Student Member, IEEE, Mounir Kaaniche, Senior Member, IEEE Tassnim Dardouri
email: tdardouri@novelis.io.

Amel Benazza-Benyahia

Gabriel Dauphin
email: gabriel.dauphin@univ-paris13.fr.

Fellow, IEEE Jean-Christophe Pesquet
email: jean-christophe@pesquet.eu.

M Kaaniche
email: mounir.kaaniche@univ-paris13.fr

A Benazza
email: benazza.amel@supcom.rnu.tn.

J.-C Pesquet

Joint Learning of Fully Connected Network Models in Lifting Based Image Coders

Keywords: Lifting schemes, adaptive wavelets, image coding, neural networks, joint learning, optimization. I

The optimization of prediction and update operators plays a prominent role in lifting-based image coding schemes. In this paper, we focus on learning the prediction and update models involved in a recent Fully Connected Neural Network (FCNN)based lifting structure. While a straightforward approach consists in separately learning the different FCNN models by optimizing appropriate loss functions, jointly learning those models is a more challenging problem. To address this problem, we first consider a statistical model-based entropy loss function that yields a good approximation to the coding rate. Then, we develop a multi-scale optimization technique to learn all the FCNN models simultaneously. For this purpose, two loss functions defined across the different resolution levels of the proposed representation are investigated. While the first function combines standard prediction and update loss functions, the second one aims to obtain a good approximation to the rate-distortion criterion. Experimental results carried out on two standard image datasets, show the benefits of the proposed approaches in the context of lossy and lossless compression.

Lifting schemes are basically composed of prediction and update steps that aim to generate detail (i.e., high frequency) and approximation (i.e., low frequency) wavelet subbands, respectively [START_REF] Sweldens | The lifting scheme: A custom-design construction of biorthogonal wavelets[END_REF], [START_REF] Daubechies | Factoring wavelet transforms into lifting steps[END_REF]. For instance, for image decomposition, two-dimensional non separable lifting structures can be used to better capture the 2D characteristics of the input image. A typical non separable LS consists of three prediction steps followed by an update step. This yields three detail subbands oriented diagonally, vertically, and horizontally as well as one approximation subband [START_REF] Sun | A two-dimensional lifting scheme of integer wavelet transform for lossless image compression[END_REF], [START_REF] Kaaniche | Two-dimensional non separable adaptive lifting scheme for still and stereo image coding[END_REF]. Generally, the coding performance of these LS depend on the design of the prediction and update operators. As a result, many efforts have been devoted to optimizing such operators while making the wavelet decomposition adaptable to the image contents [START_REF] Kaaniche | Two-dimensional non separable adaptive lifting scheme for still and stereo image coding[END_REF], [START_REF] Hampson | M -band nonlinear subband decompositions with perfect reconstruction[END_REF], [START_REF] Solé | Generalized lifting prediction optimization applied to lossless image compression[END_REF], [START_REF] Liu | Weighted adaptive lifting-based wavelet transform for image coding[END_REF]. In fact, the optimization of the prediction operator is often achieved by minimizing the 2 -norm of the detail coefficients [START_REF] Gouze | Design of signaladapted multidimensional lifting schemes for lossy coding[END_REF]. To further promote sparsity of the wavelet coefficients, 1 and weighted 1 -based minimization techniques have also been investigated in [START_REF] Kaaniche | Adaptive lifting scheme with sparse criteria for image coding[END_REF]. In addition, an entropy measure has also been used in [START_REF] Solé | Generalized lifting prediction optimization applied to lossless image compression[END_REF], [START_REF] Benazza-Benyahia | Block-based adaptive vector lifting schemes for multichannel image coding[END_REF] for the same purpose. The corresponding optimization problem is solved empirically using the Nelder-Mead simplex algorithm. While the optimization of the predictor has been widely studied, optimizing the update filter is less obvious and few studies have pursued this direction [START_REF] Gouze | Design of signaladapted multidimensional lifting schemes for lossy coding[END_REF], [START_REF] Pesquet-Popescu | Two-stage adaptive filter bank[END_REF], [START_REF] Kaaniche | Non separable lifting scheme with adaptive update step for still and stereo image coding[END_REF]. These studies rely on two main techniques. The first one consists in minimizing the reconstruction error while synthesizing the reconstructed image from the approximation coefficients [START_REF] Gouze | Design of signaladapted multidimensional lifting schemes for lossy coding[END_REF], [START_REF] Pesquet-Popescu | Two-stage adaptive filter bank[END_REF]. However, this technique results in a complex linear system of equations. To overcome this issue, the second technique aims to minimize the error between the approximation subband and the decimated version of the output of an ideal loss-pass filter applied to the input image [START_REF] Kaaniche | Non separable lifting scheme with adaptive update step for still and stereo image coding[END_REF]. Motivated by the success of neural networks and their advantages in achieving accurate nonlinear approximation, the prediction and update lifting stages have been recently implemented using Convolutional Neural Network (CNN) [START_REF] Ma | iWave: CNN-based wavelet-like transform for image compression[END_REF], [START_REF] Ma | End-to-end optimized versatile image compression with wavelet-like transform[END_REF] and Fully Connected Neural Network (FCNN) [START_REF]Dynamic neural network for lossy-to-lossless image coding[END_REF], [START_REF] Dardouri | A neural network approach for joint optimization of predictors in lifting-based image coders[END_REF]. These neural networks-based lifting architectures will be further described and discussed in Section II. In addition to this class of methods, other studies have sought to improve DCT (Discrete Cosine Transform) and Discrete Wavelet Transform (DWT)-based coding schemes [START_REF] Liu | CNN-based DCT-like transform for image compression[END_REF], [START_REF] Ahanonu | Lossless image compression using reversible integer wavelet transforms and convolutional neural networks[END_REF], [START_REF] Akyazi | Learning-based image compression using convolutional autoencoder and wavelet decomposition[END_REF]. In fact, in [START_REF] Liu | CNN-based DCT-like transform for image compression[END_REF], a CNN architecture is exploited to develop a DCT-like transform for image coding. In [START_REF] Ahanonu | Lossless image compression using reversible integer wavelet transforms and convolutional neural networks[END_REF], the authors apply a DWT to the original image, and the generated wavelet subbands are fed into a CNN to produce the final detail coefficients.

Other research efforts have been devoted to intra-prediction coding techniques using both CNN and FCNN [START_REF] Li | Fully connected networkbased intra prediction for image coding[END_REF], [START_REF] Dumas | Context-adaptive neural network-based prediction for image compression[END_REF], [START_REF] Yilmaz | Effect of architectures and training methods on the performance of learned video frame prediction[END_REF]. For instance, in [START_REF] Dumas | Context-adaptive neural network-based prediction for image compression[END_REF], the authors proposed to apply the FCNN to small image blocks and the CNN to large blocks. Moreover, most of the remaining existing NN-based image compression methods follow the same kind of procedure. First, a nonlinear analysis transform is applied to the input image. The generated feature maps are then quantized and entropy encoded. Finally, a nonlinear synthesis transform is performed to reconstruct the image. These methods are optimized using an end-to-end procedure and mainly differ in the employed NN models [START_REF] Toderici | Full resolution image compression with recurrent neural networks[END_REF], [START_REF] Ballé | End-to-end optimized image compression[END_REF], [START_REF] Rippel | Real-time adaptive image compression[END_REF], [START_REF] Li | Learning content-weighted deep image compression[END_REF] and/or the loss function used for training [START_REF] Ballé | Variational image compression with a scale hyperprior[END_REF], [START_REF] Minnen | Joint autoregressive and hierarchical priors for learned image compression[END_REF], [START_REF] Agustsson | Generative adversarial networks for extreme learned image compression[END_REF]. In this category of learned image compression techniques, context model for entropy coding is also investigated to further boost the rate-distortion performance [START_REF] Minnen | Joint autoregressive and hierarchical priors for learned image compression[END_REF], [START_REF] Cheng | Learned image compression with discretized Gaussian mixture likelihoods and attention modules[END_REF], [START_REF] He | Checkerboard context model for efficient learned image compression[END_REF]. It is worth pointing out that most of the developed NN-based image compression methods are devoted to lossy compression [START_REF] Hu | Learning end-to-end lossy image compression: A benchmark[END_REF] and only a few studies are dealing with lossless compression [START_REF] Mentzer | Practical full resolution learned lossless image compression[END_REF], [START_REF] Mentzer | Learning better lossless compression using lossy compression[END_REF], [START_REF] Schiopu | A study of prediction methods based on machine learning techniques for lossless image coding[END_REF]. The objective of this paper is to develop new learning techniques for optimizing neural network models in lifting schemebased image coders. More precisely, we focus on a recent non separable lifting architecture involving three FCNN-based prediction steps followed by an FCNN-based update one [START_REF]Dynamic neural network for lossy-to-lossless image coding[END_REF]. While these different FCNN models have been separately learned in [START_REF]Dynamic neural network for lossy-to-lossless image coding[END_REF], we propose here to investigate joint learning approaches to find the optimal FCNN models. Note that a preliminary version of this work has been presented in [START_REF] Dardouri | A neural network approach for joint optimization of predictors in lifting-based image coders[END_REF]. For instance, unlike [START_REF]Dynamic neural network for lossy-to-lossless image coding[END_REF] and [START_REF] Dardouri | A neural network approach for joint optimization of predictors in lifting-based image coders[END_REF] where 2 -and 1 -norm based loss functions are employed, we resort to an entropy based loss function in this paper. The loss function is grounded on a Generalized Gaussian probabilistic model which has been widely employed for wavelet coefficients modeling. Moreover, this work aims to learn both the FCNN prediction and update models simultaneously, while the joint learning of three FCNN prediction models is achieved independently of that of the FCNN update one in [START_REF] Dardouri | A neural network approach for joint optimization of predictors in lifting-based image coders[END_REF]. To this end, we propose a multi-level optimization technique. This technique consists in interpreting the lifting-based multiresolution decomposition as a full architecture whose involved FCNN models are globally learned at the same time through a unique loss function. In this respect, two new loss functions will be investigated. While the first one resorts to a weighted sum of the loss functions used to optimize the prediction and update stages, the second one aims to obtain a good approximation of rate-distortion functions. The remainder of this paper is organized as follows. In Section II, we describe the related neural networks-based lifting schemes and then focus on the FCNN-based structure we recently proposed. Our statistical model-based entropy loss function is introduced in Section III. The developed learning approaches for the different FCNN models are described in Section IV. Finally, extensive experiments are shown and discussed in Section V, and our conclusions are presented in Section VI.

