Wael Saideni
email: wael.saideni@xlim.fr

Fabien Courreges
email: fabien.courreges@unilim.fr

David Helbert
email: david.helbert@univ-poitiers.fr

Jean Pierre Cances
email: cances@ensil.unilim.fr

ViT-SCI: Video Transformer is all you need for Video Compressive Sensing

Keywords: video compressive sensing, Transformers, attention layer, deep learning, computer vision

In this work, we study a complete framework of Video Compressive Sensing (VCS), from capturing a sequence of video frames in one single compressed measurement to reconstructing the original frames. To our best knowledge, we present the first end-to-end sampling and recovery network built upon video Transformers, widely explored in vision related tasks, to capture long-range spatio-temporal relations. Our proposed Video Transformer for Snapshot Compressive Imaging recovery (ViT-SCI) is based on Spatio-temporal Conventional Multi-Head Attention (ST-ConvMHA) which is an extended version of the fully-connected attention adapted for vision problems. Our comprehensive qualitative and quantitative experiments on several datasets demonstrate that ViT-SCI outperforms previous stateof-the-art methods with much faster reconstruction capacities, which pave the way for applying our algorithm in real-time applications. Indeed, ViT-SCI achieves good quality results in only 8ms to reconstruct video clips of 8 frames. It is able to perform real-time reconstruction of up to 125 measurements per second. Finally, an important ablation study on the Transformer network is provided to inspire future research works aiming to test the abilities of Transformers in vision tasks.

I. INTRODUCTION

With the huge demand for data acquisition and processing, Video Compressive Sensing or precisely Video Snapshot Compressive Imaging (SCI) becomes a promising research direction. It is the task to indirectly capture high dimensional data and encode it into one single 2D compressed measurement to optimize the memory storage of the system and its transmission bandwidth. Then, an efficient reconstruction algorithm is needed to reconstruct the original video from the compressed measurement. For the last decades, practical video recovery approaches are mainly based on convolutional and recurrent neural networks [START_REF] Iliadis | Deep fully-connected networks for video compressive sensing[END_REF]- [START_REF] Iliadis | Deepbinarymask: Learning a binary mask for video compressive sensing[END_REF]. While these models achieve practical performances, the recovery process in video compressive sensing remains very challenging in terms of flexibility, scalability and speed of the training and the testing phases [START_REF] Yuan | Snapshot compressive imaging: Theory, algorithms, and applications[END_REF]. On the one hand, recurrent neural networks are designed to process data sequentially which makes the implementation of parallel computing very difficult and slows down the training phase. Also, processing long sequences through recurrent networks leads to a loss of information and causes the vanishing gradient problem [START_REF] Hochreiter | The vanishing gradient problem during learning recurrent neural nets and problem solutions[END_REF]. To deal with the vanishing gradient problem, one of the most impactful papers in Deep Learning [START_REF] Vaswani | Attention is all you need[END_REF] has proposed the attention mechanism which manages and quantifies the interdependence between input elements. This attention mechanism has contributed towards the designing and the implementation of transformer models. In fact, these models enable the efficient utilization of GPUs by parallelly processing input sequences and then speed up the training phase considerably. In addition, it is challenging to use transfer learning on recurrent models. However, it is practical to use pretrained transformers to reduce the training cost. On the other hand, convolutional neural networks (CNN) are simple to parallelize. Also, for various applications, CNN based models are fast to train but for short input sequences. For long sequences, convolutional models are unable to learn different dependencies among all the possible combinations of the input elements. That's why, it is practical to process long sequences as a whole using transformers. Transformers are thus better than recurrent neural networks and convolutional neural networks for the following reasons:

• Computational complexity per layer: Self-attention layers O(n 2 .d) are faster than recurrent layers O(n.d 2 ) and convolutional layers O(k.n.d 2 ) when the dimensionality d is bigger than the input sequence length n (which is the case in NLP models) [START_REF] Vaswani | Attention is all you need[END_REF]. • The computation can be parallelized: Recurrent networks need O(n) sequential operations. However, self-attention layers can be computed in a parallel manner. • The path length between long-range dependencies: it is more important with recurrent and convolutional layers than with self-attention layers [START_REF] Vaswani | Attention is all you need[END_REF] Bearing the above problems in mind, in this work, we intend to enhance the reconstruction performances by proposing an endto-end transformers model for SCI video reconstruction trying to solve the trilemma of flexibility, scalability and speed. However, applying efficient transformers for various computer vision applications such as SCI reconstruction is still facing several challenges. In fact, famous vision transformers (e.g. ViT [START_REF] Dosovitskiy | An image is worth 16x16 words: Transformers for image recognition at scale[END_REF]) divide input 2D images into several patches which may threaten the local spatial information [START_REF] Li | Localvit: Bringing locality to vision transformers[END_REF] because some low level visual features (e.g. edges, shapes) are divided into different patches. After the patch embedding step, global fully connected self-attention is applied to extract the global interactions between different tokens which ignores local details. Then, for video recovery problems, temporal data may be the key for better performances since missing information in one frame can be reconstructed from adjacent frames. As a result, the idea is to come up with a new transformer based architecture for video snapshot compressive imaging (SCI) with an attention layer that exploit local and spatiotemporal data information.

In a nutshell, our contributions are summarized as follows:

• To the best of our knowledge, the proposed algorithm (ViT-SCI) is the first video SCI reconstruction method built upon Transformers. • We used a convolutional attention mechanism in order to exploit spatiotemporal information instead of global fully connected attention layers used in recent vision transformers.

• We provide detailed explanation of our architecture with detailed results and analysis which may be used as reference in future research works, especially on video transformers.

II. BACKGROUND AND RELATED WORKS

A. Video Snapshot Compressive Imaging

Compressing high-speed videos is already possible due to the huge research work done in video snapshot compressive imaging (SCI). The video SCI system is composed of two main networks: the hardware encoder and the software reconstruction (decoder) network [START_REF] Yuan | Snapshot compressive imaging: Theory, algorithms, and applications[END_REF]. The hardware encoder represents the optical imaging framework and the software decoder denotes the reconstruction algorithm. The hardware encoder aims to compress the 3D video signal into a 2D measurement matrix and the compression is done across the temporal dimension. This compression aims to avoid huge memory storage and transmission bandwidth. The optical system is called the coded aperture compressive temporal imaging (CACTI) [START_REF] Llull | Coded aperture compressive temporal imaging[END_REF] system (Figure 1). In this system, and during one exposure time, the video scene is gathered by an objective lens and then coded by a temporal-variant mask (shifting physical mask [START_REF] Llull | Coded aperture compressive temporal imaging[END_REF], [START_REF] Koller | High spatio-temporal resolution video with compressed sensing[END_REF] or different patterns on a Digital Micromirror Device (DMD) [START_REF] Reddy | P2C2: Programmable pixel compressive camera for high speed imaging[END_REF], [START_REF] Sun | Compressive high-speed stereo imaging[END_REF]). Then, the output is detected by a Charge Coupled Device (CCD) and then integrated into one single measurement frame. From a mathematical perspective, a video SCI system captures a dynamic scene of B frames X ∈ R h×w×B (h and w are the height and the weight of the frame, respectively) which is modulated by a number of masks (B) noted C k ∈ R h×w , k = 1 . . . B, before being integrated into one single measurement frame Y ∈ R h×w by a camera sensor in one exposure time (B frame). This operation is expressed as follows:

