Yassine Ouali
email: yassine.ouali@centralesupelec.fr

Céline Hudelot
email: celine.hudelot@centralesupelec.fr

Myriam Tami
email: myriam.tami@centralesupelec.fr

Autoregressive Unsupervised Image Segmentation

Keywords: Image segmentation, Autoregressive models, Unsupervised learning, Clustering, Representation learning

In this work, we propose a new unsupervised image segmentation approach based on mutual information maximization between different constructed views of the inputs. Taking inspiration from autoregressive generative models that predict the current pixel from past pixels in a raster-scan ordering created with masked convolutions, we propose to use different orderings over the inputs using various forms of masked convolutions to construct different views of the data. For a given input, the model produces a pair of predictions with two valid orderings, and is then trained to maximize the mutual information between the two outputs. These outputs can either be low-dimensional features for representation learning or output clusters corresponding to semantic labels for clustering. While masked convolutions are used during training, in inference, no masking is applied and we fall back to the standard convolution where the model has access to the full input. The proposed method outperforms current state-of-the-art on unsupervised image segmentation. It is simple and easy to implement, and can be extended to other visual tasks and integrated seamlessly into existing unsupervised learning methods requiring different views of the data.

Introduction

Supervised deep learning has enabled great progress and achieved impressive results across a wide number of visual tasks, but it requires large annotated datasets for effective training. Designing such fully-annotated datasets involves a significant effort in terms of data cleansing and manual labeling. It is especially true for fine-grained annotations such as pixel-level annotations needed for segmentation tasks, where the annotation cost per image is considerably high [5,[START_REF] Girshick | Rich feature hierarchies for accurate object detection and semantic segmentation[END_REF]. This hurdle can be overcome with unsupervised learning, where unknown but useful patterns can be extracted from the easily accessible unlabeled data. Recent advances in unsupervised learning [START_REF] Hjelm | Learning deep representations by mutual information estimation and maximization[END_REF][START_REF] Ji | Invariant information clustering for unsupervised image classification and segmentation[END_REF][START_REF] Caron | Unsupervised pre-training of image features on non-curated data[END_REF][START_REF] Oord | Representation learning with contrastive predictive coding[END_REF], that closed the performance gap with its supervised counterparts, make it a strong possible alternative.

Recent works are mainly interested in two objectives, unsupervised representation learning and clustering. Representation learning aims to learn semantic Fig. 1. Overview. Given an encoder-decoder type network F and two valid orderings (o1, o2) as illustrated in (c). The goal is to maximize the Mutual Information (MI) between the two outputs over the different views, i.e. different orderings. (a) For Autoregressive Clusterings (AC), we output the cluster assignments in the form of a probability distribution over pixels, and the goal is to have similar assignments regardless of the applied ordering. (b) For Autoregressive Representation Learning (ARL), the objective is to have similar representations at each corresponding spatial location and its neighbors over a window of small displacements Ω.

features that are useful for down-stream tasks, be it classification, regression or visualization. In clustering, the unlabeled data points are directly grouped into semantic classes. In both cases, recent works showed the effectiveness of maximizing Mutual Information (MI) between different views of the inputs to learn useful and transferable features [START_REF] Hjelm | Learning deep representations by mutual information estimation and maximization[END_REF][START_REF] Oord | Representation learning with contrastive predictive coding[END_REF][START_REF] Tian | Contrastive multiview coding[END_REF][START_REF] Federici | Learning robust representations via multi-view information bottleneck[END_REF] or discover clusters that accurately match semantic classes [START_REF] Ji | Invariant information clustering for unsupervised image classification and segmentation[END_REF][START_REF] He | k-anmi: A mutual information based clustering algorithm for categorical data[END_REF].

Another line of study in unsupervised learning is generative modeling. In particular, for image modeling, generative autoregressive models [START_REF] Oord | Pixel recurrent neural networks[END_REF][START_REF] Van Den Oord | Conditional image generation with pixelcnn decoders[END_REF][START_REF] Salimans | Pixelcnn++: Improving the pixelcnn with discretized logistic mixture likelihood and other modifications[END_REF]9], such as PixelCNN, are powerful generative models with tractable likelihood computation. In this case, the high-dimensional data, e.g., an image x, is factorized as a product of conditionals over its pixels. The generative model is then trained to predict the current pixel x i based on the past values x ≤i-1 in a raster scan fashion using masked convolutions [START_REF] Van Den Oord | Conditional image generation with pixelcnn decoders[END_REF] (Fig. 3 (a)).

In this work, instead of using a single left to right, top to bottom ordering, we propose to use several orderings obtained with different forms of masked convolutions and attention mechanism. The various orderings over the input pixels, or the intermediate representations, are then considered as different views of the input image * , and the model is then trained to maximize the MI between the outputs over these different views.

Our approach is generic, and can be applied for both clustering and representation learning (see Fig. 1). For a clustering task (Fig. 1 (a)), we apply a pair of distinct orderings over a given input image, producing two pixel-level predictions in the form of probability distribution over the semantic classes. We then maximize the MI between the two outputs at each corresponding spatial location and its intermediate neighbors. Maximizing the MI helps avoiding degeneracy (e.g., uniform output distributions) and trivial solutions (e.g., assigning all of the pixels to the same cluster). For representation learning (Fig. 1 (b)), we maximize a lower bound of MI between the two output feature maps over the different views.

We evaluate the proposed method using standard image segmentation datasets: Potsdam [START_REF] Gerke | Use of the stair vision library within the isprs 2d semantic labeling benchmark[END_REF] and COCO-stuff [5], and show competitive results. We present an extensive ablation study to highlight the contribution of each component within the proposed framework, and emphasizing the flexibility of the method.

To summarize, we propose following contributions: (i) a novel unsupervised method for image segmentation based on autoregressive models and MI maximization; (ii) various forms of masked convolutions to generate different orderings; (iii) an attention augmented version of masked convolutions for a larger receptive field, and a larger set of possible orderings; (iv) an improved performance above previous state-of-the-art on unsupervised image segmentation.

Related Works

Autoregressive models. Many autoregressive models [START_REF] Larochelle | The neural autoregressive distribution estimator[END_REF][START_REF] Germain | Made: Masked autoencoder for distribution estimation[END_REF][START_REF] Van Den Oord | Conditional image generation with pixelcnn decoders[END_REF][START_REF] Salimans | Pixelcnn++: Improving the pixelcnn with discretized logistic mixture likelihood and other modifications[END_REF][START_REF] Parmar | Image transformer[END_REF]9,[START_REF] Child | Generating long sequences with sparse transformers[END_REF] for natural image modeling have been proposed. They model the joint probability distribution of high-dimensional images as a product of conditionals over the pixels. PixelCNN [START_REF] Van Den Oord | Conditional image generation with pixelcnn decoders[END_REF][START_REF] Oord | Pixel recurrent neural networks[END_REF] specifies the conditional distribution of a sub-pixel (i.e., a color channel of a pixel) as a full 256-way softmax, while PixelCNN++ [START_REF] Salimans | Pixelcnn++: Improving the pixelcnn with discretized logistic mixture likelihood and other modifications[END_REF] uses a mixture of logistics. In both cases, masked convolutions are used to process the initial image x in an autoregressive manner. In Image [START_REF] Parmar | Image transformer[END_REF] and Sparse [START_REF] Child | Generating long sequences with sparse transformers[END_REF] transformers, self-attention [START_REF] Vaswani | Attention is all you need[END_REF] is used over the input pixels, while PixelSNAIL [9] combines both attention and masked convolutions. Clustering and unsupervised representation learning. Recent works in clustering aim at combining traditional clustering algorithms [START_REF] Hartigan | Direct clustering of a data matrix[END_REF] with deep learning, such as using K-means style objectives when training deep nets training [START_REF] Caron | Deep clustering for unsupervised learning of visual features[END_REF]18,[START_REF] Fard | Deep k-means: Jointly clustering with kmeans and learning representations[END_REF]. However, such objective can lead to trivial and degenerate solutions [START_REF] Caron | Deep clustering for unsupervised learning of visual features[END_REF]. IIC [START_REF] Ji | Invariant information clustering for unsupervised image classification and segmentation[END_REF] proposed to use a MI based objective which is intrinsically more robust to such trivial solutions. Unsupervised learning of representations [START_REF] Hjelm | Learning deep representations by mutual information estimation and maximization[END_REF][START_REF] Bachman | Learning representations by maximizing mutual information across views[END_REF][START_REF] Oord | Representation learning with contrastive predictive coding[END_REF][START_REF] Gidaris | Unsupervised representation learning by predicting image rotations[END_REF] rather aims to train a model, mapping the unlabeled inputs into some lower-dimensional space, while preserving semantic information and discarding instance-specific details. The pre-trained model can then be fine-tuned on a down-stream task with fewer labels.