II. FULLY CONNECTED NEURAL NETWORK BASED LIFTING ARCHITECTURE

A. Related neural networks-based lifting schemes

The use of neural networks for the design of lifting based image coding schemes have been mainly addressed in recent works conducted by Ma et al. [START_REF] Ma | iWave: CNN-based wavelet-like transform for image compression[END_REF], [START_REF] Ma | End-to-end optimized versatile image compression with wavelet-like transform[END_REF] and Dardouri et al. [START_REF]Dynamic neural network for lossy-to-lossless image coding[END_REF], [START_REF] Dardouri | A neural network approach for joint optimization of predictors in lifting-based image coders[END_REF]. In [START_REF] Ma | iWave: CNN-based wavelet-like transform for image compression[END_REF], the authors have considered a separable lifting structure where the prediction stage is achieved using a CNN model and the update one is performed by a mean operation. The corresponding network parameters are then learned by optimizing a distortion criterion. The latter scheme (called iwave), has been extended in [START_REF] Ma | End-to-end optimized versatile image compression with wavelet-like transform[END_REF] (named iwave++) by applying CNNs to both prediction and update stages and optimizing the architecture in an end-to-end fashion using a rate-distortion-based loss function. It must be emphasized that the use of neural networks has been investigated in three different modules [START_REF] Ma | End-to-end optimized versatile image compression with wavelet-like transform[END_REF]: lifting decomposition, entropy coding, and post-processing, resulting in a high computational complexity. Four variants have been developed. While the first one is dedicated to lossless compression, the second and third ones are designed for lossy coding and consider single and multiple NN models, respectively. The last variant is a universal scheme which uses a single model for both lossy and lossless compression. However, the latter suffers from two main drawbacks. First, while the best lossy compression performance is obtained with the multi-model configuration requiring a separate NN model for each point of the R-D curve, the universal scheme results in a significant performance drop. Moreover, iwave and iwave++ rely on the concept of a one-dimensional (1D) LS-based decomposition, which yield an increase of the number of employed NN models, and so the number of parameters, in the whole multiresolution architecture.

To alleviate these shortcomings, we have proposed in [START_REF]Dynamic neural network for lossy-to-lossless image coding[END_REF], [START_REF] Dardouri | A neural network approach for joint optimization of predictors in lifting-based image coders[END_REF] to design a single FCNN model for both lossy and lossless compression while focusing on a Non-Separable Lifting Scheme (NSLS) [START_REF] Kaaniche | Adaptive lifting scheme with sparse criteria for image coding[END_REF] which better captures the 2D characteristics of the data and reduces the number of lifting stages performed in the image decomposition. More precisely, both prediction and update steps are achieved using an FCNN. To this end, new loss functions, defined in the wavelet transform domain, for learning the FCNN prediction and update models have been proposed. While all the FCNN models are learned in a separate manner in [START_REF]Dynamic neural network for lossy-to-lossless image coding[END_REF], joint learning restricted to prediction models is developed in [START_REF] Dardouri | A neural network approach for joint optimization of predictors in lifting-based image coders[END_REF].

B. FCNN based wavelet representation

To perform wavelet decomposition, we have retained the conventional NSLS composed of three prediction steps followed by an update one. In conventional image coders, these steps are often achieved using linear operators. To go beyond these linear models and obtain more accurate nonlinear approximation properties, we have recently proposed performing the prediction and update using neural networks, and more specifically FCNN models. The analysis structure of this new FCNN based NSLS is shown in Fig. 1. More precisely, x j denotes the approximation subband at resolution level j where x 0 = x represents the input image. In the first split stage, the input x j is decomposed into four subsets of samples denoted by

       x 0,j (m, n) = x j (2m, 2n), x 1,j (m, n) = x j (2m, 2n + 1), x 2,j (m, n) = x j (2m + 1, 2n), x 3,j (m, n) = x j (2m + 1, 2n + 1). (1)
Then, three prediction stages, based on three FCNN models f

∀ o ∈ {HH, LH, HL}, x (o) j+1 (m, n) = x i,j (m, n) -xi,j (m, n) = x i,j (m, n) -f (o) j (x (o) j (m, n)) (2)
where, for each i ∈ {1, 2, 3}, x i,j (m, n) is the sample to be predicted, x(o) j (m, n) is the input reference vector containing the samples used to generate the detail coefficients x (o) j+1 , and xi,j (m, n) is the predicted value that corresponds to the output of the FCNN model f As shown in Fig. 2, the reference vector x(o) j (m, n) is first fed to the input layer of an FCNN model. Then, a few hidden layers with various dimensions (i.e., number of neurons) are employed. The output values of these neurons are computed based on a linear combination (with bias) followed by a nonlinear activation function. Finally, the output layer, with a single neuron, generates the predicted value xi,j (m, n) based on a linear combination of the neuron values associated with the last hidden layer. Following the three prediction steps, an update step using an xi,j (m, n)

(o) j (x (o) j (m, n)). xj(m, n) • ¹ ¹ ¹ x3,j(m, n) x2,j(m, n) x1,j(m, n) x0,j(m, n) xj+1(m, n) x (HL) j+1 (m, n) x (LH) j+1 (m, n) x (HH) j+1 (m, n) f (HH) j f (LH) j f (HL) j f (LL) j ×ÔÐ Ø
À Ò Ð Ý Ö× Θ (o) j x(o) j (m, n) ÇÙØÔÙØ Ð Ý Ö ÁÒÔÙØ Ð Ý Ö Fig. 2. FCNN-based prediction stage. FCNN model f (LL) j
is performed to produce the approximation coefficients x j+1 :

x j+1 (m, n) = x 0,j (m, n) + tj (m, n) = x 0,j (m, n) + f (LL) j (x j+1 (m, n)) (3)
where xj+1 (m, n) is the input reference vector containing the diagonal, vertical, and horizontal detail coefficients. Similar to the FCNN-based prediction stage, xj+1 (m, n) serves as an input vector for the FCNN update model f (LL) j . A stack of hidden layers followed by an output layer with a single neuron are then used to produce tj (m, n) and deduce the approximation coefficients x j+1 according to (3).

C. Independent learning approaches

To learn the four FCNN models, f (o) j with o ∈ {HH, LH, HL, LL}, a simple approach, adopted in [START_REF]Dynamic neural network for lossy-to-lossless image coding[END_REF], trains each model defined at a given resolution level j. We recall the two optimization techniques which have been employed for learning the prediction and update models. is learned by performing several forward and backward propagation passes while minimizing a loss function. In this manner, the Mean Square Error (MSE) loss function has been widely used in regression tasks. As a result, we proposed in [START_REF]Dynamic neural network for lossy-to-lossless image coding[END_REF] to train each FCNN-based prediction model using the 2 -norm of the prediction error between the target pixels x i,j (m, n) and the predicted ones xi,j (m, n):

∀ o ∈ {HH, LH, HL}, L (p) (Θ (o) j ) = 1 M j N j Mj m=1 Nj n=1 x i,j (m, n) -xi,j (m, n) 2 (4)
where M j and N j represent the dimensions of the image x j divided by 2. Note that the superscript 'p' is used to emphasize that the above loss function is specific to the prediction model. To optimize this loss function, a Mini-Batch Gradient Descent (MBGD) algorithm has been employed [START_REF] Kingma | Adam: A method for stochastic optimization[END_REF]. For instance, the gradient of the loss function is computed for each mini-batch and the model parameters are then updated from one minibatch to the next one. By repeating this process over several epochs until convergence of the algorithm, the optimal weights Θ (o) j are obtained. The optimal weights are finally applied to the test images to find the predicted pixels xi,j and produce the detail coefficients x (o) j+1 using (2).

2) Learning the FCNN-based update model: Once the three FCNN prediction models are learned and their respective detail subbands are produced, one can focus on learning the FCNN update model. However, since the generation of the approximation coefficients is quite different from that of the detail ones, a more appropriate loss function should be employed for the update stage. In this respect, we proposed to use a simple and efficient update optimization design method which consists in minimizing the error between the approximation subband and the decimated version of the output of an ideal low pass filter (whose impulse response will be denoted by h) applied to the input x j [START_REF] Kaaniche | Non separable lifting scheme with adaptive update step for still and stereo image coding[END_REF]. This error is given by

e j (m, n) = y j+1 (m, n) -x j+1 (m, n) = y j+1 (m, n) -x 0,j (m, n) -f (LL) j (x j+1 (m, n)) (5)
where

y j+1 (m, n) = ( h * x j )(2m, 2n). (6)
Thanks to [START_REF] Xing | Adaptive non separable vector lifting scheme for digital holographic data compression[END_REF], the update learning task can be seen as a prediction learning task that aims to predict, from the input reference vector xj+1 (m, n), the target signal t j (m, n) defined as

t j (m, n) = y j+1 (m, n) -x 0,j (m, n). (7)
Therefore, similar to (4), the FCNN-based update model is trained by minimizing the MSE between t j (m, n) and

tj (m, n) = f (LL) j (x j+1 (m, n))
. Thus, the loss function used for training the FCNN update model is given by

L (u) (Θ (LL) j ) = 1 M j N j Mj m=1 Nj n=1 y j+1 (m, n) -x 0,j (m, n) -tj (m, n) 2 (8)
where the superscript 'u' is employed to refer to the loss function used with the update model. Once the update network is trained, its optimal vector of parameters Θ (LL) j is employed to compute the output tj (m, n) from the input vector xj+1 (m, n) and generate the approximation coefficients using (3). For the test phase, and in order to generate integer wavelet coefficients, which are mandatory for lossless compression, a rounding operator is applied to the second terms (i.e., f 2) and (3).

