the shift map only:

Let us consider the first part, concerning the image colour values. Since we have u(p) = u(p + φ(p)), we obtain again:

Thus, the shift map cost function may be seen as a special case of the non-local patch-based formulation of Arias et al. under the following conditions :

• The intra-patch weighting function g a is uniform; • u(p) = u(p + φ(p)), that is, u(p) is reconstructed using its correspondent according to a single shift map. Arias et al. [START_REF] Arias | Analysis of a variational framework for exemplar-based inpainting[END_REF] have shown the existence of optimal correspondance maps for a relaxed version of Wexler's energy (Equation A.1), and also that the minima of Equation A.1) converge to the minima of Equation A.5 as γ → 0. Such results also highlight the link between different inpainting energies.

However, one should keep in mind that the two formulations which we have considered (that of Arias et In this Appendix, we look in further detail at the reasons why textures may pose a problem when comparing patches for the purposes of inpainting. Liu and Caselles noted in [START_REF] Liu | Exemplar-based image inpainting using multiscale graph cuts[END_REF] that the subsampling necessary for the use of multi-resolution pyramids inevitably entails a loss 1 We note that using the reconstruction of Equation A.7 poses problems on the occlusion border, but we ignore this here for the sake of simplicity and clarity.


L Fournier

D Delafosse

C Bosch

T Magnin

STRESS CORROSION CRACKING OF NICKEL BASE SUPERALLOYS IN AERATED SUPERCRITICAL WATER

Keywords: stress corrosion cracking, nickel base superalloy, supercritical water oxidation (SCWO), alloy 718, alloy 690, oxygen embrittlement

The stress corrosion cracking behaviour of two nickel base superalloys (A718 and A690) is investigated in aerated supercritical water by means of constant extension rate tensile (CERT) tests at 400°C and 25 MPa. Alloy 718 is observed to be extremely sensitive to intergranular stress corrosion cracking (IGSCC) while alloy 690 is found to be immune to IGSCC. CERT tests are also carried out in air and under vacuum on both alloys at 600°C, i.e. in a temperature domain where nickel base superalloys exhibit oxygen embrittlement. The results obtained strongly suggest that oxidation is controlling both the stress corrosion crack initiation and propagation stages of nickel base superalloys in aerated supercritical water. In the light of these results, the main issue of material selection for the construction of supercritical water oxidation (SCWO) reactors is discussed.

INTRODUCTION

In the SCWO process, organic wastes are oxidised into carbon dioxide and water by the molecular oxygen added to water in the supercritical state (T>374°C and P>22 MPa). Due to its low values of density and dielectric constant, supercritical water acts as a non-polar solvent in which organic compounds and oxygen exhibit a total miscibility. Therefore, conversion rates typically greater than 99.99 % are obtained, making SCWO a very effective and promising technology. However, the reactor is subjected to a highly aggressive environment and material selection, through corrosion resistance, is a major problem in the context of development of SCWO. Nickel base superalloys, such as alloy 625 and alloy 718, have been selected for the construction of the first pilot reactors. Under SCWO conditions, these materials have been shown to exhibit significant general corrosion rates ranging from 2 mm/year to 12 mm/year (1-2). Furthermore, stress corrosion cracking (SCC) has also been observed [START_REF] Latanision | ″Corrosion in supercritical water oxidation systems[END_REF] and is believed to be a key issue. In this context, the aim of this work was to investigate, for the first time, the SCC behaviour of nickel base superalloys in aerated supercritical water. A test autoclave was designed and CERT tests were performed at 400°C and 25 MPa on smooth specimens of two different alloys: A718 and A690. Alloy 718 is a nickel base superalloy (18 wt% chromium) mainly strengthened by γ '' precipitates and widely used for highly stressed components in high temperature applications as well as in pressurised water reactor (PWR) systems. Alloy 690 (29 wt% chromium) is a solution hardened nickel base superalloy. Additionally, our objective was to gain insights in the mechanisms that may be responsible for SCC of nickel base superalloys in supercritical water. Kriksunov and Macdonald (4) have emphasized that the transition from subcritical to supercritical water was accompanied by a transition from the corrosion mechanisms characteristic of a liquid phase to those of a gas phase, i.e. from an ionic mechanism to a molecular mechanism. Nickel base superalloys being sensitive to oxygen embrittlement in the temperature range 500°C to 700°C (5-10), the possible role played by oxygen in the SCC process of nickel base superalloys in aerated supercritical water was examined here. For this purpose, CERT tests were performed in air and under vacuum at 600°C on both A718 and A690. Results were compared with the CERT tests carried out in aerated supercritical water.