Unsupervised learning and MI maximization. Maximizing MI for unsupervised learning is not a new idea [START_REF] Hartigan | Direct clustering of a data matrix[END_REF][START_REF] Becker | Self-organizing neural network that discovers surfaces in random-dot stereograms[END_REF], and recent works demonstrated its effectiveness for unsupervised learning. For representation learning, the training objective is to maximize a lower bound of MI over continuous random variables between distinct views of the inputs. These views can be the input image and its representation [START_REF] Hu | Learning discrete representations via information maximizing self-augmented training[END_REF], the global and local features [START_REF] Hjelm | Learning deep representations by mutual information estimation and maximization[END_REF], the features at different scales [START_REF] Bachman | Learning representations by maximizing mutual information across views[END_REF], a sequence of extracted patches from an image in some fixed order [START_REF] Oord | Representation learning with contrastive predictive coding[END_REF] or different modalities of the image [START_REF] Tian | Contrastive multiview coding[END_REF]. For a clustering objective, with discrete random variables as outputs, the exact MI can be maximized over the different views, e.g., IIC [START_REF] Ji | Invariant information clustering for unsupervised image classification and segmentation[END_REF] maximizes the MI between the image and its augmented version. Unsupervised Image Segmentation. Methods that learn the segmentation masks entirely from data with no supervision can be categorized as follows: (1) GAN based methods [8,4] that extract and redraw the main object in the image for object segmentation. Such methods are limited to only instances with two classes, a foreground and a background. The proposed method is more generalizable and is independent of the number of ground-truth classes; (2) Iterative methods [START_REF] Hwang | Segsort: Segmentation by discriminative sorting of segments[END_REF] consisting of a two-step process. The features produced by a CNN are first grouped into clusters using spherical K-means. The CNN is then trained for better feature extraction to discriminate between the clusters. We propose an end-to-end method simplifying both training and inference; (3) MI maximization based methods [START_REF] Ji | Invariant information clustering for unsupervised image classification and segmentation[END_REF] where the MI between two views of the same instance at the corresponding spatial locations is maximized. We propose an efficient and effective way to create different views of the input using masked convolutions. Another line of work consists of leveraging the learned representations of a deep network for unsupervised segmentation, e.g., CRFs [START_REF] Kanezaki | Unsupervised image segmentation by backpropagation[END_REF] and deep priors [START_REF] Kanezaki | Unsupervised image segmentation by backpropagation[END_REF].

Method

Our goal is to learn a representation that maximizes the MI, denoted as I, between different views of the input. These views are generated using various orderings, capturing different aspects of the inputs. Formally, let x ∼ X be an unlabeled data point, and F : X → Y be a deep representation to be learned as a mapping between the inputs and the outputs. For clustering, Y is the set of possible clusters corresponding to semantic classes, and for representation learning, Y corresponds to a lower-dimensional space of the output features. Let (o i , o j ) ∈ O be two orderings o i and o j obtained from the set of possible and valid orderings O (Fig. 2). For two outputs y ∼ F(x; o i ) and y ∼ F(x; o j ), the objective is to maximize the predictability of y from y and vice-versa, where F(x; o i ) corresponds to applying the learning function F with a given ordering o i to process the image x. This objective is equivalent to maximizing the MI between the two encoded variables: We start by presenting different forms of masked convolutions to generate various raster-scan orderings, and propose an attention augmented variant (Section 3.1). We then formulate the training objective for maximizing Eq. ( 1) (Section 3.2). We finally conclude with a flexible design architecture for the function F (Section 3.3).

Orderings

Masked Convolutions In neural autoregressive modeling [START_REF] Van Den Oord | Conditional image generation with pixelcnn decoders[END_REF][START_REF] Salimans | Pixelcnn++: Improving the pixelcnn with discretized logistic mixture likelihood and other modifications[END_REF]9], for an input image x ∈ R H×W ×3 with 3 color channels, a raster-scan ordering is first imposed on the image (see Fig. 2, ordering o 1 ). Such an ordering, where the pixel x i only depends on the pixels that come before it, is maintained using masked convolutions

Our proposition is to use all 8 possible raster-scan type orderings as the set of valid orderings O as illustrated in Fig. 2. A simple way to obtain them is to use a single ordering o 1 with the standard masked convolution (Fig. 3 (a)), along with geometric transformations g (i.e., image rotations by multiples of 90 degrees and horizontal flips), resulting in 8 versions of the input image. We can then maximize the MI between the two outputs, i.e., I(y; g -1 (y )) with y ∼ F(g(x); o j ). In this case, since the masked weights are never trained, we cannot fall-back to the normal convolution where the function F has access to the full input during inference, greatly limiting the performance of such approach.

This point motivates our approach. Our objective is to learn all the weights of the masked convolution during training, and use an unmasked version during inference. This can be achieved by using a normal convolution, and for a given ordering o i , we mask the corresponding weights during the forward pass to construct the desired view of the inputs. Then in the backward pass, we only update the unmasked weights and the masked weights remain unchanged. In this case, all of the weights will be learned and we will converge to a normal convolution given enough training iterations. During inference, no masking is applied, giving the function F full access to the inputs.

A straight forward way to implement this is to use 8 versions of the standard masked convolution to create the set O (Fig. 3 (d)). However, for each forward pass, the majority of the weights are masked, resulting in a reduced receptive field and a fewer number of weights will be learned at each iteration, leading to some disparity between them.

Given that we are interested in a discriminative task, rather than generative image modeling where the access to the current pixel is not allowed. We start by relaxing the conditional dependency, and allow the model to have access to the current pixel, reducing the number of masked locations by one (Fig. 3 (b)).