(o) j (•)) of (

D. Analysis of the FCNN-based lifting structure

The employed loss functions and learning strategy present some advantages and drawbacks. One of these advantages is that the defined loss functions are computed in the transform domain and so, they do not require to perform image reconstruction as often considered in the existing learned image compression methods. Moreover, the adopted learning strategy can be seen as the straightforward solution which consists in training the different FCNN models in a separate manner. This allows to learn some models (in particular those related to the second and third prediction stages) in parallel. However, such an independent learning approach does not take into account two main aspects inherent to NSLS based coding schemes. The first aspect concerns the dependencies existing between these models at a given resolution level j. As it can be seen in Fig. 1, the output of the first FCNN prediction model, and more specifically the diagonal detail signal x (HH) j+1 , is used as the reference signal (i.e., as an input) of the second and third FCNN prediction models to produce the vertical and horizontal detail signals. The second aspect is related to the multiresolution form of the NSLS architecture. A decomposition carried out over J resolution levels is iteratively performed, and yields one approximation subband and 3J detail subbands. This means that the coefficients computed at a given level will impact those generated at coarser levels. For this reason, we propose in this paper to resort to novel joint learning approaches while investigating different loss functions, as it will be described in Sections III and IV.

III. PROPOSED STATISTICAL MODEL BASED LOSS

FUNCTION

While the mean square prediction error has been widely used as a loss function for learning predictive models, we propose here to explore an entropy-based loss function. This choice is motivated by the fact that the entropy represents a good approximation to the final bitrate [START_REF] Gish | Asymptotically efficient quantizing[END_REF].

A. Statistical model

The analysis of wavelet coefficients and their probability distributions have often been exploited in various image processing tasks including denoising and compression. For instance, the Generalized Gaussian Distribution (GGD) has been extensively used to model the subband coefficients x (o) j+1 of a given wavelet representation. These coefficients can be seen as realizations of a random variable X j+1 is given by ∀o ∈ {HH, LH, HL}, ∀ξ ∈ R,

g (o) j+1 (ξ; α (o) j+1 , β (o) j+1 ) = β (o) j+1 2α (o) j+1 Γ( 1 β (o) j+1 ) e - |ξ| α (o) j+1 β (o) j+1 ( 9
)
where Γ is the Gamma function, α

j+1 ∈]0, +∞[ is the scale parameter, and

β (o)
j+1 ∈]0, +∞[ is the shape parameter of the GGD. The particular case when β

(o) j+1 = 2 (resp. β (o) j+1 = 1)
corresponds to the Gaussian distribution (resp. the Laplacian one). Let us also recall that the use of 2 (resp. 1 )-norm based loss function suggests that the data distribution follows such a Gaussian (resp. Laplacian) model. However, multiresolution representations have the advantage of producing sparse coefficients whose distribution shape parameters β (o) j+1 are smaller than 1. To confirm this property, we have conducted a distribution analysis of the detail coefficients, and more specifically the shape parameters β (o) j+1 of their associated GGD models, for a large set of samples extracted from the popular CLIC database described in Section V. Note that the parameters of the GGD models are estimated by applying the maximum likelihood technique [START_REF] Do | Wavelet-based texture retrieval using generalized Gaussian density and Kullback-Leibler distance[END_REF] to the subband coefficients produced by the independent MSE-based learned FCNN model (see Section II-C). Fig. 3 illustrates the distribution of the estimated

β (o)
j+1 values obtained with the three detail subbands at the first three resolution levels. We also provide the average values

β(o) j+1 = 1 K K k=1 β (o,k) j+1 ( 10
)
where K is the number images. Thus, it can be observed that typical values of the shape parameters β (o) j+1 range from 0.2 to 1 and their mean values are around 0.6. Moreover, these values become smaller at coarser levels. Based on this analysis, instead of using a given value of β

B. Entropy-based loss function

Let

x(o) j+1 (m, n) be the quantized coefficients using a uniform scalar quantizer with a quantization step q (o) j+1 . These coefficients can also be seen as realizations of a random variable X(o) j+1 . At high bitrates, it has been shown in [START_REF] Gish | Asymptotically efficient quantizing[END_REF] that the discrete entropy of X(o) j+1 can be approximated as follows:

H( X(o) j+1 ) ≈ h(X (o) j+1 ) -log 2 (q (o) j+1 ) (11)
where h(X

(o) j+1 ) is the differential entropy of the variable X (o)
j+1 . Thus, it can be seen that the discrete entropy of the quantized source is (up to an additive constant) approximately equal to the differential entropy of the original (i.e., non-quantized) source. For such a source following a GGD, the law of large numbers yields the following expression of the differential entropy:

h(X (o) j+1 ) ≈ 1 M j N j ln(2)(α (o) j+1 ) β (o) j+1 Mj m=1 Nj n=1 x (o) j+1 (m, n) β (o) j+1 + log 2 2α (o) j+1 Γ 1 β (o) j+1 β (o) j+1 . ( 12
)
Therefore, with the ultimate goal of approximating the bitrate of the subbands to be generated, we propose to use the differential entropy of the detail coefficients as a learning loss for the different FCNN-based prediction models. In this context, a possible choice for setting the β

α(o) j+1 = 1 K K k=1 β(o) j+1 M j N j Mj m=1 Nj n=1 |x (o,k) j+1 (m, n)| β(o) j+1 1/ β(o) j+1 , (13)
where

x(o,k) j+1 is the detail subband of the k-th training image whose GGD shape parameter is

β(o) j+1 . The expression of β(o) j+1
is still given by [START_REF]Dynamic neural network for lossy-to-lossless image coding[END_REF]. By omitting constant terms, this approach leads to the following loss function, which in turn is used to learn the FCNN prediction weight parameters

Θ (o) j : ∀ o ∈ {HH, LH, HL}, L (p,1) j,o (Θ (o) j ) = 1 M j N j ln(2)(α (o) j+1 ) β(o) j+1 Mj m=1 Nj n=1 x (o) j+1 (m, n) β(o) j+1 . (14)
This loss function provides a natural extension of both 1 and 2 losses. It relies however on a relatively rough averaged estimate of α (o) j+1 . A second approach for estimating the α (o) j+1 parameter relies on the maximum likelihood estimate that would be obtained from the generated subband coefficients:

α(o) j+1 = β (o) j+1 M j N j Mj m=1 Nj n=1 |x (o) j+1 (m, n)| β (o) j+1 1/β (o) j+1 . ( 15
)
Although this estimate cannot be practically calculated, its expression can be injected into [START_REF] Daubechies | Factoring wavelet transforms into lifting steps[END_REF], thus leading to the definition of an alternative form of the loss function for minimizing the entropy of the wavelet coefficients:

∀ o ∈ {HH, LH, HL}, L (p,2) j,o (Θ (o) j ) = 1 β(o) j+1 log 2 1 M j N j Mj m=1 Nj n=1 x (o) j+1 (m, n) β(o) j+1 . (16)
Because of the behaviour of the log 2 function around 0, this loss function has however a tendency to be sensitive to the dynamics of the subband coefficients.

To benefit from the advantages of the two previous approaches, we propose to define the loss function as the arithmetic mean of the two previous expressions in ( 14) and ( 16), i.e.

∀ o ∈ {HH, LH, HL}, L (p) j,o (Θ (o) j ) = 1 2 (L (p,1) j,o (Θ (o) j ) + L (p,2) j,o (Θ (o) j )). ( 17
)
Note that this loss function depends on the averaged GGD parameters β(o) j+1 and α(o) j+1 given by ( 10) and ( 13), respectively. In this respect, and for the sake of simplicity (to avoid generating the wavelet coefficients of each image and estimating the GGD parameters of the resulting subbands for several epochs), β(o) j+1 and α(o) j+1 are computed over the training dataset only once time before starting the learning process (i.e., without updating them after each epoch). We will see in the next section how the defined loss function can be employed to develop a new rate-distortion strategy.

IV. TOWARDS JOINT LEARNING PREDICTION AND UPDATE FCNN MODELS

A. Motivation

To learn the FCNN models involved in our NSLS architecture, different approaches can be envisaged. A straightforward solution, adopted in [START_REF]Dynamic neural network for lossy-to-lossless image coding[END_REF], consists of independently learning the FCNN models as described in Section II-C. To overcome the aforementioned limitations of such approach, a single level optimization technique could be used. This technique aims at jointly learning the FCNN models by optimizing a loss function performed on the outputs of a given resolution level. Such a learning approach, where only the three prediction models are jointly trained, independently of the update, has been developed in [START_REF] Dardouri | A neural network approach for joint optimization of predictors in lifting-based image coders[END_REF]. However, in order to deal with the strong dependencies existing between the different wavelet subbands (intra and inter scales), it appears more judicious to learn the involved FCNN prediction and update models while taking into account all the generated coefficients. Such an approach, performed across the different resolution levels, will be referred to as a multi-scale optimization technique.