EXPERIMENTAL PROCEDURE

The chemical compositions of the investigated alloys are given in Table 1. Alloy 718 was conventionally heat treated as follows: 955°C for 1 hour then furnace cooled; 720°C for 8 hours then cooled at a rate of 50°C per hour; 620°C for 8 hours and then air cooled. The average grain size was about 20 µm. Blocky primary niobium carbide precipitates were observed, mostly at grain boundaries (Figure 1.a) which is consistent with the findings of other authors [START_REF] Fox | ″Influence of carbides on the mechanical properties on inconel 718[END_REF][START_REF] Sundararaman | [END_REF]. TEM examination showed γ '' (Ni 3 Nb) precipitates with a mean diameter between 30 nm and 50 nm. Alloy 690 was studied in the solution annealed conditions with a grain size of about 60 µm. TEM examination revealed discrete precipitation of M 23 C 6 carbides at grain boundaries. CERT tests in supercritical water were performed in a test autoclave attached to a SCHENCK RMC100 mechanical test machine. A schematic of the test autoclave is given in Figure 2.a. It consists of a tubular pressure vessel [1] and two smooth cylindrical axes [2 and 3]. The tensile specimen [START_REF] Kriksunov | [END_REF] is attached to one of the ends of each of these axes by means of a threaded nut [5] which hold two alumina clevices. The other ends of the axes are fixed to the frame (axis 2) or to the cross-head (axis 3) of the testing machine. The pressurised volume around the test specimen is about 40 cm 3 and the pressure confinement is handled by both a nitrile mobile seal that is water cooled and a metal-metal gasket. A height cartridge heaters system, a type K thermocouple, and a PID regulator were used to monitor and control the temperature of the fluid. Smooth tensile specimens with 10 mm gage length and 5 mm diameter (Figure 2.b) were polished to a 3 µm finish before testing. CERT tests in air and under vacuum (2×10 -5 mbar) were conducted on a different system already described elsewhere (13). For these tests, smooth tensile specimens with 15 mm gage length and 4 mm diameter (Figure 2.c) also polished to a 3 µm finish were used.

RESULTS

Stress corrosion cracking

Alloy 718. Figure 3 displays the stress-strain curves of alloy 718 specimens strained at 10 -6 s -1 and 400°C in air and in supercritical water at 25 MPa. In supercitical water, the ductility of alloy 718 is significantly decreased in comparison to air. Interestingly, the final fracture in supercritical water does not occur in an abrupt way but by a continuous decrease of the flow stress down to zero. It may also be noted that the specimen strained in air exhibits serrated flow, i.e. Portevin -Le Chatelier effect. The serrations occur after a finite amount of plastic strain close to 5% and persist up to fracture. Following the nomenclature proposed by Chihab et al. (14), the plastic instabilities observed can be designated as type C. In contrast, as its fracture strain is below the critical strain for the occurrence of plastic instabilities, the specimen deformed in supercritical water exhibits a continuous plastic flow. As shown in Figure 4, the specimen strained in supercritical water fails in a brittle manner, with almost no reduction in area. The whole fracture surface of the specimen is intergranular. The absence of ductile tearing may explain the final fracture by the continuous decrease of the flow stress down to zero mentioned earlier. Several secondary cracks can also be noticed on the fracture surface. The examination of the surface of the specimen strained in supercritical water shows that crack initiation occurs at the vicinity of oxidized particles (Figure 5.a). These particles were identified as niobium carbides by EDX analysis (Figure 5.b), which suggests that oxidation and swelling of the primary niobium carbides is responsible for crack initiation in supercritical water. In contrast, at 400°C in air, for the same strain rate of 10 -6 s -1 , fracture occurs in a ductile way by shearing on a macroscopic plane 45° to the tensile axis (Figure 6). This particular mode of failure has already been observed for alloy 718, and was found to coincide with the manifestation of type C plastic instabilities (7).

Alloy 690. The stress-strain curves of alloy 690 specimens strained at 10 -6 s -1 and 400°C, in air and in supercritical water at 25 MPa, are plotted in Figure 7. Clearly, the environment has no significant effect on the fracture strain. Additionally, alloy 690 strained in supercritical water fails in a ductile way with an important reduction in area (Figure 8). In contrast to alloy 718, in the same experimental conditions, alloy 690 seems to be immune to SCC.

Oxygen embrittlement

Alloy 718. Figure 9 displays the stress-strain curves of alloy 718 specimens strained at 5×10 -7 s -1 in air and under vacuum at 600°C. The oxidising environment leads to a significant decrease in the ductility of alloy 718, in comparison to the inert environment. The specimen strained under vacuum exhibits a ductile mode of failure, while the specimen strained in air fails in an intergranular way (Figure 10). The SEM examination of the surface of the specimen strained in air shows that crack initiation occurs, like in aerated supercritical water, at the vicinity of oxidised primary niobium carbides (Figure 11).