To further reduce the number of masked weights, for an F × F convolution, instead of masking the lower rows, we can simply shift the input by the same amount and only mask the weights of the last row. We thus reduce the number of masked weight from F 2 /2 (Fig. 3 Attention Augmented Masked Convolutions As pointed out by [START_REF] Van Den Oord | Conditional image generation with pixelcnn decoders[END_REF], the proposed masked convolutions are limited in terms of expressiveness since they create blind spots in the receptive field (Fig. 6). In our case, by applying dif- † e.g., for Shift1 and a 3 × 3 convolution, an image of spatial dimensions H × W is first padded on the top resulting in (H + 1) × W , the last row is then cropped, going back to H × W . ferent orderings, we will have access to all of the input x over the course of training, and this bug can be seen as a feature where the blind spots can be considered as an additional restriction. This restricted receptive filed, however, can be overcome using the self-attention mechanism [START_REF] Vaswani | Attention is all you need[END_REF]. Similar to previous works [45,[START_REF] Zhang | Self-attention generative adversarial networks[END_REF]3], we propose to add attention blocks to model long range dependencies that are hard to access through standalone convolutions. Given an input tensor of shape (H, W, C in ), after reshaping it into a matrix X ∈ R HW ×Cin , we can apply a masked version of attention [START_REF] Vaswani | Attention is all you need[END_REF] in a straight forward manner. The output of the attention operation is:

A = Softmax((QK ) M oi )V (2)
with Q = XW q , K = XW k and V = XW v , where W q , W k ∈ R Cin×d and W v ∈ R Cin×d are learned linear transformations that map the input X to queries Q, keys K and values V , and M oi ∈ R HW ×HW corresponds to a masking operation to maintain the correct ordering o i . The output is then projected into the output space using a learned linear transformation W O ∈ R d×Cin obtaining X att = AW O . The output of the attention operation X att is concatenated channel wise with the input X, and then merged using a 1 × 1 convolution resulting in the output of the attention block.

Zigzag Orderings. Using attention gives us another benefit, we can extend the set of possible orderings to include zigzag type orderings introduced in [9] (Fig. 4). With zigzag orderings, the outputs at each spatial location will be mostly influenced by the values of the corresponding neighboring input pixels, which can give rise to more semantically meaningful representations compared to that of raster-scan orderings. This is done by simply using a mask M oi corresponding to the desired zigzag ordering o i . Resulting in a set O of 16 possible and valid orderings o i with i ∈ {1, . . . , 16} in total. See Fig. 5 for an example.

Training Objective

In information theory, the MI I(X; Y ) between two random variables X and Y measures the amount of information learned from the knowledge of Y about X and vice-versa. The MI can be expressed as the difference of two entropy terms: Intuitively, I(X; Y ) can be seen as the reduction of uncertainty in one of the variables, when the other one is observed. If X and Y are independent, knowing one variable exposes nothing about the other, in this case, I(X; Y ) = 0. Inversely, if the state of one variable is deterministic when the state of the other is revealed, the MI is maximized. Such an interpretation explains the goal behind maximizing Eq. ( 1). The neural network F must be able to preserve information and extract semantically similar representations regardless of the applied ordering o i , and learn representations that encode the underlying shared information between the different views. The objective can also be interpreted as having a regularization effect, forcing the function F to focus on the different views and subparts of the input x to produce similar outputs, reducing the reliance on specific objects or parts of the image.

I(X; Y ) = H(X) -H(X|Y ) = H(Y ) -H(Y |X) (3)
Let p(y, y ) be the joint distribution produced by sampling examples x ∼ X and then sampling two outputs y ∼ F(x; o i ) and y ∼ F(x; o j ) with two possible orderings o i and o j . In this case, the MI in Eq. ( 1) can be defined as the Kullback-Leibler (KL) divergence between the joint and the product of the marginals:

I(y, y ) = D KL (p(y, y ) p(y)p(y )) (4)
To maximize Eq. (4), we can either maximize the exact MI for a clustering task over discrete predictions, or a lower bound for an unsupervised learning of representations over the continuous outputs. We will now formulate the loss functions L AC and L ARL of both objectives for a segmentation task.

Autoregressive clustering (AC).

In a clustering task, the goal is to train a neural network F to predict a cluster assignment corresponding to a given semantic class k ∈ {1, . . . , K} with K possible clusters at each spatial location. In this case, the encoder-decoder type network F is terminated with K-way softmax, outputting y ∈ [0, 1] H×W ×K of the same spatial dimensions as the input. Concretely, for a given input image x and two valid orderings (o i , o j ) ∈ O, we forward pass the input through the network producing two output probability distributions F(x; o i ) = p(y|x, o i ) and F(x; o j ) = p(y |x, o j ) over the K clusters and at each spatial location. After reshaping the outputs into two matrices of shape HW × K, with each element corresponding to the probability of assigning pixel x l with l ∈ {1, . . . , HW } to cluster k, we can compute the joint distribution p(y, y ) of shape K × K as follows:

p(y, y ) = F(x; o i ) F(x; o j ) (5)
The marginals p(y) and p(y ) can then be obtained by summing over the rows and columns of p(y, y ). Similar to IIC [START_REF] Ji | Invariant information clustering for unsupervised image classification and segmentation[END_REF], we symmetrize p(y, y ) using [p(y, y )+ p(y, y ) ]/2 to maximize the MI in both directions. The clustering loss L AC in this case can be written as follows:

L AC = E x∼X E p(y,y ) log p(y, y ) p(y)p(y ) (6)
In practice, instead of only maximizing the MI between two corresponding spatial locations, we maximize it between each spatial location and its intermediate neighbors over small displacements u ∈ Ω (see Fig. 1). This can be efficiently implemented using a convolution operation as demonstrated in [START_REF] Ji | Invariant information clustering for unsupervised image classification and segmentation[END_REF].

Autoregressive representation learning (ARL).

Although the clustering objective in Eq. ( 6) can also be used as a pre-training objective for F, Tschannen et al . [43] recently showed that maximizing the MI does not often results in transferable and semantically meaningful features, especially when the downstream task is a priori unknown. To this end, we follow recent representation learning works based on MI maximization [START_REF] Oord | Representation learning with contrastive predictive coding[END_REF][START_REF] Hjelm | Learning deep representations by mutual information estimation and maximization[END_REF][START_REF] Bachman | Learning representations by maximizing mutual information across views[END_REF][START_REF] Tian | Contrastive multiview coding[END_REF], where a lower bound estimate of MI (e.g., InfoNCE [START_REF] Oord | Representation learning with contrastive predictive coding[END_REF], NWJ [START_REF] Nguyen | Estimating divergence functionals and the likelihood ratio by convex risk minimization[END_REF]) is maximized between different views of the inputs. These estimates are based on the simple intuitive idea, that if a critic f is able to differentiate between samples drawn from the joint distribution p(y, y ) and samples drawn from the marginals p(y)p(y ), then the true MI is maximized. We refer the reader to [43] for a detailed discussion.