B. Multi-scale optimization techniques for joint learning of prediction and update models

To achieve this goal, our lifting-based multiresolution structure will be interpreted as a global architecture whose different FCNN prediction and update models will be jointly learned. In this respect, and instead of using different loss functions specific to the prediction and update models, these models will be learned at once through a unique loss function. Two expressions of the loss functions will be investigated subsequently.

1) Combined predict-update loss functions: Let Θ denote the set of weights associated with all the FCNN prediction and update models used across the J resolution levels:

Θ = Θ (o) j o∈{HH,LH,HL,LL} j∈{0,...,J-1} . ( 18
)
To learn the resulting vector of parameters, a first approach consists in combining the two loss functions previously used for optimizing the prediction and update models, respectively. Thus, the corresponding loss function is a weighted sum of two terms evaluated on the multiresolution representation of each training image. The first term represents the sum of the entropy of all the detail subbands while the second one aims at ensuring that the subband x J , obtained at the resolution level J, represents a coarse approximation to the original image. Therefore, our first loss function used for jointly learning the prediction and update models is given by

L (p,u) 1 (Θ) = J-1 j=0 1 4 j+1 o∈{HH,LH,HL} L (p) j,o (Θ (o) j ) + λ 1 4 J L(u) (Θ (LL) J-1 ) (19)
with λ 1 is a positive constant weighting the predict and update loss functions. Note that the factor 1 4 j+1 corresponds to the ratio between the size of the wavelet subband x (o) j+1 and that of the original image, so as to weight properly the contribution of this subband in the final image representation. While the predict loss function L (p) j,o is defined in [START_REF] Liu | Weighted adaptive lifting-based wavelet transform for image coding[END_REF], the update one L(u) will be expressed as

L(u) (Θ (LL) J-1 ) = M J-1 m=1 N J-1 n=1 y J (m, n) -x 0,0 (m, n) -tJ-1 (m, n) 2 (20)
where y J is obtained by applying J successive operations of convolution to the original image x 0 , using the ideal low-pass filter h (see Eq. ( 6)), followed by a decimation of factor 2.

2) Rate-Distortion approximation based loss function: In the context of lossy coding application, a more appealing learning approach would consist in optimizing all the FCNN models by minimizing a Rate-Distortion (R-D) based cost function. Note that most of the existing end-to-end learning image compression methods use a R-D based loss function where the quantization is replaced with an additive uniform noise on the unit interval. This is achieved while assuming a fixed uniform scalar quantizer. However, such a common approach cannot be easily exploited in our subband-based image coding context where variable quantization steps are generally assigned to the different subbands in the test coding phase. As a result, we propose an alternative solution based on a coarse-to-fine coding/decoding strategy. More precisely, while the rate can again be approximated by the weighted sum of the entropies of all the subbands, a both simple and efficient distortion evaluation will be performed. First, we compute the reconstruction error (MSE) between the original image and the reconstructed one from the approximation coefficients at the coarsest resolution level J, i.e by assuming that the remaining detail coefficients are set to zero. Then, similar reconstruction errors are evaluated and accumulated while progressively adding the detail subbands. Therefore, our new R-D based loss function, denoted by L (p,u) 2

, can be defined as follows:

L (p,u) 2 (Θ) = R(Θ) + λ 2 D(Θ) (21)
where λ 2 is a positive constant weighting the rate and distortion criteria, given respectively by

R(Θ) = J-1 j=0 1 4 j+1 o∈{HH,LH,HL} L (p) j,o (Θ (o) j ) + 1 4 J L (p) J-1,LL (Θ (LL) J-1 ), (22)
and

D(Θ) = 3J i=1 MSE(x, xi (Θ)). (23)
Hereabove, xi corresponds to the reconstructed image obtained through the synthesis stage of the FCNN structure while using the first i-th subbands of the multiresolution representation.

To evaluate [START_REF] Ma | iWave: CNN-based wavelet-like transform for image compression[END_REF], the subbands are added following a coarseto-fine resolution order as shown in Fig. 4. Note that the diagonal detail subband obtained at the finest resolution level is not considered in the last synthesis stage. This is because adding the last subband will result in a reconstruction error between x and x3J+1 equal to zero in theory, due to the perfect reconstruction property of the lifting scheme. The parameters α(o) j+1 and β(o) j+1 , used in the above two loss functions, have been defined in [START_REF] Sun | A two-dimensional lifting scheme of integer wavelet transform for lossless image compression[END_REF] and depend on the orientation of the detail subband as well as its resolution index. Moreover, whatever the employed loss function (L

(p,u) 1 or L (p,u) 2
), it is minimized using the MBGD algorithm. After training, the optimized weight vector Θ j containing the learned FCNN prediction and update models is applied to the test images to generate their respective multiresolution representations (using Eqs.

(2) and ( 3)), and they are subsequently encoded.

V. EXPERIMENTAL RESULTS

In this section, the proposed joint learning of the prediction and update FCNN models in lifting based coding schemes will be evaluated. Our approach will be compared to different state-of-the-art neural networks based compression methods.

A. Experimental settings

To evaluate the performance of the proposed approach, our FCNN-based lifting architectures have been trained using the Challenge on Learned Image Compression (CLIC) database 1 .

The training dataset consists of 585 images of various sizes. The compression methods are then validated on the test CLIC dataset by randomly selecting 40 crop images of size 512 × 512. In addition, we have also considered 30 images, of size 1200 × 1200, taken from the Tecnick sampling dataset 2 [START_REF] Asuni | Test images: a large-scale archive for testing visual devices and basic image processing algorithms[END_REF], [START_REF]Test images: A large data archive for display and algorithm testing[END_REF]. Note that all the tested methods will be evaluated on the luminance component of the CLIC and Tecnick images since our FCNN-based LS is designed to process 2D images (i.e., a single component). Regarding the different FCNN prediction and update models, we have employed: 4 hidden layers of size 128 × 64 × 32 × 16, Parametric Rectified Linear Unit (PReLU) activation functions, and a learning rate equal to 10 -3 while applying a decay of 10 -4 . The simulations were carried out by using Keras and TensorFlow on an NVIDIA Tesla V100 32 GB GPU. Finally, the initialization steps of the joint learning approaches used the pre-trained FCNN models obtained with the independent learning strategy. 3

B. Comparison methods

First, the prediction and update steps of the proposed lifting structure have been performed using the 2D spatial supports shown in Fig. 5. Recall that these supports represent the samples assigned to the input layers of the different FCNN models.

(b) (a) (c) (d)
x 0,j (m, n)

x 1,j (m, n)

x 2,j (m, n) j+1 , and (d) the approximation coefficients x j+1 . Note that for every step, the pixels to be predicted and updated are highlighted in black. 1 http://www.compression.cc/2018/challenge/ 2 https://testimages.org/ 3 The source code of the proposed methods as well as the trained FCNN prediction and update models will be made publicly available.

x 3,j (m, n) x (HH) j+1 (m, n) x (HL) j+1 (m, n) x (LH) j+1 (m, n)
Our proposed FCNN-based LS decompositions, carried out over three resolution levels, will be designated as follows.

• FCNN-LS-JL-ML1 represents the first improved version of the FCNN-based LS method, described in Section IV-B, where both FCNN prediction and update models are jointly learned using a multi-level optimization technique based on the combined predict-update loss function L

(p,u) 1 in ( 19). • FCNN-LS-JL-ML2 is the second improved version of the FCNN-based LS. It is similar to the previous one, except that we resort to the rate-distortion based loss function

L (p,u) 2
in [START_REF] Pesquet-Popescu | Two-stage adaptive filter bank[END_REF]. Note that these two approaches are tested using four λ 1 and λ 2 values (as discussed at the beginning of Section V-D) and the best model in terms of R-D performance is selected. The above methods will be compared to the following stateof-the-art deep learning-based coding methods:

• AE-Fact [START_REF] Ballé | End-to-end optimized image compression[END_REF] is among the first reference end-to-end optimized image compression methods and relies on a nonlinear transform composed of three successive stages of linear filters (convolutions) and nonlinear activation functions. It uses a non-adaptive distribution model, based on piecewise linear functions, referred to as factorizedprior model. • AE-Hyp [START_REF] Ballé | Variational image compression with a scale hyperprior[END_REF] corresponds to an extension of the previous one and aims to integrate a hyperprior model to capture the spatial dependencies of the latent representation. To do so, a zero-mean Gaussian distribution with standard deviation parameter σ 2 is considered. • AE-Hyp-GMM [START_REF] Cheng | Learned image compression with discretized Gaussian mixture likelihoods and attention modules[END_REF] is also an improved version of the previous architectures and uses a Gaussian mixture model (GMM) with an attention module.

• iwave [START_REF] Ma | iWave: CNN-based wavelet-like transform for image compression[END_REF] is a recent method more related to this work.

It resorts to a neural network-based LS where the update step is a mean filter and the prediction one is performed using a CNN. • iwave++ [START_REF] Ma | End-to-end optimized versatile image compression with wavelet-like transform[END_REF] corresponds to an extended version of the "iwave" by applying CNN to both prediction and update steps and optimizing the architecture in an end-to-end fashion. Note that the tested method corresponds to the lossy multi-model "iwave++", which is the only version available on GitHub. All these NN-based coding methods (except "iwave++ [START_REF] Ma | End-to-end optimized versatile image compression with wavelet-like transform[END_REF]") have been developed for lossy image compression purposes and as such, they will be considered as benchmarks in terms of R-D performance. However, "iwave [START_REF] Ma | iWave: CNN-based wavelet-like transform for image compression[END_REF]" can be easily exploited as a lossless compression method. Furthermore, the proposed methods will also be compared to our previous FCNN-based lifting schemes [START_REF]Dynamic neural network for lossy-to-lossless image coding[END_REF], [START_REF] Dardouri | A neural network approach for joint optimization of predictors in lifting-based image coders[END_REF], through an ablation study, to show the benefits of the developed joint multi-scale learning techniques compared to the independent and single level learning approaches. In addition to the aforementioned deep learning methods, the proposed ones will be compared to some popular image coding standards including JPEG2000 and BPG. It should be noted that for the proposed FCNN-LS and "iwave [START_REF] Ma | iWave: CNN-based wavelet-like transform for image compression[END_REF]" methods, JPEG2000 has been used only as an entropy encoder. For instance, the analysis and synthesis stages of JPEG2000 (based on 5/3 and 9/7 transforms) have been disabled and replaced by those of the tested FCNN-LS or "iwave".