Alloy 690. Alloy 690 specimens strained at 5×10 -7 s -1 in air and under vacuum at 600°C exhibit similar fracture strain (Figure 12) and a ductile mode of failure (Figure 13).

DISCUSSION

Stress corrosion cracking mechanism

It is well established that SCC results from local corrosion -deformation interactions at the crack tip [START_REF] Magnin | Advances in Corrosion-Deformation Interactions[END_REF]. In aqueous solution, the hydrogen produced by the cathodic reaction of the corrosion process (ionic mechanism) inside a crack is believed to contribute to its propagation for various materialenvironment systems, such as austenitic stainless steel in boiling MgCl 2 [START_REF] Delafosse | [END_REF] and nickel base alloys in the primary water of PWR [START_REF] Magnin | ″Hydrogen effects on PWR SCC mechanisms in monocrystalline and polycrystalline alloy 600[END_REF]. However, in supercritical water, as emphasized by Kriksunov and Macdonald (4), the corrosion process is controlled by a molecular mechanism rather than by an ionic mechanism. As a consequence, the study of the SCC process of nickel base superalloys in supercritical water requires the investigation of possible molecular mechanisms.

In the present study, we examined the role that oxygen embrittlement may play in the SCC process of nickel base superalloys in aerated supercritical water. Interestingly, A718 and A690 were found to exhibit similar behaviour when strained in aerated supercritical water (400°C and 25 MPa) and in air at 600°C. For alloy 718, in both environments, crack initiation resulted from oxidation and swelling of primary niobium carbides and crack propagation was purely intergranular. Conversely, alloy 690 was found to be immune to any environmental effect in both environments. Oxygen embrittlement of nickel and its alloys has been first documented by Bricknell and Woodford (5). They observed that specimens exposed to air at a temperature typically greater than 1000°C exhibit an important loss of ductility, coupled with intergranular fracture when straining is performed at lower temperature. In the temperature range 500°C to 650°C, nickel base superalloys have also been shown to be sensitive to oxygen embrittlement under tensile dynamic loading. Pandey et al. (6) have reported an important loss of ductility in alloy X-750 strained in air at 625°C, in comparison to vacuum. Several authors (8)(9)(10) have also observed that crack growth rates in air were several orders of magnitude higher than under vacuum, and that fracture mode changed from transgranular under vacuum to intergranular in air, during low cycle fatigue tests performed on compact tension specimens of alloy 718 at 650°C. Finally, recent CERT tests carried out in air and under vacuum at 5×10 -7 s -1 showed that alloy 718 was embrittled by oxygen at 500°C (7), i.e. at a temperature relevant to the SCWO process. From a mechanistic point of view, Bricknell and Woodford (5) clearly demonstrated that grain boundary penetration by oxygen is responsible for oxygen embrittlement above 1000°C. At lower temperature, Molins et al. (8) proposed a different mechanism to explain alloy 718 oxygen embrittlement. These authors have investigated the influence of oxygen partial pressure on both crack growth rate of compact tension specimens during creep-fatigue tests at 650°C, and composition of the oxide formed during the early stages of the oxidation process. They observed the existence of a slow crack growth rate regime at an oxygen partial pressure below 10 -3 mbar and a three orders of magnitude higher crack growth rate regime beyond 10 -3 mbar. The increase in crack growth rate was found to coincide with a transition in the cracking path from transgranular to intergranular. They also observed that at oxygen partial pressures in excess of 10 -3 mbar, the formation of a nickel-rich oxide was favoured in the early stages of the oxidation process, while at oxygen partial pressure below 10 -3 mbar, the formation of a chromiumrich oxide was favoured. On the basis of the observation by Hancock (18) that metal oxidation by cationic diffusion through the oxide results in vacancy generation, Molins et al. (8) suggested that nickel oxide formation at the crack tip and the resulting vacancy injection were responsible for the intergranular embrittlement by oxygen. The application of these mechanisms to explain SCC in supercritical water is certainly not straightforward. However, both mechanisms may be considered. Internal oxidation has been recently proposed as a plausible mechanism to explain IGSCC of alloy 600 in the primary water of PWR [START_REF] Scott | ″Some possible mechanisms of intergranular stress corrosion cracking of alloy 600 in PWR primary water[END_REF] and several studies have been carried out to test its validity [START_REF] Thomas | ″Insights into environmental degradation mechanisms from analytical transmission electron microscopy of SCC cracks[END_REF][START_REF] Newman | ″Internal oxidation and embrittlement of alloy 600[END_REF]. Additional experiments aimed at determining the composition and the mechanism of growth (anionic or cationic) of the oxide films formed on nickel base superalloys in supercritical water would allow to establish whether vacancy injections can be invoked.