In our case, with image segmentation as the target down-stream task, we maximize the InfoNCE estimator [START_REF] Oord | Representation learning with contrastive predictive coding[END_REF] over the continuous outputs. Specifically, with two outputs (y, y ) ∈ R H×W ×C as C-dimensional feature maps. The training objective is to maximize the infoNCE based loss L ARL :

L ARL = E x∼X log e f (y l ,y l ) 1 N N m=1 e f (y l ,y m ) (7)
For an input image x and two outputs y and y . Let y l and y m correspond to C-dimensional feature vectors at spatial positions l and m in the first and second outputs respectively. We start by creating N pairs of feature vectors (y l , y m ), with one positive pair drawn from the joint distribution and N -1 negative pairs drawn from the marginals. A positive pair is a pair of feature vectors corresponding to the same spatial locations in the two outputs, i.e., a pair (y l , y m ) with m = l. The negatives are pairs (y l , y m ) corresponding to two distinct spatial positions m = l. In practice, we also consider small displacements Ω (Fig. 1) when constructing positives. Additionally, the negatives are generated from two distinct images, since two feature vectors might share similar characteristics even with different spatial positions. By maximizing Eq. ( 7), we push the model F to produce similar representations for the same spatial location regardless of the applied ordering, so that the critic function f is able to give high matching scores to the positive pairs and low matching to the negatives. We follow [START_REF] Hjelm | Learning deep representations by mutual information estimation and maximization[END_REF] and use separable critics f (y, y ) = φ 1 (y) φ 2 (y ), where the functions φ 1 /φ 2 nonlinearly transform the outputs to a higher vector space, and f (y l , y m ) produces a scalar corresponding to a matching score between the two representations at two spatial positions l and m of the two outputs.

Note that both losses L AC and L ARL can be applied interchangeably for both objectives, a case we investigate in our experiments (Section 4.1). For L AC , we can consider the clustering objective as an intermediate task for learning useful representations. For L ARL , during inference, K-means [START_REF] Johnson | Billion-scale similarity search with gpus[END_REF] algorithm can be applied over the outputs to obtain the cluster assignments.

Model

The representation F can be implemented in a general manner using three subparts, i.e., F = h • g ar • d, with a feature extractor h, an autoregressive encoder g ar and a decoder d. With such a formulation, the function F is flexible and can take different forms. With h as an identity mapping, F becomes a fully autoregressive network, where we apply different orderings directly over the inputs. Inversely, if g ar is an identity mapping, F becomes a generic encoder-decoder network, where h plays the role of an encoder. Additionally, h can be a simple convolutional stem that plays an important role in learning local features such as edges, or even multiple residual blocks [START_REF] He | Deep residual learning for image recognition[END_REF] to extract higher representations. In this case, the orderings are applied over the hidden features using g ar . g ar is similar to h, containing a series of residual blocks, with two main differences, the proposed masked convolutions are used, and the batch normalization [START_REF] Ioffe | Batch normalization: Accelerating deep network training by reducing internal covariate shift[END_REF] layers are omitted to maintain the autoregressive dependency, with an optional attention block. The decoder d can be a simple conv1 × 1 to adapt the channels to the number of cluster K, followed by bilinear upsampling and a softmax operation for a clustering objective. For representation learning, d consists of two separable critics φ 1 /φ 2 , which are implemented as a series of conv3 × 3 -BN -ReLU and conv1 × 1 for projecting to a higher dimensional space. See sup. mat. for the architectural details.

Experiments

Datasets. The experiments are conducted on the newly established and challenging baselines by [START_REF] Ji | Invariant information clustering for unsupervised image classification and segmentation[END_REF]. Potsdam [START_REF] Gerke | Use of the stair vision library within the isprs 2d semantic labeling benchmark[END_REF] with 8550 RGBIR satellite images of size 200 × 200, of which 3150 are unlabeled. We experiment on both the 6-labels variant (roads and cars, vegetation and trees, buildings and clutter) and Potsdam-3, a 3-label variant formed by merging each of the pairs. We also use COCO-Stuff [5], a dataset containing stuff classes. Similarly, we use a reduced version of COCO-Stuff with 164k images and 15 coarse labels, reduced to 52k by taking only images with at least 75% stuff pixel. In addition to COCO-Stuff-3 with only 3 labels, sky, ground and plants. Evaluation Metrics. We report the pixel classification Accuracy (Acc). For a clustering task, with a mismatch between the learned and ground truth clusters. We follow the standard procedure and find the best one-to-one permutation to match the output clusters to ground truth classes using the Hungarian algorithm [START_REF] Kuhn | The hungarian method for the assignment problem[END_REF]. The Acc is then computed over the labeled examples.

Implementation details. The different variations of F are trained using ADAM with a learning rate of 10 -5 to optimize both objectives in Eqs. ( 6) and [START_REF] Caron | Unsupervised pre-training of image features on non-curated data[END_REF]. The training is conducted on NVidia V100 GPUs, and implemented using the Py-Torch framework [START_REF] Paszke | Automatic differentiation in pytorch[END_REF]. For more experimental details, see sup. mat. Overclustering. The Acc obtained when using a number of output clusters greater than the number of ground truth classes K > Kgt. With variable number of images used to find the best many-to-one matching between the outputs and targets.

Ablation Studies

We start by performing comprehensive ablation studies on the different components and variations of the proposed method. Table 1 and Fig. 7 show the ablation results for AC, and Table 2 shows a comparison between AC and ARL, analyzed as follows: Variations of F. Table 2a compares different variations of the network F. With a fixed decoder d (i.e., a 1 × 1 Conv followed by bilinear upsampling and softmax function), we adjust h and g ar going from a fully autoregressive model (F 1 ) to a normal decoder-encoder network (F 4 and F 5 ). When using masked versions, we see an improvement over the normal case, with up to 8 points for Potsdam, and to a lesser extent for Potsdam-3 where the task is relatively easier with only three ground truth classes. When using a fully autoregressive model (F 1 ), and applying the orderings directly over the inputs, maximizing the MI becomes much harder, and the model fails to learn meaningful representations. Inversely, when no masking is applied (F 4 and F 5 ), the task becomes comparatively simpler, and we see a drop in performance. The best results are obtained when applying the orderings over low-level features (F 2 and F 3 ). Interestingly, the unmasked versions yield results better than random, and perform competitively with 3 output classes for Potsdam-3, validating the effectiveness of maximizing the MI over small displacements u ∈ Ω. For the rest of the experiments we use F 2 as our model. Attention and different orderings. Table 2c shows the effectiveness of attention. With a single attention block added at a shallow level, we observe an improvement over the baseline, for both raster-scan and zigzag orderings, and their combination, with up to 4 points for Potsdam. In this case, given the quadratic complexity of attention, we used an output stride of 4. Data augmentations. For a given training iteration, we pass the same image two times through the network, applying two different orderings at each forward pass. We can, however, pass a transformed version of the image as the second input. We investigate using photometric (i.e., color jittering) and geometric (i.e., ). Table 2d shows the obtained results. We see 2 points improvement when using hard sampling for Potsdam. For simplicity, we use random sampling for the rest of the experiments. Additionally, to investigate the effect of the number of orderings (i.e., the cardinality of O), we compute the Acc over different choices and sizes of O. Table 2b shows best results are obtained when using all 8 raster-scan orderings. Interestingly, for some choices, we observe better results, which may be due to selecting orderings that do not share any receptive fields, as the ones used in hard sampling. Overclustering. To compute the Acc for a clustering task using linear assignment, the output clusters are chosen to match the ground truth classes K = K gt . Nonetheless, we can choose a higher number of clusters K > K gt , and then find the best many-to-one matching between the output clusters and ground truths based a given number of labeled examples. In this case, however, we are not in a fully unsupervised case, given that we extract some information, although limited, from the labels. Fig. 7 shows that, even with a very limited number of labeled examples used for mapping, we can obtain better results than the fully unsupervised case. AC and ARL To compare AC and ARL, we apply them interchangeably on both clustering and representation learning objectives. In clustering, for ARL, after PCA Whitening, we apply K-means over the output features to get the cluster assignments. In representation learning, we evaluate the quality of the learned representations using both linear and non-linear separability as a proxy for disentanglement, and as a measure of MI between representations and class labels. Table 2 shows the obtained results.