C. Performance metrics

The proposed methods have been evaluated in the context of lossy as well as lossless compression using various criteria. In the context of lossless compression, we have considered the entropy of the multiresolution representation as well as the real bitrate of the encoded image. While the bitrate is measured using the JPEG2000 encoding module, the entropy value of a given wavelet representation is expressed as

H = J j=1 o∈{HL,LH,HH} 1 4 j H (o) j + 1 4 J H (LL) J (24)
where

H (o) j
is the entropy of the wavelet subband x (o) j at resolution level j and orientation o. The lossy compression performance are illustrated in terms of R-D results. To assess the quality of reconstructed images at different bitrates, three metrics are employed. The first and second ones are the widely used Peak Signal-to-Noise Ratio (PSNR) and Structural SIMilarity (SSIM) metrics [START_REF] Wang | Image quality assessment: from error visibility to structural similarity[END_REF]. The third one is the Perceptual Image-Error Assessment through Pairwise Preference (PieAPP) metric [START_REF] Prashnani | PieAPP: Perceptual imageerror assessment through pairwise preference[END_REF], which was found to be better-correlated with human perception than its counterparts such as PSNR and SSIM. It is worth pointing out that such traditional metrics, and more specifically the PSNR, were found to be much less accurate to assess the visual quality of the reconstructed images in recent studies devoted to the quality assessment aspects of deep learning based compressed images [START_REF] Valenzise | Quality assessment of deep learning-based image compression[END_REF], [START_REF] Cheng | Perceptual quality study on deep learning based image compression[END_REF], [START_REF] Khan | NNCD-IQA: A new neural networks based compressed database for image quality assessment[END_REF]. Finally, the Bjøntegaard metric [START_REF] Bjøntegaard | Calculation of average PSNR differences between RD curves[END_REF] has been employed to evaluate the R-D performance in terms of bitrate saving and image quality enhancement.

D. Results and discussion

1) Impact of parameters:

• Influence of λ 1 and λ 2 values: Our proposed FCNN-LS-JL-ML1 and FCNN-LS-JL-ML2 methods, whose loss functions are given by ( 19) and ( 21), were first evaluated using different λ 1 and λ 2 values. More precisely, for each k ∈ {1, 2}, we considered λ k ∈ {10 -4 , 10 -3 , 10 -2 , 5×10 -1 }. The impact of these λ k values on the coding performance is shown in Fig. 6 and Table I in terms of R-D and entropy results, respectively. For the first multi-scale optimization approach, the best results are obtained with λ 1 = 10 -2 at low bitrates and λ 1 = 5×10 -1 at higher bitrates. However, based on the R-D plots obtained with the different images of both datasets, we observed that the appropriate choice for the λ 1 value may change from one image to another. Regarding the second-multi-scale optimization technique, it is important to note that using a single model obtained with λ 2 = 10 -3 leads to a good coding performance in terms of SSIM at different bitrates. However, and similar to the first approach, it has been observed with the PieAPP metric that the different test images may have various appropriate λ 2 values. For these reasons, we propose to apply the four models obtained with λ k ∈ {10 -4 , 10 -3 , 10 -2 , 5 × 10 -1 } and then select the best one for each target bitrate of the test images. Note that these hybrid approaches, designated as FCNN-LS-JL-ML1 and FCNN-LS-JL-ML2 in what follows, will result in a negligible overhead (2 bits per image) that needs to be sent to the decoder. Unlike the R-D coding performance, the results of the proposed multi-scale optimization techniques in terms of entropy, shown in Table I, indicate that setting λ 1 (or λ 2 ) to 10 -4 leads to the lowest entropy values for all the dataset images. Therefore, it will be enough to employ a single model (corresponding to λ 1 = λ 2 = 10 -4 ) in the context of lossless compression.

• Influence of β(o) j+1 parameters: We have also studied the impact of the β(o) j+1 parameter, used in the β(o) j+1 -norm arising in [START_REF] Liu | Weighted adaptive lifting-based wavelet transform for image coding[END_REF], on the proposed FCNN-LS-JL-ML1 and FCNN-LS-JL-ML2 methods. While the most commonly used criterion consists in setting β(o) j+1 to 2, we have proposed here to use adaptive β(o) j+1 values, provided in the captions of Fig. 3, which depend on the orientation of the detail wavelet subbands as well as their resolution levels. Table II and Fig. 7 show the coding performance of these methods in terms of entropy and R-D results. Observe that using adaptive β(o) j+1 parameters yields better coding performance, especially in the context of lossy compression.

2) Lossy coding performance: An objective evaluation of the aforementioned coding methods is first performed in the context of lossy compression. The average R-D results, obtained with CLIC and Tecnick datasets, are shown in Fig. 8. Note that higher SSIM values and lower PieAPP values indicate better quality of reconstruction. According to the PSNR values, JPEG2000 appears more performant than some deep learning based image coding methods. On the other hand, the SSIM plots show that most of these deep learning approaches yields better results compared to JPEG2000. For this reason, and based on the recent image quality assessment studies showing the limitations of these traditional metrics (as discussed in Section V-C), we proposed to resort to a more recent perceptual metric (PieAPP), which was found to be more accurate to judge the visual quality of the decoded images [START_REF]Dynamic neural network for lossy-to-lossless image coding[END_REF]. Thus, the PieAPP metric confirms that most of deep learning based image compression methods outperforms JPEG2000. Most importantly, the two proposed multi-scale learning approaches, and in particular the second (i.e., FCNN-LS-JL-ML2), lead to better results compared to different existing deep learning techniques (except iwave++ [START_REF] Ma | End-to-end optimized versatile image compression with wavelet-like transform[END_REF]). Note that the good performance obtained with iwave++ [START_REF] Ma | End-to-end optimized versatile image compression with wavelet-like transform[END_REF] is due to the integration of a postprocessing model in the developed compression method. For fairer evaluation, the proposed methods should be compared to iwave [START_REF] Ma | iWave: CNN-based wavelet-like transform for image compression[END_REF] (rather than iwave++ [START_REF] Ma | End-to-end optimized versatile image compression with wavelet-like transform[END_REF]) since iwave only focuses on the use of neural networks to improve lifting based decomposition (as investigated in this paper). For instance, by removing the Post-Processing (PP) module in iwave++ [START_REF] Ma | End-to-end optimized versatile image compression with wavelet-like transform[END_REF], the latter (designated by "iwave++ (w/o PP) [START_REF] Ma | End-to-end optimized versatile image compression with wavelet-like transform[END_REF]") becomes less performant than the proposed method FCNN-LS-JL-ML2 as depicted in Fig. 9. Finally, our FCNN-LS-JL-ML2 method is slightly worse than BPG in terms of PieAPP metric. However, our method achieves a significant gain compared to BPG in the lossless compression context as it will be shown later. In addition, the proposed method FCNN-LS-JL-ML2 has been compared to the closely related work "iwave [START_REF] Ma | iWave: CNN-based wavelet-like transform for image compression[END_REF]" using the Bjøntegaard metric. Table III.(a) shows the relative gains at low, middle and high bitrates corresponding respectively to the bitrates {0.07, 0.1, 0.15, 0.2}, {0.25, 0.3, 0.4, 0.5} and {0.7, 0.8, 0.9, 1} bpp. Note that a negative value in terms of bitrate saving (resp. PieAPP difference) indicates a decrease of bitrate (resp. PieAPP) for the same PieAPP (resp. bitrate). Thus, it can be seen that our proposed method achieves significant bitrate savings at different bitrates. These bitrate savings reach about 35%, 25%, and 16% at low, middle, and high bitrates. A subjective evaluation of the decoded images at different bitrates is also conducted to confirm the good performance of the proposed FCNN-based LS coding methods. Figures 10 and11 illustrate some reconstructed images for various methods. It can be seen that the proposed FCNN-LS-JL-ML2 based coding method leads to a better visual reconstruction quality compared to the other methods. In particular, our reconstructed images present better contrast while preserving sharp edges.

3) Lossless coding performance: Since our FCNN-LS based methods are applicable to lossy-tolossless coding, they have also been evaluated in the context of lossless compression. Table IV provides the average bitrate values of different lossless coding methods for the CLIC, Tecnick and Kodak image datasets. It can be first observed that the modified version of iwave [START_REF] Ma | iWave: CNN-based wavelet-like transform for image compression[END_REF] is less performant than JPEG2000 and so, it is not efficient for lossless image coding. However, our proposed joint learning approaches significantly outperform iwave [START_REF] Ma | iWave: CNN-based wavelet-like transform for image compression[END_REF]. Moreover, they achieve a gain of about 0.15 bpp compared to the JPEG2000 coding standard. Finally, our methods are more performant than BPG yielding a gain of about 0.1 bpp, 0.45 bpp and 0.25 bpp on CLIC, Tecnick and Kodak datasets, respectively. It should be emphasized that the two variants of the proposed multiscale optimization technique (FCNN-LS-JL-ML1 and FCNN-LS-JL-ML2) lead to similar lossless coding performance. Note that the comparison with other variants of FCNN-based lifting schemes will be later discussed in the ablation study.