Application to material selection for the SCWO process

As shown in the result section, the presence of oxidation sensitive particles is highly detrimental to the resistance of nickel base superalloys to stress corrosion crack initiation in aerated supercritical water. This suggests that the choice of nickel base superalloy hardened by precipitates, which either contain TiC or NbC, is not suitable for the construction of SCWO reactors. In contrast, the good resistance to SCC exhibited by alloy 690 during CERT test in aerated supercritical water suggests that solid solution hardened nickel base superalloy may be suitable as a construction material. At this stage, use of fracture mechanics testing techniques would be particularly interesting in order to get information on the resistance to stress corrosion crack propagation of these nickel base superalloys in supercritical water. However, Speidel and Magdowski [START_REF] Speidel | ″Stress corrosion cracking of nickel base alloys in high temperature water[END_REF] have shown that an increase in yield strength significantly increases the stress corrosion crack growth rates in alloy 600 specimens with different degree of cold-work in de-aerated water at 350°C. Therefore, it is very likely that precipitation hardened nickel base superalloys would exhibit a much higher stress corrosion crack growth rate in supercritical water than solid solution hardened nickel base superalloys, which have a much lower yield strength. Additionally, the chromium content of the nickel base superalloy may be of particular importance. Molins et al. (8) have performed crack propagation tests under various oxygen partial pressure on binary Ni-Cr alloys with 20 wt% and 30 wt% of chromium, respectively. Like for alloy 718, for the alloy with 20 wt% of chromium, they observed the existence of a transition partial pressure between a slow crack growth rate regime and a high crack growth rate regime at 10 -3 mbar. Conversely, the alloy with 30 wt% of chromium only exhibited a slow crack growth rate regime suggesting that it was immune to oxygen embrittlement. Moreover, an important chromium content, through the formation of a protective oxide film, may guarantee a better resistance to general corrosion as well as a better resistance to crack initiation in supercritical water.

CONCLUSIONS

In aerated supercritical water (400 °C, 25 MPa), alloy 718 strained at 10 -6 s -1 is sensitive to SCC. Crack initiation results from oxidation and swelling of primary niobium carbides and crack propagation is purely intergranular. In the same experimental conditions, alloy 690 is immune to SCC. At 600°C, alloy 718 is also sensitive to oxygen embrittlement. CERT test performed at 5×10 -7 s -1 showed that this environmental effect is accompanied by a significant decrease of the fracture strain in air compared to vacuum, and a change in the fracture mode from transgranular under vacuum to intergranular in air. Like in supercritical water, crack initiation results from oxidation and swelling of primary niobium carbides. Conversely, in the same experimental conditions, alloy 690 is not sensitive to oxygen embrittlement. These similarities suggest that oxygen may play a key role in both the stress corrosion crack initiation and propagation mechanisms of nickel base superalloys in aerated supercritical water. Air. 600 °C Vacuum. 600 °C

FIGURE 1 -FIGURE 2 -

 12 FIGURE 1 -Microstructure of (a) alloy 718 and (b) alloy 690.

1 FIGURE 3 -FIGURE 4 -FIGURE 5 -

 1345 FIGURE 3 -Stress-strain curves from CERT tests conducted at 10 -6 s -1 on alloy 718 specimens in aerated supercritical water at 400°C and 25 MPa, and in air at 400°C.

FIGURE 6 -FIGURE 7 -

 67 FIGURE 6 -Fracture surface of the alloy 718 specimen deformed at 10 -6 s -1 in air at 400°C.

FIGURE 8 - 1 FIGURE 9 -FIGURE 10 -FIGURE 11 -

 8191011 FIGURE 8 -Fracture surface of the alloy 690 specimen deformed at 10 -6 s -1 in supercritical water.

FIGURE 12 -FIGURE 13 -

 1213 FIGURE 12 -Stress-strain curves from CERT tests conducted at 5×10 -7 s -1 on alloy 690 specimens in air and under vacuum at 600°C.

TABLE 1 -

 1 Chemical composition (in weight %) of the alloy used in this study.

	Ni	Cr	Fe	Nb	Mo	Al	Ti	Cu	Mn	Si	Co	C	S	B	P
	A718 53.67 17.95 bal. 5.27 2.98 0.52 0.92	0.1	0.04 0.07 0.04 0.033 0.006 0.005 0.002
	A690 bal. 28.64 10.23	-	-	1.27 0.26	-	0.3 0.256	-	0.023 <0.005	-	<0.020

ACKNOWLEDGMENTS

The authors gratefully acknowledge Anjou Recherche, CNIM, CNRS and Ecodev for their financial support as well as Aubert et Duval for supplying the materials.