Clustering. As expected, AC outperforms ARL on a clustering task, given that the clusters are directly optimized by computing the exact MI during training. Quality of the learned representations. Surprisingly, AC outperforms ARL on both linear and non-linear classifications. We hypothesize that unsupervised representation learning objectives that work well on image classification, fail in image segmentation due to the dense nature of the task. The model in this case needs to output distinct representations over pixels, rather than the whole image, which is a harder task to optimize. This might also be due to using only a small number of features (i.e., N pairs) for each training iteration.

Comparison with the state-of-the-art

Table 3 shows the results of the comparison. AC outperforms previous work, and by a good margin for harder segmentation tasks with a large number of output classes (i.e., Potsdam and COCO-Stuff), highlighting the effectiveness of maximizing the MI between the different orderings as a training objective. We note that no regularization or data augmentation were used, and we expect that better results can be obtained by combining AC with other procedures as demonstrated in the ablation studies.

Conclusion

We presented a novel method to create different views of the inputs using different orderings, and showed the effectiveness of maximizing the MI over these views for unsupervised image segmentation. We showed that for image segmentation, optimizing over the discrete outputs MI works better for both clustering and unsupervised representation learning, due to the dense nature of the task. Given the simplicity and ease of adoption of the method, we hope that the proposed approach can be adapted for other visual tasks and used in future works.

Fig. 2 .

 2 Fig. 2. Raster-scan type orderings.

Fig. 3 .

 3 Fig. 3. Masked Convolutions. (a) Standard masked convolution used in autoregressive generative modeling, yielding an ordering o1. (b) A relaxed version of standard masked convolution where we have access to the current pixel at each step. (c) A simplified version of masked convolution with a reduced number of masked weights. (d) The 8 versions of the standard masked convolution to construct all of the possible raster-scan type orderings. (e) The proposed types of masked convolutions with the corresponding shifts to obtain all of the 8 desired raster-scan types orderings. F = 3 in this case.

Fig. 4 .

 4 Fig. 4. Zigzag type orderings.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Attention Masks. Examples of the different attention masks Mo i of shape HW × HW applied for a given ordering oi. With HW = 9.

  Fig.7. Overclustering. The Acc obtained when using a number of output clusters greater than the number of ground truth classes K > Kgt. With variable number of images used to find the best many-to-one matching between the outputs and targets.

Table 1 .

 1 AC Ablations. Ablations studies conducted on Potsdam (POS) and Potsdam-3 (POS3) for Autoregressive Clusterings. We show the pixel classification accuracy (%). Sampling of oi.

	Network F = h • gar • d
	h	gar Random		POS POS3 28.5 38.2	|O|	POS	POS3
	F1 Id		5 Res. blocks 39.3	56.3	2	43.2±2.19	59.5±5.12
	F2 Stem		5 Res. blocks 46.4	66.4	4	45.6±3.22 63.55±3.52
	F3 Res. block F4 5 Res. blocks Id 4 Res. blocks 47.9 35.1	64.5 63.4	8	46.4	66.4
	F5 ResNet-18	Id		40.7	51.9	(b) Number of orderings.
		(a) Variation of F.
		Orderings
	Raster-Scan Zigzag Attention POS POS3	Sampling oi POS POS3
			×	×	45.2		61.0	Random	46.4	66.4
			×		47.9		66.3	No Rep.	48.6	64.8
	×				47.8		66.5	Hard	48.9	65.2
					49.3		65.4
		(c) Attention. (d) Type Transf. POS POS3
		None		-	46.4	66.4	p	POS POS3
	Photometric Col. Jittering 47.9	65.5	0	46.4	66.4
	Geometric	Flip	46.7	68.0	0.1 47.9	64.7
	Geometric	Rot.	48.5	68.3	0.2 46.9	65.1
	Geo. & Pho.	All	48.5 68.3
		(e) Transformations.		(f) Dropout.

Table 2 .

 2 Comparing ARL and AC. We compare ARL and AC on a clustering task (left). And investigate the quality of the learned representations by freezing the trained model, and reporting test Acc obtained when training a linear (center) and non-linear (right) functions trained on the labeled training examples.rotations and H-flips) transformations. For geometric transformations, we bring the outputs back to the input coordinate space before computing the loss. Results are shown in Table2e. As expected, we obtain relative improvements with data augmentations, highlighting the flexibility of the approach. Dropout. To add some degree of stochasticity to the network, and as an additional regularization, we apply dropout to the intermediate activations within residual blocks. Table2fshows a small increase in Acc for Potsdam. Orderings. Until now, at each forward pass, we sample a pair of possible orderings with replacement from the set O. With such a sampling procedure, we might end-up with the same pair of orderings for a given training iteration. As an alternative, we investigate two other sampling procedures. First, with no repetition (No Rep.), where we choose two distinct orderings for each training iteration. Second, using hard sampling, choosing two orderings with opposite receptive fields (e.g., o 1 and o 6

	Clustering		Linear Evaluation	Non-Linear Evaluation
	Method	POS POS3	Method POS POS3	Method POS POS3
	Random CNN 28.5 AC 46.4 66.4 38.2	AC	23.7 41.4	AC	68.0 81.8
	ARL	45.1	57.1	ARL	23.7	38.5	ARL	47.6	63.5

Table 3 .

 3 Unsupervised image segmentation. Comparison of AC with state-ofthe-art methods on unsupervised segmentation.

		COCO-Stuff-3 COCO-Stuff	Potsdam-3 Potsdam
	Random CNN	37.3	19.4	38.2	28.3
	K-means [40]	52.2	14.1	45.7	35.3
	SIFT [33]	38.1	20.2		28.5
	Doersch 2015 [11]	47.5	23.1	49.6	37.2
	Isola 2016 [26]	54.0	24.3	63.9	44.9
	DeepCluster 2018 [6]	41.6	19.9	41.7	29.2
	IIC 2019 [27]	72.3	27.7	65.1	45.4
	AC	72.9	30.8	66.5	49.3

Acknowledgments. We gratefully acknowledge the support of Randstad corporate research chair, Saclay-IA platform of and Mésocentre computing center.


R Nez

M Gonzalez-Flesca

David Marchal
email: david.marchal@centralesupelec.fr

T Schmitt

P Scouflaire

S Candel

S Ducruix

Experimental and numerical characterizations of acoustic damping rates in a coupled-cavity configuration

This work aims at a better understanding of damping in coupled-cavities configurations such as liquid rocket engines (LRE). A cold flow rig called NPCC (New Pressurized Couple Cavities) is employed. It consists of two cavities mimicking dome and combustion chamber, linked by three injectors. The cavities can be either continuously excited or successively excited then relaxed by two different perforated rotating wheels placed at the outlets. Eigenmodes are first characterized. The rig is then excited at three successive eigenfrequencies using continuous modulation. The modes structures are checked using pressure transducers. Velocities at the injector's exits are measured with hot wires to characterize the injector's response. Forcing followed by relaxation (alternating forcing) is then used to study the damping rate of each mode, which is retrieved from the pressure transducers signal. 2D unsteady numerical simulations are performed to retrieve these values, both for continuous and alternating modulation. In the damping rate study, at least 10 excitation/relaxation cycles are necessary to achieve good convergence. The results are shown to be independant from the location of the probe used to derive the damping rate. The results are in reasonable agreement with the experiments for the three modes studied with continuous modulation. The study of the damping rate will however require 3D simulations to achieve good results with the methods presented here.