4) Ablation study: This study aims to illustrate the role of the different loss functions used to learn the involved prediction and update models in our FCNN-based lifting architecture. In this respect, the proposed joint learning techniques FCNN-LS-JL-ML1 and FCNN-LS-JL-ML2, based on the loss functions L (p,u) 1 [START_REF] Kaaniche | Adaptive lifting scheme with sparse criteria for image coding[END_REF] and L (p,u) 2 [START_REF] Pesquet-Popescu | Two-stage adaptive filter bank[END_REF], have been compared to the two following approaches:

• FCNN-LS-IL where an Independent Learning approach [START_REF]Dynamic neural network for lossy-to-lossless image coding[END_REF] is applied using the prediction and approximation errors based loss functions L (p) and L (u) defined in ( 4) and [START_REF] Huang | LINN: Lifting inspired invertible neural network for image denoising[END_REF], respectively. • FCNN-LS-JL-SL where the FCNN prediction models are jointly learned using a single level optimization tech-nique [START_REF] Dardouri | A neural network approach for joint optimization of predictors in lifting-based image coders[END_REF] (i.e., by minimizing a weighted sum of the loss functions L (p) evaluated on a given resolution level).

For fair comparison, since the proposed joint learning techniques have been designed using adaptive β(o) j+1 values, the independent and single level optimization techniques have also been evaluated using the same β(o) j+1 values instead of setting them to 2 as performed in [START_REF]Dynamic neural network for lossy-to-lossless image coding[END_REF] and [START_REF] Dardouri | A neural network approach for joint optimization of predictors in lifting-based image coders[END_REF]. Let us recall that the benefits of using adaptive β(o) j+1 parameters has been shown at the beginning of Section V-D and Fig. 7. Fig. 12 illustrates the R-D performance of the different learning strategies. It can be noticed that the single level learning approach is slightly less performant than the independent strategy, which suggests that the joint optimization of only the prediction models (per resolution level) is still suboptimal. Thus, by resorting to a multi-level optimization technique and combining the prediction and update loss functions, the FCNN-LS-JL-ML1 improves the R-D results. Finally, further improvements are achieved using the R-D approximation based loss function (i.e., FCNN-LS-JL-ML2). The latter has also been compared to the independent learning approach FCNN-LS-IL using the Bjøntegaard metric. Table III.(b) shows the relative gains at low, middle, and high bitrates (using the same bitrates defined with Table III.(a)). The proposed method outperforms the independent learning approach and reaches a bitrate saving of about 6%, 9%, and 14% at low, middle, and high bitrates. Moreover, in the context of lossless compression, Table IV shows that the single level optimization technique (i.e "FCNN-LS-JL-SL") results in a slight improvement of 0.02 bpp compared to the independent learning approach (i.e "FCNN-LS-IL"). This gain reaches 0.1 bpp by applying the multi-level optimization technique.

5) Computational complexity analysis:

Finally, the proposed method is evaluated and compared to the recent related work "iwave++ [START_REF] Ma | End-to-end optimized versatile image compression with wavelet-like transform[END_REF]" in terms of encoding/decoding time, number of model parameters, and number of floating point operations per second (FLOPs). Table V illustrates this comparison for an image of size 1200 × 1200 using an Intel Xeon(R) processor (4 GHz) and a Python implementation. It can be observed that "iwave++ [START_REF] Ma | End-to-end optimized versatile image compression with wavelet-like transform[END_REF]" employs a large amount of parameters (17.91 M) and FLOPs (5595 G). However, the proposed FCNN-LS-JL-ML2 method uses only 167244 trainable parameters and 43,2 GFLOPs. Moreover, the proposed method requires 8.5/3.2 seconds for the encoding/decoding process, which is about 2.7 times faster than "iwave++ [START_REF] Ma | End-to-end optimized versatile image compression with wavelet-like transform[END_REF]". The runtime of our method is reduced to 0.3/0.08 seconds when the code is executed on an NVIDIA Tesla V100 32Gb GPU. The main difference in the computational complexity between the two architectures is explained by the fact that "iwave++ [START_REF] Ma | End-to-end optimized versatile image compression with wavelet-like transform[END_REF]" uses neural networks for three main modules including lifting based decomposition, entropy coding, and post-processing, while our method only focuses on the first module. It should be noted here that the computational complexity has been only given for FCNN-LS-JL-ML2 in Table V since it is the same for all the proposed FCNN-LS. Indeed, the different FCNN-LS schemes use the same architecture and the main difference between them only concerns the learning approach of the FCNN prediction and update models. While the different end-to-end learning methods often use a specific model for each point of the R-D curve which will yield multiple models covering low, middle and high bitrates, it is worth pointing out that the proposed method has the additional main advantage of learning a single model that can be used at different bitrates as shown at the beginning of Section V-D and Fig. 6. Therefore, the proposed method corresponds to a light model and allows a fast encoding/decoding process while being suitable for lossyto-lossless compression. All these results confirm the effectiveness of the proposed FCNN design strategies, and the flexibility offered by the joint learning approaches.

VI. CONCLUSION AND PERSPECTIVES

Novel learning approaches for the design of FCNN based lifting coding schemes have been proposed in this paper. While a straightforward approach consists in separately learning all the involved FCNN models, we have investigated the benefits of joint learning approaches. These approaches allow us to take into account the dependencies existing between the different models as well as the multiresolution aspect of the LS architecture. In doing so, an entropy based loss function and a multi-scale optimization technique have been developed. Experimental results have shown the effectiveness of the proposed approaches in the context of both lossy and lossless compression. In future work, an extension of the proposed FCNN-LS architecture to more sophisticated structures, like vector lifting scheme [START_REF] Dhifallah | Efficient joint multiscale decomposition for color stereo image coding[END_REF], could be envisaged to deal with multi-component images. Another road of investigation could be an end-to-end learning approach taking into account the quantization and entropy coding modules. PieAPP iwave++ (w/o PP) [START_REF] Ma | End-to-end optimized versatile image compression with wavelet-like transform[END_REF] iwave++ [START_REF] Ma | End-to-end optimized versatile image compression with wavelet-like transform[END_REF] FCNN-LS-JL-ML2 Fig. 9. Removing the post-processing (PP) module in iwave++ [START_REF] Ma | End-to-end optimized versatile image compression with wavelet-like transform[END_REF] and comparison with the proposed method: R-D results for the CLIC dataset.

  j+1 , respectively. These detail coefficients are computed as follows:

Fig. 1 .

 1 Fig. 1. Analysis structure of the FCNN-based NSLS architecture.

1 )

 1 Learning FCNN-based prediction models: Let Θ (o) j , with o ∈ {HH, LH, HL} denote the vector of parameters associated with the FCNN model f (o) j . Thus, for each detail subband x (o) j+1 , the vector Θ (o) j

  e., β (o) j+1 = 2 or β (o) j+1 = 1) as is often considered in previous studies, we propose here to adaptively select the values of β (o) j+1 that depend on the subband orientation o as well as the resolution level j.

  j+1 parameters consists of using values β(o) j+1 and α(o) j+1 estimated from K training images. Their subband coefficients are computed by the independent MSE-based learned FCNN model. By using an averaged maximum likelihood estimator, we obtain

Fig. 3 .

 3 Fig. 3. Distribution of β (o) j+1 values for the horizontal (first column), vertical (second column) and diagonal (third column) detail subbands at the first three resolution levels (from top to bottom).

1 Fig. 4 .

 14 Fig.[START_REF] Kaaniche | Vector lifting schemes for stereo image coding[END_REF]. The order of the retained i wavelet subbands (with i ∈ {1, . . . , 3J}, with J = 2) during reconstruction error evaluation.

Fig. 5 .

 5 Fig. 5. Spatial supports of the prediction and update operators used to generate: (a) the diagonal detail coefficients x (HH) j+1 , (b) the vertical detail coefficients x (LH) j+1 , (c) the horizontal detail coefficients x (HL)

  MethodCLIC Tecnick FCNN-LS-JL-ML1 (λ1 = 5 × 10 -1 ) 4.14 3.77 FCNN-LS-JL-ML1 (λ1 = 10 -2 )

SSIMFCNN- 4 )Fig. 6 .Fig. 7 .

 467 Fig. 6. Average R-D results of FCNN-LS-JL-ML1 (first row) and FCNN-LS-JL-ML2 (second row) with different λ k values, with k ∈ {1, 2}, for the CLIC (first column) and Tecnick (second column) image datasets.

  0

Fig. 8 .

 8 Fig. 8. Average R-D results of the CLIC (first column) and Tecnick (second column) image datasets using PieAPP, SSIM and PSNR metrics.

  : SSIM=0.81, PieAPP=3.79 (c): SSIM=0.85, PieAPP=1.61 (d): SSIM=0.84, PieAPP=1.47 (e): SSIM=0.86, PieAPP=1.24 (f): SSIM=0.89, PieAPP=1.13

Fig. 10 .

 10 Fig. 10. (a) Original test image (image 21 of the CLIC dataset). The reconstructed ones at 0.1 bpp using: (b) JPEG2000, (c) CNN-LS [23], (d) FCNN-LS-JL-SL [25], (e) FCNN-LS-JL-ML1, (f) FCNN-LS-JL-ML2.

  : SSIM=0.84, PieAPP=3.86 (c): SSIM=0.85, PieAPP=2.79 (d): SSIM=0.86, PieAPP=2.47 (e): SSIM=0.88, PieAPP=1.48 (f): SSIM=0.89, PieAPP=1.42

Fig. 11 .

 11 Fig. 11. (a) Original test image (image 3 of the CLIC dataset). The reconstructed ones at 0.2 bpp using: (b) JPEG2000, (c) CNN-LS [23], (d) FCNN-LS-JL-SL [25], (e) FCNN-LS-JL-ML1, (f) FCNN-LS-JL-ML2.