Y = B k=1 X k • C k + G, (1)
where X k and denotes the k th frame, • and G ∈ R h×w denote the Hadamard product and noise, respectively. Then, we define y = V ec(Y ) ∈ R hw and g = V ec(G) ∈ R hw , where V ec represents the vectorization operator. Correspondingly, we define x ∈ R hwB as:

x = [V ec(X 1 ) T , ..., V ec(X B ) T ] T . (2)
The measurement y can then be expressed as:

y = [D 1 , ..., D B ]x + g, (3)
where

D b = diag(V ec(C b )) ∈ R hw×hw , for b = 1 . . . B
denotes a diagonal matrix. We have in this case a matrix [D 1 , ..., D B ] that is highly structured and sparse. Depending on the theoretical study in [START_REF] Jalali | Snapshot compressed sensing: Performance bounds and algorithms[END_REF], the original video can be reconstructed from a single compressed measurement frame y and the coding patterns {C k } B k=1 [START_REF] Jalali | Snapshot compressed sensing: Performance bounds and algorithms[END_REF] with a sampling rate of 1 B . The second important part of video SCI is the reconstruction process which aims to recover the original video from the 2D measurement frame and the masks. This process is crucial to have a practical and efficient video SCI system. In the literature, the reconstruction algorithms could be classified into two categories: optimization based methods and Deep Learning based algorithms. The optimization based algorithms, such as GAP-TV [START_REF] Yuan | Generalized alternating projection based total variation minimization for compressive sensing[END_REF], GMM [START_REF] Yang | Video compressive sensing using Gaussian mixture models[END_REF], DeSCI [START_REF] Liu | Rank minimization for snapshot compressive imaging[END_REF], and PnP-FFDNet [START_REF] Yuan | Plug-and-play algorithms for large-scale snapshot compressive imaging[END_REF], require huge computational resources and large reconstruction time. For instance, DeSCI takes hours to generate a 256 × 256 × 8 video from one single measurement frame). In Deep Learning based methods [START_REF] Iliadis | Deep fully-connected networks for video compressive sensing[END_REF]- [START_REF] Iliadis | Deepbinarymask: Learning a binary mask for video compressive sensing[END_REF], this computational problem has been ameliorated. However, some architectures need a large memory and huge time for the training phase. BIRNAT [START_REF] Cheng | BIRNAT: Bidirectional recurrent neural networks with adversarial training for video snapshot compressive imaging[END_REF], for example, can take weeks to train a model of size 256 × 256 × 8 [START_REF] Cheng | Memory-efficient network for largescale video compressive sensing[END_REF]. Obviously, both categories have their advantages and drawbacks, which make this research direction challenging and very promising for the future if we aim to come up with a memory friendly model that consumes less computational cost for our daily life applications.

B. From NLP to computer vision

Since there is various high-level analogies between video processing and NLP, we decided to take advantage of this architecture for our video reconstruction purpose. In fact, video and sentences have sequential features. In addition, if a word can be understood from the context in a sentence, patches could be reconstructed based on the contextual features gathered from the rest of the video or to be precise from the tokens having similar features based on the computations of the attention layer.

C. Transformers in computer vision

Transformers are originally proposed in 2017 [START_REF] Vaswani | Attention is all you need[END_REF] as a simple and scalable architecture in language translation and successfully dominate natural language processing (NLP) tasks [START_REF] Devlin | Bert: Pre-training of deep bidirectional transformers for language understanding[END_REF], [START_REF] Radford | Improving language understanding by generative pre-training[END_REF]. Indeed, transformers are based on self attention mechanism which is a highly efficient technique to learn the correlations between input features and update the embeddings in parallel. Thus, in contrast to recurrent architectures, transformers based models allow modelling long dependencies between input data components and handle parallel processing. Thus, they are known by their scalability to very highcomplexity models. Recently, transformers start to improve computer vision tasks. They have been used in various computer vision applications such as classification [START_REF] Dosovitskiy | An image is worth 16x16 words: Transformers for image recognition at scale[END_REF], [START_REF] Wang | Non-local neural networks[END_REF], video segmentation [START_REF] Carion | End-to-end object detection with transformers[END_REF], object detection [START_REF] Wang | End-to-end video instance segmentation with transformers[END_REF] and video inpainting [START_REF] Liu | Fuseformer[END_REF].