Introduction

Since the Apollo program in the 1960s, high-frequency combustion instabilities have been responsible for many delays in the access to space [START_REF] Oefelein | Comprehensive review of liquid-propellant combustion instabilities in F-l engines[END_REF] . In liquid rocket engines, they are the result of the complex interaction between a turbulent flow, a confined chamber with its acoustic eigenmodes and the combustion that takes place in it. This interaction being not yet completely understood, engineers often rely on costly trial and error processes, with minor increments between two full scale tests. However, the increase of computing power in the recent years gives us an alternative to tackle the problem of combustion instabilities : by correctly simulating the system, one could get rid of the uncertainty behind the development of the next generation of rocket engines. In particular, large eddy simulation (LES) has enlightened several mechanisms that take place in transcritical and supercritical cryogenic flames found in liquid rocket engines. [START_REF] Hakim | Dynamics of a transcritical coaxial flame under high-frequency transverse acoustic forcing: The effect of a modulation frequency[END_REF][START_REF] Oefelein | Thermophysical characteristics of shear-coaxial LOx-H2 flames at supercritical pressure[END_REF][START_REF] Urbano | Study of flame response to transverse acoustic modes from the LES of a 42-injector rocket engine[END_REF] Injector dynamics, flame anchoring and flame-acoustic field interaction simulations help us understand the complex mechanisms leading to combustion instabilities. Lab-scale experiments are crucial to validate these new codes : non-reacting test rigs, [START_REF] Richecoeur | Experimentations et simulations numeriques des interactions entre modes acoustiques transverses et flammes cryotechniques[END_REF][START_REF] Méry | Mecanismes d'instabilites de combustion haute-frequence et application aux moteurs-fusees[END_REF][START_REF] Hakim | Dynamics of Transcritical Coaxial Flames in High-Frequency Transverse Acoustic Fields: Application to Liquid Rocket Engine Instabilities[END_REF] simplified hot flow configurations [START_REF] Vingert | MASCOTTE, a test facility for cryogenic combustion research at high pressure[END_REF][START_REF] Richecoeur | High-frequency transverse acoustic coupling in a multiple-injector cryogenic combustor[END_REF] and even a reduced-scale rocket engine [START_REF] Groning | Influence of hydrogen temperature on the acoustics of a rocket engine combustion chamber operated with LOx/H2 at representative conditions[END_REF][START_REF] Groning | Influence of hydrogen temperature on the stability of a rocket engine combustor operated with hydrogen and oxygen[END_REF][START_REF] Schmitt | Large-eddy simulations of a sub-scale liquid rocket combustor: influence of fuel injection temperature on thermo-acoustic stability[END_REF] have been designed in the past few years to study combustion instabilities, in parallel with their simulation. With the help of theoretical works, [START_REF] Méry | Impact of heat release global fluctuations and flame motion on transverse acoustic wave stability[END_REF] experiments and simulations have shown that fluctuations of chamber pressure as well as changes of the injection mass flow rate could lead to an unsteady release of the heat by flames. These are examples of source terms that drive combustion instabilities. They are in competition with damping terms, which will tend to decrease the amplitude of the pressure oscillations. This damping will be studied here, in a cold flow test rig developed by Méry et al. [START_REF] Mery | Injection coupling with high amplitude transverse modes: experimentation and simulation[END_REF] and experimented by Gonzalez-Flesca et al. [START_REF] Gonzalez-Flesca | Reduced order modeling approach to combustion instabilities of liquid rocket engines[END_REF] This paper presents ongoing results obtained with unsteady simulations of this experiment. First, Sec. 2 summarizes the theory and modelling of the acoustic damping rate. Sec. 3 presents the NPCC test rig used in the experiments. Then, Sec. 4 shows how this test rig is simulated in 2D using the AVBP software from CERFACS and IFPEN. The comparison between simulation and experimental results is displayed in Sec. 5, as well as the post-processing methods used. Conclusion and perspectives are drawn in sec. 6.

Study of the acoustic damping rate 2.1 Theory and modelling

The study of the acoustic damping is made in the framework of reduced order modeling. [START_REF] Gonzalez-Flesca | Reduced order modeling approach to combustion instabilities of liquid rocket engines[END_REF] The state variables are projected on the eigenmodes of the system. The relaxation of an acoustic eigenmode m, in the absence of thermoacoustic source term and external forcing is given by :

ηm (t) + ω 2 m η m (t) = -D m (t) (1)
where η m is the amplitude of the mode, ω m its angular frequency and D m the damping term corresponding to that mode. Assuming linear damping, D m is expressed as :

D m (t) = 2α m ηm (t) (2)
where α m is called the damping rate of mode m. The general solution of ( 1) is:

η m (t) = e -α m t (a exp[it(ω 2 m -α 2 m ) 1/2 ] + b exp[it(ω 2 m -α 2 m ) 1/2 ]) (3)
where a and b are integration constants. The response of the system is therefore pseudo-harmonic, with a decaying envelope described by the factor e -α m t . The object of this study is to determine α m , in both the experiment and the simulation. Two methods described below are used in the experiments : the alternating forcing method and the Full Width at Half Maximum (FHWM) method.

Only the alternating forcing method is used in the simulations.

Alternating forcing method

In order to study the damping rate of the acoustic modes, the NPCC test bed described in Sec.3 will undergo acoustic modulation followed by relaxation. The forcing is applied using a half-perforated wheel described below. During the relaxation, there is no more external forcing which corresponds to the framework of Eq. ( 1). This modulation will be referred to as 'alternating forcing' in the following.

FWHM method

Another way to determine α is to perform a continuous modulation by exciting the NPCC test bench with a fully perforated wheel described in Sec.3 and progressively increase the rotation speed of the wheel. This method uses the result of Siebert 16 :

α = π∆ f ( 4
)
where α is the damping rate and ∆ f the Full Width at Half Maximum (FWHM), i.e. the width of the power spectrum of the pressure signal taken at half the peak amplitude. This method was applied by Webster et al. [START_REF] Webster | Analysis of pressure dynamics, forced excitation and damping in a high pressure LOX/H2 combustor[END_REF] in their analysis of the BKH model combustor. This method will not be used in the following study.

The NPCC test rig

In a typical liquid rocket engine, the propellants are conveyed through domes to the coaxial injectors: the oxidizer circulates through the central part of the injector, whereas the fuel comes through the annular part. Therefore, the possible coupling between the combustion chamber and the injection system can be the root cause of many combustion instabilities problems. The New Pressurized Coupled Cavities (NPCC) test rig is a cold flow experiment which allow intensive study of this coupling in a simplified configuration.

Configuration

The NPCC test rig is presented in Fig. 1. It is made of aluminium and stainless steel rectangular cavities and mounted on a rail. From left to right : the first cavity is the dome fed by compressed air (3.5 bar) at ambient temperature. The following plate contains the three injectors, all with a 6mm diameter. The chamber is made of three parts. Parts 1 and 3 are fixed, whereas part 2 is interchangeable, adding the possibility to change the length of the chamber. At the end of the chamber, another plate contains two exit nozzles. These nozzles can be obturated with a partially perforated wheel driven by an electric motor, this setup is called the Very High Amplitude Modulator (VHAM). [START_REF] Méry | Mecanismes d'instabilites de combustion haute-frequence et application aux moteurs-fusees[END_REF] Figure 1: NPCC test rig

The electric motor has a maximum rotation speed of 6000 rpm. The program controlling the engine has two modes: it can run at a constant speed or progressively increase in speed, doing a ramp. Fig. 2 presents the two different perforated wheels that have been designed: one fully perforated to constantly excite the system, one partially perforated to alternate between excitation and relaxation. With 69 perforations on the standard wheel, the maximum modulation frequency is 6900 Hz. The test rig is equipped with probes for pressure measurements: 4 in the dome (Hfdn in Fig. 1) and 6 in the chamber (HFcn in Fig. 1). Their positions are chosen to clearly identify the acoustic modes studied (Sec 3.2). Hot wire sensors (calibrated before performing any experiment) are used at the end of the injectors to measure the injection velocities. The operating pressure is 3.5 bar. The air stream is controlled by a mass flow meter.