Fig. 12 .

 12 Fig. 12. R-D results for the CLIC dataset using different learning techniques of the proposed FCNN based lifting architecture.

TABLE I AVERAGE

 I ENTROPY RESULTS (IN BPP) OF FCNN-LS-JL-ML1 AND FCNN-LS-JL-ML2 WITH DIFFERENT λ k VALUES, WITH k ∈ {1, 2}, FOR THE CLIC AND TECNICK (SECOND COLUMN) IMAGE DATASETS.


G.-M Roux

R Billardon

IDENTIFICATION OF THERMAL BOUNDARY CONDITIONS AND THERMO-METALLURGICAL BEHAVIOUR OF X10CrMoVNb9-1 STEEL DURING SIMPLE TIG WELDING TESTS

To validate the thermo-metallurgical models that are used during finite element simulations of multi-pass TIG welding, two simple experiments -called "disk-spot" and "disk-cycle" TIG welding tests-have been designed. A procedure has been developed to identify the thermal boundary conditions that are imposed to the specimen during these experiments. These boundary conditions include the heat source that is equivalent to the TIG torch as well as the heat flux by convection (-radiation) at the surface of the specimen. A thermo-metallurgical model has been built and identified to represent most important phase changes occurring in a X10CrMoVNb9-1 martensitic steel -also known as ASTM A387 or T91 steel-during the welding process. The results of the thermo-metallurgical analysis of a "disk-spot" TIG mono-pass welding test without metal deposit is compared with experimental observations.

INTRODUCTION

The nominal temperature in the pressure vessels of future nuclear reactors -or so-called Very High Temperature Reactors using gas coolant at low pressure-should be around 450 • C. Because of its microstructural stability at such a temperature, X10CrMoVNb9-1 martensitic steel -also known as ASTM A387 or T91 steel-is considered as a promising candidate material for these structures. Since, the manufacturing process of such thick pressure vessels will most probably involve TIG multi-pass welding, different studies are presently in progress in different laboratories to develop reliable tools for the failure assessment of multi-pass weld joints in structures made of X10CrMoVNb9-1 martensitic steel.

The work presented herein is part of a study the aim of which is to develop finite element simulations of TIG multi-pass welding to predict the residual stresses induced by the process in the vicinity of welded joints in thick structures made of X10CrMoVNb9-1 341 martensitic steel. Since the accuracy of such complex thermo-metallo-mechanical finite element analyses strongly depends on many inputs, two simple experiments, that are called "disk-spot" and "disk-cycle" TIG welding tests, have been designed to validate the numerical tool that is currently in development.

The thermal boundary conditions that are applied to the specimen during a disk-spot TIG welding test without metal deposit are identified in the first part of this paper. The model that is proposed to model the thermo-metallurgical behaviour of X10CrMoVNb9-1 steel during the welding process is discussed and identified in the second part of this paper. The third and last part of this paper is dedicated to the first comparisons between experimental observations and results of thermo-metallurgical numerical simulations of a disk-spot TIG mono-pass welding test without metal deposit.

THERMAL BOUNDARY CONDITIONS

DISK-SPOT AND DISK-CYCLE TIG WELDING TESTS

The development of a reliable finite element tool to simulate TIG multi-pass welding requires a validation procedure including comparisons between experimental results obtained during well controlled tests and the numerical results of the corresponding simulations. Such structural analyses involve so many different thermo-metallo-mechanical aspects that the complete validation procedure of the predictive capabilities of the numerical tool must be a multi-step process, the first steps corresponding to problems of limited complexity. The so-called disk-spot TIG welding test has been designed to validate simulations of TIG mono-pass welding without metal deposit. The so-called disk-cycle TIG welding test has been designed to validate simulations of TIG mono-or multi-pass welding without or with metal deposit.

The disk-spot TIG welding test consists in heating in its centre a circular disk with a TIG spot torch without filler material. The disk rests on three point supports that are made of long ceramic pins in order to minimize contact heat conduction as well as spurious radiation effects. As shown on Fig. 1, different thermocouples and displacement sensors can be used to measure the temperature and the vertical or radial displacements at different points of the disk. A global view of the experimental set up is given in Fig. 2 where the disk is 8 mm thick and 50 mm in diameter.

The disk-cycle TIG welding test can be performed on the same experimental set-up, first, by locating the TIG torch at a given non zero radius and then, by controlling the horizontal displacement of the torch so that its trajectory is a circle. The first circle corresponds to the first welding pass. After a full circle has been completed, the following circles correspond to successive passes. Disk-cycle TIG welding tests can be performed without or with filler material.

The TIG source used for the experiments that are presented in this paper is an automatic Polysoude TIG source and the welding process parameters used for these experiments are given in Tab. 1. Because of the low weight of the disk, "sticking" between the disk and the electrode must be avoided and non-contact priming is obtained by using high frequency current. The TIG torch -without filler material-used for the experiments discussed herein is modelled as a surfacic heat source of given analytical form. The heat flux at the surface of the specimen is modelled by a temperature dependent convection-radiation coefficient. In the following, these thermal boundary conditions are identified while assuming that the thermal properties of the welded material are known.

A typical plot of the evolution during heating of the specific heat, thermal conductivity and density of the base material used for the experiments presented herein is given in Fig. 3. One can notice the non-monotonous evolution of the thermal properties due to a solid-solid phase change that will be discussed in next section of this paper.

During heat transfer analyses, it is common to model, as a first approximation, the heat flux at the surface of solids as:

q = -h(T -T ∞ ) (1)
where h, T and T ∞ respectively denote the convection-radiation coefficient, the local surface temperature and the temperature in the atmosphere "far" from the solid. The convection-radiation coefficient obviously depends on the nature of the surface, its local temperature T and many environmental factors. Among other analytical models, it has been chosen to represent the TIG torch -without filler material-as a surfacic heat source represented by a truncated infinite Gaussian function:

P S = Qexp -3 r 2 r 2 0 with 0 ≤ r ≤ r M ax (2)
where Q, r, r 0 and r M ax respectively denote the intensity of the TIG torch, the current radius and characteristic radii of the torch -measured from the axis of the torch. These choices imply that the thermal boundary conditions that are imposed to the disk during the welding process are defined by the values of the variable convection-radiation coefficient h at any point on the surface of the disk as well as the values of the parameters of the heat source, viz. Q, r 0 and r M ax .

A sensitivity analysis has shown that the accuracy of the heat transfer numerical analysis of this experiment strongly depends on the convection-radiation coefficient [START_REF] Roux | Identification of Thermal Boundary Conditions and Thermo-Metallurgical Behaviour of X10CrMoVNb9-1 Steel. Application to a Disk-Spot Welding Experiment[END_REF]. For instance, it can be shown that the influence on the surface temperature far from the centre of the disk on the surface opposite to the TIG torch of geometric heat source parameter r 0 is negligible compared to the influence of convection-radiation coefficient h. However, to simplify the identification process of the thermal boundary conditions -from a limited number of local temperature measurements-it is proposed to assume that:

• the convection-radiation coefficient is solely temperature dependant -for given material and environment-; • the value of r M ax can be a priori fixed -for given welding parameters. Hence, the process is restricted to the identification of the relationship h(T ) -that is supposed to be valid at any point on the surface of the disk-and of the values of parameters Q and r 0 (see Fig. 4). In practise, the identification procedure that has been applied is a two-step process.

The first step consists in the analysis of the cooling of the disk after its heating in an oven up to a uniform temperature of 800 • C and the identification of the evolution of the convection-radiation coefficient from 700 • C to room temperature.

The inverse identification of the evolution of the convection-radiation coefficient has been performed by using a Levenberg-Marquardt gradient minimization algorithm [START_REF] Levenberg | A Method for the Solution of Certain Problems in Least Squares[END_REF][START_REF] Marquardt | An Algorithm for Least Squares Estimation of Non Linear Parameters[END_REF] and Cast3M [START_REF] Charras | L'Optimisation Dans Cast3M[END_REF] 2-D axisymmetric finite element heat transfer analyses. It consists in the minimization of the following quadratic function of measured and numerically predicted temperatures, respectively denoted by T e and T n , at given control points -of radius rand given time steps t:

J = 1 M 0 M t M0 1 Mt 1 [T e (r, t) -T n (r, t)] 2 (3)
where M 0 and M t respectively denote the number of control points and the number of time steps. The identified temperature evolution of coefficient h -while assuming

T ∞ = 20 • C -is given in Tab. 2.
Tab. 2 Values of the convection-radiation coefficient h to be associated to T∞ = 20 • C as identified from the cooling of an oven heated disk. To minimize the number of parameters to be identified, it is assumed that the evolution of convection-radiation coefficient h at temperatures higher than 700 • C can be interpolated from its values up to 700 • C and its value at a single higher temperature, say 1400 • C.

However, it appears that the non-convexity of the problem does not allow the inverse identification of the value of convection-radiation coefficient h(1400 • C) together with the heat source parameters Q and r 0 , from the heat transfer analysis of a single experiment. Conversely, convergence of the minimization algorithm can be achieved by coupling the analyses of two disk-spot tests that are performed with different TIG torches, i.e. with different heat source parameters Q and r 0 , whereas the heat flux at the surface of the disk is supposed to be controlled by the same temperature evolution of the convectionradiation coefficient h(T ).

Hence, the second step of the identification procedure consists in the inverse identification of the convection-radiation coefficient h(1400 • C) together with the two sets of parameters (Q 1 , r 01 ) and (Q 2 , r 02 ) that define the heat sources respectively associated to TIG torches that are chosen 12 and 16 mm in diameter (see Fig. 5). The same gradient minimization algorithm has been used -with identical sets of time steps t and control points of radius r for both experiments-.