D. Challenges in computer vision applications

Although transformers are becoming a research trend in the last two years because of their excellent performances, they are facing some crucial challenges in the computer vision field including the fact that they require a large amount of data to be trained and the associated high computational costs (in terms of computational time and memory resources needed for processing) [START_REF] Liu | Video swin transformer[END_REF].

III. OVERVIEW OF THE PROPOSED ARCHITECTURE:

VIT-SCI As illustrated in Figure 2, ViT-SCI consists of three main modules: low frequency feature extraction module, deep feature extraction module and a video reconstruction module. These 3 modules are preceded by a measurement normalization phase aiming to generate a preprocessed video. The entire training process is shown in Algorithm 1.

A. Preprocessed Video and Measurement Energy Normalization

The output of the compressive sensing acquisition phase is the measurement matrix Y ∈ R h×w . Having the measurement matrix Y and the coding patterns (masks) C ∈ R h×w×B , we preprocessed the training data before feeding our deep learning algorithm. One of the preprocessing techniques that have been applied recently [START_REF] Cheng | BIRNAT: Bidirectional recurrent neural networks with adversarial training for video snapshot compressive imaging[END_REF], [START_REF] Cheng | Memory-efficient network for largescale video compressive sensing[END_REF], [START_REF] Wang | Metasci: Scalable and adaptive reconstruction for video compressive sensing[END_REF] is measurement energy normalization. In fact, the measurement matrix Y is not usually energy normalized which requires a normalization process to fit into the neural network. Technically, the energynormalized measurement matrix Ȳ can be expressed as:

Ȳ = Y B k=1 C k , (4)
where represents the matrix dot division. Figure 3, which describes the preprocessing approach (the illustrated frames are extracted from the training dataset), shows that the energynormalized measurement matrix Ȳ presents more visual information than the initial measurement matrix Y . Obviously, Ȳ can be defined as the estimated average of the original B high-speed frames X ∈ R h×w×B . Then, in order to generate a preprocessed video from the energy-normalization measurement matrix Ȳ and the coding patterns C ∈ R h×w×B , we process the following concatenation along the 3 rd dimension:

I = [ Ȳ • C 1 , ..., Ȳ • C B ] 3 ∈ R h×w×B . (5)
The preprocessed video I, preserving the background and some main objects of the frames as illustrated in Figure 3, will feed the reconstruction network.

B. Low Frequency Feature Extraction Module

Given the preprocessed video I ∈ R B×c×h×w (c denotes the number of channels), we use 5 residual blocks followed by LeakyReLU activation function in order to learn low frequency features F low ∈ R B×c×h×w as:

F low = N f e (I), (6)
where N f e denotes the network designed to extract low frequencies features from the input video I.

C. Positional encoding

In contrast to standard neural networks, Transformer based models are permutation-invariant. However, ViT-SCI necessitates accurate position information. As a result, we add a fixed 3D positional encodings, including spatial and temporal information, to the features generated by the low frequency features extraction module of the input I. The 3D positional encoding (PE) [START_REF] Wang | End-to-end video instance segmentation with transformers[END_REF] is defined as:

P E 3D (pos, i) = sin(β k .pos) for i = 2k cos(β k .pos) for i = 2k + 1, (7)
where β =

D. Deep Feature Extraction Module

We have specifically developed a new transformer encoder for video SCI recovery that achieves deep features extraction. The idea behind the deep feature extraction module is to build a network aiming to learn non-linear mapping to enable video reconstruction. This transformer encoder maps the input video space to a higher dimensional feature space. The deep features F deep ∈ R B×c×h×w , extracted by the encoder, can be expressed as:

F deep = N transf ormer (F low ). ( 8
)
where N transf ormer denotes the application the deep feature extraction module. 1) Spatio-Temporal Convolutional Multi-Head Attention (ST-ConvMHA) It has been proved in previous research works [START_REF] Cao | Video superresolution transformer[END_REF], [START_REF] Wu | Cvt: Introducing convolutions to vision transformers[END_REF] that fully-connected self-attention originally developed in [START_REF] Vaswani | Attention is all you need[END_REF] is not suitable for computer vision tasks and especially for video reconstruction models. In fact, fully-connected self-attention is used to extract global interactions between different tokens which neglects local information. Also, it ignores the temporal dimension which is a crucial information in video processing related tasks. In addition, in [START_REF] Cao | Video superresolution transformer[END_REF], it has been theoretically proved that fully-connected self-attention layers used for vision tasks may cause the vanishing gradient problem destabilizing the training process. Bearing in mind the aforementioned limitation of the fullyconnected self-attention layer, deep feature module, which enables to map the features to a series of continuous models, is mainly based on the Spatio-Temporal Convolutional Multi-Head Attention (ST-ConvMHA) layer designed to extract spatial-temporal information and the similarities between different tokens. Our proposed ST-ConvMHA is a stack of parallel convolu- tional multi-head attention layers that allow a better understanding of the different aspects of the input feature maps F low ∈ R B×c×h×w . ST-ConvMHA is based on convolutional projections applied for Query(Q), Key(K) and Value(V) embeddings, respectively and a patch-wise non-local attention model using unfold and fold operations inspired from [START_REF] Mou | COLA-Net: Collaborative attention network for image restoration[END_REF]. The first step in calculating ST-ConvMHA is replacing the existing position-wise linear projections in the fully-connected self-attention mechanism [START_REF] Vaswani | Attention is all you need[END_REF] with convolutional projections using three different convolutional layers with trainable elements W Q , W K and W V . This embedding step, aiming to learn the spatial features of the different frames, can be expressed as follows:

Q = EmbQ(F low ) = W Q ⊛ F low K = EmbK(F low ) = W K ⊛ F low V = EmbV (F low ) = W V ⊛ F low , (9)
where ⊛ denotes the convolution operation and Emb is the embedding step. The second step in the calculation process is using the unfold operation to extract sliding local tokens from Q, K and V tensors. The kernel size used in this operation is H patch × W patch and the stride is s = H patch or s = W patch . As illustrated in Figure 4, the output of the unfolding operation is three groups of 3D tokens. Each group contains N 3D tokens (N = BW H W patch ×H patch ). Each token has the size of dim patch = c × W patch × H patch . This process is expressed as follows:

Q 1 , Q 2 , ..., Q N = Θ(Q) = Θ(W Q ⊛ F low ) K 1 , K 2 , ..., K N = Θ(K) = Θ(W K ⊛ F low ) V 1 , V 2 , ..., V N = Θ(V ) = Θ(W V ⊛ F low ), ( 10
)
where Θ is the unfolding operation and Q, K, V ∈ R B×c×W patch ×H patch The third step is to reshape the Query and the Key tensors into 1D vector of size dim patch × N . The reshaping operator is subsequently denoted by ∆. Then, we calculate the score (the similarity matrix) by calculating the dot product of the matrix of the reshaped query and the matrix of the reshaped key. This score (similarity matrix) is related to all embedding patches of the video which guarantee the learning of the spatial-temporal details. Then, the obtained scores are divided by the square root of each patch in the current head layer since we are implementing a muti-head attention layer, motivated by [START_REF] Vaswani | Attention is all you need[END_REF]. Then, we pass the result through a softmax operation. In fact, the softmax layer will determine the importance of patches corresponding to other patches. The fourth step is to multiply each value vector by the output of the softmax layer. The third and fourth steps are expressed as follows:

Attention(Q i , K i , V i ) = sof tmax( Q i Ki T dim patch n heads )V i (11)
Finally, we apply the folding operation Γ in order to combine the sliding local blocks of size N × c × W patch × H patch into one large containing tensor (feature map) of size B × c × W × H. Then, we apply a convolutional layer W f to generate the final feature map.