The compressed air in the dome flows through the injectors to the chamber, where the internal pressure is modulated by the VHAM. The perforated wheel is alternatively obturating each nozzle, maintaining a constant escaping flow rate and thus a constant pressure in the chamber. This modulation creates acoustic waves inside the system, leading to the installation of acoustic standing modes, whose shapes depend on the frequencies of excitation.

Eigenmodes and eigenfrequencies

The eigenmodes of the system define the resonant frequencies at which the system will respond with the highest intensity. The ramp mode of the motor is well-suited for recovering these frequencies: the frequency of modulation is slowly increased from 0 Hz to 3000 Hz during a ramp of 99s. The pressure signal recovered at HFc1 and HFd1 is shown in Fig. 3a. A time-frequency analysis (Fig. 3b) provides the eigenfrequencies of the system by associating the times of maximum response with the modulation frequency. The AVSP Helmholtz solver [START_REF] Nicoud | Acoustic modes in combustors with complex impedances and multidimensional active flames[END_REF] is used to check the eigenfrequencies with a simplified geometry (no nozzles). Results are close to those obtained using the ramp mode of operation (Table 1). Three eigenmodes will be studied in the present work: 1T, 1T1L and 1T2L. The 1T mode is fully transverse, while the 1T2L mode shows 2 pressure nodes along its longitudinal axis. The inner dimensions of the dome and chamber were chosen to match the first transverse eigenmode frequency of the dome (1T d ) with the first transverse first longitudinal mode of the chamber (1T1L c ). The 1T1L mode of the chamber is thus a coupled mode. Fig. 4 presents the shape of these three modes, as found by AVSP. The simulations clearly confirm that the 1T1L mode of the chamber is coupled with the 1T mode of the dome, while the 1T and 1T2L modes of the chamber are not coupled with the dome.

2D Numerical simulations

By considering a longitudinal cut through the plane containing the injectors, a 2D domain simplifying the experiment was created. This will not strictly account for the physics of the three dimensional problem, but allows for fast calculations and testing of the methodology which will be used later on the much more costly 3D simulations.

Governing equations, models and numerics

The Favre-filtered, fully compressible Navier-Stokes equations are used to formulate the Large Eddy Simulation (LES) approach. The fluid is air, its viscosity is calculated with the Sutherland method. The sub-grid scale stress tensor is modeled with the dynamic Smagorinsky model. [START_REF] Germano | A dynamic subgridscale eddy viscosity model[END_REF] The SGS energy and species fluxes are modeled using the gradient transport assumption, introducing turbulent Prandtl and Schmidt numbers, both set to 0.6. Thermodynamics are modeled by the perfect gas equation. The compressible unstuctured solver AVBP, [START_REF] Schonfeld | Steady and unsteady flows simulations using the hybrid flow solver avbp[END_REF][START_REF] Moureau | High-order methods for DNS and LES of compressible multi-component reacting flows on fixed and moving grids[END_REF] developed by CERFACS and IFPEN, is used. The numerical scheme is a third order in time and space Taylor-Galerkin scheme, [START_REF] Colin | Development of high-order Taylor-Galerkin schemes for unsteady calculations[END_REF] called TTGC.

Computational domain, mesh and boundary conditions

The computational domain is shown in Fig. 5. It reproduces an axial cut of the geometry of the experimental setup. The dome is 60x110 mm, the chamber is 210x140 mm, the injectors are 6 mm in diameter and 50 mm long. The nozzles at the outlet are a bit different than the ones in the test rig. They were designed this way to better absorb the vortices that appear in the 2D simulation and to optimize the pressure modulation amplitude at the outlet without encountering any backflow. The inlet boundary is on the left of the dome, contrary to the experiment where it is located on the upper side. The 2D mesh contains 128000 nodes and 253000 cells. The smallest cells are located in the injectors, which contain 20 cells in their diameter (6 mm wide, cell size: 0.30 mm ). The inlet boundary condition is fully reflective for all frequencies. All solid boundary conditions are modeled with an adiabatic wall law. [START_REF] Schmitt | Simulation aux grandes échelles de la combustion étagée dans les turbines à gaz et son interaction stabilité -polluants-thermique[END_REF] A relaxation method on the pressure is used at the outlet, where superimposed ingoing sine acoustic waves are used to model the fully perforated wheel.

Operating conditions

In this 2D simulation, bulk velocities in the injectors (u in j ) are conserved with respect to the experiment. The corresponding inlet velocity u inlet is enforced on the left boundary of the dome which serves as the inlet here, due to the impossibility to reproduce the lateral feeding pipe in this plane. The mean chamber pressure p c is maintained at 3.5 bar by enforcing a mean static pressure p out = 2.95 bar at the outlet boundaries (isentropic expansion to Mach 0.45). The amplitude of the modulation is imposed at 0.4 bar. The three eigenmodes studied by Gonzalez-Flesca et al. [START_REF] Gonzalez-Flesca | Reduced order modeling approach to combustion instabilities of liquid rocket engines[END_REF] are simulated by imposing alternatively pulsed boundary conditions at the outlet. The operating conditions are summarized in Table 2.

Implementation of the alternating forcing

To reproduce the alternating forcing method used for the damping rate study in the simulation, sine ingoing acoustic waves at the outlet boundaries are modulated with a square function, equal to 1 during the excitation and 0 during the relaxation.

It was observed that the signals gathered from the experiment at the HFc1 probe did not reach the limit cycle after one excitation cycle (corresponding to the rotation of the part containing 16 holes in the perforated part of the wheel). [START_REF] Gonzalez-Flesca | Reduced order modeling approach to combustion instabilities of liquid rocket engines[END_REF] Therefore, it was decided to increase the duration of the excitation-relaxation cycle to 64 periods in the simulations: 32 for the excitation, 32 for the relaxation. Fig. 6 illustrates this choice by showing a few cycles as measured by a probe located at the bottom nozzle's end (where the boundary condition is applied), during a 1T excitation-relaxation calculation.

The operating conditions are the same as in the continuous modulation case (Table 2).

Figure 6: Signal measured at the outlet boundary (bottom nozzle) for the 1T alternating forcing method. 32 periods of modulation are followed by a time of relaxation corresponding to the 32 periods again.