The values gathered in Tab. 3 illustrate the convergence of the minimization algorithm during the second step of the identification procedure and the final results of this identification procedure are given in Fig. 6.

The quality of this identification is illustrated by the comparison that is given in Fig. 7 (for the disk-spot TIG welding test with a ø16 mm TIG torch) of the experimentally mea- sured and numerically simulated time evolutions of the temperature at different locations on the surface of the disk -in fact the locations of the thermocouples-.

1st test 1st test h [W/m 2 ] (ø12 mm TIG torch) (ø16 mm TIG torch) at 1400 • C Q 1 [W] r 01 [10 -3 m] Q 2 [W] r 02 [10

THERMO-METALLURGICAL MODEL

X10CrMoVNb9-1 MARTENSITIC STEEL

The microstructural stability of X10CrMoVNb9-1 steel is explained by its chemical composition and in particular the high percentage of chromium and the presence of vanadium and niobium (see Tab. 4).

Tab. 4 Composition of X10CrMoVNb9-1 steel (wt%) (from [START_REF] Duthilleul | Etude des Evolutions Microstructurales Lors D'Un Traitement Thermique Rapide á Haute Température D'Un Acier de Type T91[END_REF]). The as-delivered microstructure of this material corresponds to tempered martensite. During heating, for instance during the welding process, depending on the maximum temperature reached, this base material may transform into austenite γ, ferrite δ or liquid (see the equilibrium pseudo-binary diagram given in Fig. 8).

C

MODELLING OF AUSTENITIZATION OF X10CrMoVNb9-1 DURING HEATING

The martensite ⇒ austenite diffusion transformation -i.e. the austenitization of the base material-corresponds to a bcc ⇒ fcc phase change. If one assumes that the important specific volume variation that is associated to this phase change is directly related to the evolution of the fraction of formed austenite, this evolution of the austenite fraction rate can be identified from standard dilatometry tests, for instance at constant heating rates. Austenitization start temperature is then naturally associated to the first detection of a volume change. The evolution of the austenite fraction y(T ) for different constant heating rates and the corresponding plot of the austenite start temperature are given in Fig. 9a.

It has been proposed in [START_REF] Brachet | Modelling of Phase Transformations Occuring in Low Activation Martensitic Steels[END_REF] to derive from the experimental results T (y, Ṫ ) given in Fig. 9b the austenite transformation close to equilibrium by using a relationship proposed in [START_REF] Zhu | Determination of Equilibrium Solid Phase Transition Temperature Using DTA[END_REF] such that

T (y eq ) = T (y, Ṫ ) -C Ṫ T (y, Ṫ )exp E R • T 1/3 (4)
In practice, for each value of austenite fraction y, T (y eq ) is identified as the average of the values computed with E [kJ/mole] ∈ [50, 450]. The results are given in Fig. 10.

The approach proposed herein to model austenitization during heating assumes that the transformation can be splitted into two successive stages, viz. an incubation stage and then an initiation-and-growth stage. The complete identification of the model is a three-step process.

It has been proposed in [START_REF] Duthilleul | Etude des Evolutions Microstructurales Lors D'Un Traitement Thermique Rapide á Haute Température D'Un Acier de Type T91[END_REF] that the transformation kinetics close to equilibrium -as derived from the dilatometry tests (see Eqn. (4) and Fig. 9)-can be modelled by the following relationship

y eq (T ) = 1 -exp (-(K 0 (T -A eq0 )) m0 ) (5)
As proposed for instance in [START_REF] Gautier Aeby | Transformations Perlitique et Martensitique Sous Contrainte de Traction Dans les Aciers[END_REF], after discretization in a set of isothermal steps ∆t i at temperatures T i , the incubation time during any anisothermal heating T (t) -for instance at a constant heating rate-, can be related to the incubation times t i (T i ) during isothermal transformation at temperatures T i , by using Scheil additivity principle such that Fig. 9 X10CrMoVNb9-1 steel (tempered martensite initial state) (from [START_REF] Duthilleul | Etude des Evolutions Microstructurales Lors D'Un Traitement Thermique Rapide á Haute Température D'Un Acier de Type T91[END_REF]).

i

∆t i t i (T i ) = 1 (6)
Besides, it has been proposed in [START_REF] Roux | Modélisation de la Transformation Austénitique au Chauffage D'Un Acier Martensitique[END_REF] to model the incubation times t i (T i ) with the following phenomenological relationship

t i (T i ) = A(A ssat -T i )exp C T i -A eq0 (7)
Last, it has been proposed in [START_REF] Brachet | Modelling of Phase Transformations Occuring in Low Activation Martensitic Steels[END_REF] to model the transformation kinetics far from equilibrium -as derived from the dilatometry tests-with a modified version of a model initially proposed in [START_REF] Holt | IAEA Specialists Meeting, IWGFPT/7[END_REF], so that

ẏ = Kexp -W R • T T (y eq ) -A eq0 n + (1 -y) (8)
Eventually, this approach requires the successive identification from experimental results such as those that are given in Fig. 2-3 of the following set of parameters (A eq0 , K 0 and m 0 ), (A, A ssat and C) and (K, W and n).

After identification with Matlab (using a 4-5 order Runge-Kutta scheme with temperature steps equal to 1 • C), the complete model is apt to describe the austenitization kinetics during any anisothermal heating loading, for instance at constant heating rates as illustrated by Fig. 9.

MODELLING OF OTHER TRANSFORMATIONS OF X10CrMoVNb9-1 DURING WELDING

In the following, as a first approximation, the austenite ⇔ ferrite δ transformation is ignored.

During cooling, whatever the cooling rate (in fact such that Ṫ ≥ 0.2 K/s), austenized material transforms into quenched martensite. Standard Koistinen-Marburger model has been used to model this transformation so that the martensite fraction y m (T ) during cooling is modelled by

y m (T ) = y 0 (1 -exp (-K m (M s -T ))) (9)
where y 0 denotes the initial volume fraction of the austenite that depends on the heating loading faced by each point in the vicinity of the weld, whereas K m and M s denote material coefficients.

NUMERICAL SIMULATION OF SIMPLE TIG WELDING TESTS

The simplified thermo-metallurgical model briefly described above has been implemented in the finite element software Cast3M. Latent heats associated to each phase transformation (martensite ⇒ austenite, austenite ⇔ (ferrite δ/)liquid and austenite ⇒ martensite) are taken into account during heat transfer analyses without specific treatment of the molten zone. An illustration of the result of the simulation of a disk-spot TIG welding test is given in Fig. 11 where experimental and simulated heat affected zones may be compared. One can notice that the size of the molten zone is underestimated by the numerical simulation: this point can be explained by the fact that the austenite ⇔ ferrite δ transformation has been ignored and that the exact values of the liquidus and solidus temperature are yet unknown for the material brand used for these experiments. Besides, it can be noticed that the ferrite δ not only appears during heating but is still present in non-negligible proportions after cooling. Last, it can be noticed that the quenched martensite in the heat affected zone exhibits two clearly different morphologies: this point may be related with phenomena that have not been discussed herein, viz. carbides dissolution and austenite grain growth. An illustration of the result of the simulation of a DISK-CYCLE test is given in Fig. 12.

CONCLUSIONS AND WORK IN PROGRESS

The identification procedure of the thermal boundary conditions that is presented in this paper has been first implemented while using temperature measurements from thermocouples located in six different points of the disk (see Fig. 1). A similar procedure is currently being developed while replacing the temperature measurements from thermocouples TC1, TC2 and TC3 by a temperature field measured by an IR camera.

The results that are briefly discussed in the last section of this paper prove that the thermo-metallurgical model that is presented in this paper can be improved, in particular to take account of the austenite ⇔ ferrite δ transformation, carbides dissolution, austenitic grain growth and thermo-metallo-mechanical couplings.

During multi-pass welding process, after reheating induced by successive welding passes, transformed material may be as-delivered material i.e. tempered martensite or quenched martensite resulting from previous pass(es). Besides, depending on the maximum temperature reached, quenched martensite may be austenized or only tempered. These complex loadings require further investigations in terms of thermo-metallurgical model.

Fig. 1 Tab. 1

 11 Fig. 1 Disk-spot TIG welding tests. Fig. 2 Experimental set up.

Fig. 3

 3 Fig. 3 Resistance spotwelding principle.

Fig. 4

 4 Fig. 4 Thermal boundary conditions during disk-spot TIG welding tests.

Fig. 5

 5 Fig. 5 Two-step identification procedure of thermal boundary conditions.

Fig. 6

 6 Fig. 6 Identified values of convection-radiation coefficient and heat source parameters.

Fig. 7

 7 Fig. 7 Experimental vs. simulated time evolutions of temperature at the locations.

Fig. 8

 8 Fig. 8 Fe-0.1 wt% C/Cr equilibrium pseudo-binary diagram.

  (a) Austenite start temperature. (b) Austenite transformation kinetics y(T )

Fig. 10

 10 Fig. 10 Austenite transformation kinetics close to (yeq(T )) and far from (y(T )) equilibrium X10CrMoVNb9-1 steel (tempered martensite initial state).

Fig. 11

 11 Fig. 11 Microstructure of the heat affected zone (experimental vs. simulated).

Fig. 12

 12 Fig. 12 Austenite fraction simulated during a disk-cycle TIG welding test.


I Abalakin
email: ilya.abalakin@gmail.com

V Bobkov

V Tsvetkova
email: valeria.tsvetkova@gmail.com

B Sauvage

F Miralles
email: florian.miralles@umontpellier.fr

T Kozubskaya
email: tatiana.kozubskaya@gmail.com

S F Wornom

Bruno Koobus
email: bruno.koobus@umontpellier.fr

Alain Dervieux
email: alain.dervieux@inria.fr

Towards efficient simulation of turbulent flows and noise in rotating machines

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