Results, comparisons between simulation and experiment 5.1 Continuous modulation

In this section, simulations are performed using a continuous modulation at a frequency that successively corresponds to the 3 modes presented. In particular, the shape of the modes, the injector velocity evolution and the acoustic pressure amplitudes are compared to the experiments. 1 for the probe positions). Both experimental and numerical signals have been filtered with a 3rd order Butterworth bandpass filter, with high and low cut-off frequencies at 200 Hz above and below the modulation frequency for each case. [START_REF] Gonzalez-Flesca | Simulation, experimentation and modeling contributions to the analysis of high frequency combustion instabilities in liquid propellant rocket engines[END_REF] The velocity fluctuations for cases 1T1L and 1T2L are qualitativelty well-recovered, amplitudes being properly reproduced. Case 1T shows large discrepancies with the experimental measurements, both in terms of phase between the signals and amplitude. This question is still under investigation. Fig. 9 presents the pressure signals measured by the HFc1 (chamber) and HFd1 (dome) probes (see Fig. 1 for the probe positioning). The signals from the simulation have been filtered to remove the mean value, using a high pass filter with a cut-off frequency of 10% of the modulation frequency.

Considering the 1T mode, the amplitude in the chamber is well-retrieved for HFc1 in the simulation but the HFd1 signal has a higher amplitude than in the experiment. For the excitation of the 1T1L coupled mode both signals are in phase with what was expected (see figs. 3 and5). The amplitude in the dome is well-retrieved, but the chamber amplitude is lower than in the experiment. The 1T2L mode is again the one that is reproduced the best by the simulation.

Study of the damping rate

In this section, the alternating forcing method is used on the NPCC test rig, as described in Sec. 3 for the experiments and Sec. 4 for the simulations.

Post-processing methods

The pressure signal is recovered from the probes, treated with a Hilbert transform to extract its envelope. Two submethods are then used : each decaying slope can be fitted with an exponential function to measure its damping rate and a statistical average of each damping rate value can be performed (statistical method), or the average cycle can be computed using phase locking and the exponential fitting can be made on its envelope (average cycle method).

Only the statistical method was performed by Gonzalez-Flesca et al. to treat the results extracted from the experiments, whereas both methods are used here for the simulation results. Fig. 10 gives an overview of the methods applied on the 1T simulated case. It was observed that at least 10 cycles were needed in order to have a good convergence of the damping rate for the cases 1T and 1T1L and even more for the case 1T2L. This is not an issue for the 2D simulations but could be problematic if this methodology is applied to more expensive 3D simulations. Fig. 11 illustrates this by showing the damping rates obtained with the average cycle method with respect to the number of cycles computed.

Comparison between experiment and simulations for the damping rate evaluation

The results of the damping rate study are presented in Table 3. Experimental results are reproduced from Gonzalez-Flesca et al. N is the number of cycles used to determine the damping rate in the statistical method, σ is the relative standard deviation in percentage.

The experiments on the NPCC test rig using alternating forcing by Gonzalez-Flesca et al. led to series of 30 cycles for each excited mode, which were then treated using the statistical method. Simulations are performed for more than 20 cycles for each mode, which gives a relevant amount of statistical points.

The damping rates retrieved are lower than the experimental ones for modes 1T and 1T2L and much lower for mode 1T1L. This is probably due to two-dimensional effects. A 3D simulation using the methodology described here is therefore needed.

Influence of the probe position

A study was conducted on the 1T mode to see whether the position of the probe had an influence on the evaluation of the damping rate. All signals from the 6 probes located in the chamber (HFcx) were studied with the average-cycle method and the statistical method presented earlier. The results are shown in Fig. 12. The damping rate value was found to be very close for each probe, with a maximum variation of 8% (2% if we consider only the average-cycle method). Therefore, the choice to study the signal from only one probe (HFc1) can be validated a posteriori. Figure 12: Dispersion with probe location of the damping rate using a) the statistical method and b) the average-cycle method. x HFc1, •HFc2, HFc3, HFc4, HFc5, +HFc6

Conclusion and perspectives

2D simulations of the NPCC test rig were presented in this paper. The objective is to study the response of a coupled cavity to modal excitation. The mode shape was recovered for the three modes studied experimentally. The results were in reasonable agreement in terms of injection velocity oscillations and acoustic pressure oscillations. Study of the damping rate led to the conclusion that at least 10 cycles were needed in order to obtain a good convergence of the results. Moreover, the damping rate was found to be independant of the position of the probe in the experiment. However, the damping rates found in the simulation were not comparable to the experimental ones, especially for the coupled mode 1T1L, despite good convergence. This is certainly due to the fact that simulations were two-dimensional. Current work focuses on the 3D simulation of the experiment, which will hopefully give better results using the methodology described here.

Figure 2 :

 2 Figure 2: Fully perforated wheel (left), partially perforated wheel (right)

Figure 3 :

 3 Figure 3: (a) Pressure signal recorded at HFc1 (chamber) and HFd1 (dome) as a function of time during a ramp test, each peak corresponds to an eigenmode in the cavity. (b) Short time Fourier transform analysis of the signal detected by HFc1.

Figure 4 :

 4 Figure 4: The selected set of eigenmodes found by the AVSP solver

Figure 5 :

 5 Figure 5: Left: Sketch of the mesh. Right: Zoom on one injector

Fig. 7

 7 Fig. 7 presents instantaneous fields of acoustic pressure. Qualitative comparisons with Fig. 3 show that the mode structures are well retrieved by the simulations. The coupling between cavities of the 1T1L mode is clearly visible.

Figure 7 :

 7 Figure 7: Instantaneous pressure fields for cases 1T, 1T1L and 1T2L given at two instants in phase opposition. Red: maximum, Blue: minimum

Figure 8 :

 8 Figure 8: Injector axial velocity component given by the 2D simulation (left) and the experiment (right). -Signal for HW1 --Signal for HW3

Figure 9 :

 9 Figure 9: Pressure signals recorded by HFc1 (blue) and HFd1 (red) probes. On the left, 2D simulations. On the right, experiments.

Figure 10 :

 10 Figure 10: Damping rate determination for the 1T simulated mode, using the pressure signal from HFc1 probe. (a) Average cycle of the pressure signal (-), envelope (--), exponential fit (-) (b) Logarithm of the average cycle (c) Damping rate determined by the average cycle method with respect to the number of cycles computed (d) Damping rate evaluated for each cycle and used by the statistical method with respect to the number of cycles computed

Figure 11 :

 11 Figure 11: Damping rates obtained by the average cycle method. (a) 1T mode (b) 1T1L mode (c) 1T2L mode

Table 1 :

 1 Eigenfrequencies obtained from AVSP and the experiment

		1T	1T1L 1T2L
	Calculated frequency, Hz	1210 1450	2009
	NPCC measured frequency, Hz 1227 1465	2002

Table 2 :

 2 Operating conditions of the 2D simulation

	p c	3.5 bar
	u in j	17.6 m.s -1
	u inlet 1.37 m.s -1
	ρ 0 inlet T 0 inlet	4.1 kg.m -3 293 K
	p out	2.95 bar

  [START_REF] Gonzalez-Flesca | Simulation, experimentation and modeling contributions to the analysis of high frequency combustion instabilities in liquid propellant rocket engines[END_REF]

Table 3 :

 3 Damping rate results

	1T	1T1L 1T2L

Acknowledgements

Support provided by ArianeGroup, the prime contractor of the Ariane launcher cryogenic propulsion system and CNES, the French National Space Agency, is gratefully acknowledged. This work was granted access to the HPC resources of IDRIS and CINES made available by GENCI (Grand Equipement National de Calcul Intensif) under the allocation A0042B06176. A part of this work was performed using HPC resources from the mesocentre computing center of Ecole CentraleSupélec and Ecole Normale Supérieure Paris-Saclay supported by CNRS and Région Ile-de-France.