The ith attention head process can be expressed:

head i = W i f ⊛ Γ(sof tmax( QK T dim patch n heads )V ),
i.e, when expanding the expressions of Q i , K i and V i :

head i = W i f ⊛Γ(sof tmax( Θ(∆(W i Q ⊛ F low )) × Θ(∆(W i K ⊛ F low )) T dim patch n heads ) Θ(W i V ⊛ F low )) (12)
And the overall process of the St-ConvMHA layer is summarized as follows:

ST -ConvM HA = Concat(head 1 , ...head h )W O , ( 13
)
where h is the number of heads or the number of parallel convolutional attention layers. The implemented ST-ConvMHA enables to deeply learn spatial-temporal features in comparison with the fullyconnected self-attention mechanism.

2) Feed-Forward Network

As shown in Figure 4, the ST-ConvMHA layer is followed by a Feed Forward Network (FFN) [START_REF] Vaswani | Attention is all you need[END_REF]. It is applied to every attention tensor to transform them into a form that can feed the next transformer encoder layer. In fact, the parallelization process is enabled by the FFN, because it processes all the attention tensors at one time.

E. Video Reconstruction Module

In the reconstruction module, we recover the video frames from processing the deep features generated by the transformer encoder as:

O rec = N rec (F deep + F low ), (14)
where N rec is the reconstruction network. O rec depends on F deep and F low to stabilize the training phase. The final output of out approach is the aggregation of the output of the reconstruction module O rec while the input preprocessed video I:

O f = O rec + I. (15)

F. Training Process and Loss Function

In our implementation (Algorithm 1), we optimize the parameters of ViT-SCI by minimizing the reconstruction error: the loss function used is the mean square error (MSE):

L = N n=1 B k=1 ∥O fn,k -G tn,k ∥ 2 2 , (16)
where O fn,k is the final output or the k th reconstructed frame of the n th training video using ViT-SCI, and G tn,k is the corresponding ground truth frame. for i = 1 : n epochs do 4:

for All training video sequences do

5: Load Y , C k , G t 6: Ȳ ← Y B k=1 C k 7: I ← [ Ȳ • C 1 , ..., Ȳ • C B ] 3 8:
F low ← N f e (I)

9:
F lowP E ← F low + P E 10:

for d = 1 : N umberOf AttentionLayers do 11:

for h = 1 : n heads do 12:

Att h ← Attention(F lowP E )

13:

end for 14:

F deep ← F F N (Att i ) 15:
end for 16:

O rec ← N rec (F deep + F lowP E ) 17: O rec ← O rec + I 18:
Obtain loss: end for 22: end while

L = N n=1 B k=1 ∥O fn,k - G tn,k ∥ 2

IV. PERFORMANCE EVALUATION, COMPARISON AND

DISCUSSION

In this section, we describe the implementation framework and compare the performances of the proposed reconstruction method with several state-of-the-art methods.

A. Datasets

To train our algorithm, we use DAVIS2017 [START_REF] Pont-Tuset | The 2017 davis challenge on video object segmentation[END_REF] dataset, designed for video object segmentation applications, since video SCI algorithms can be applied on any video scene and there is no specific dataset for the training phase. The original DAVIS2017 dataset has only 90 video scenes (6242 frames of size 854 × 480). For an efficient training in a video SCI context, we prepare the dataset by transforming and reformatting it. In fact, we generate 6516 video scenes of size 8 × 256 × 256. Then, we test our trained model on six evaluation datasets: Aerial, Drop, Kobe, Runner, Traffic, and Vehicle.

B. Data Augmentation

In order to deal with the problem of overfitting, data augmentation is a commonly used pre-processing technique aiming to generate more data than RNN and CNN based models, becoming greater in terms of size. Since transformerbased models in general require more data, augmenting the diversity of the training dataset will enhance the performances of our proposed ViT-SCI [START_REF] Touvron | Training data-efficient image transformers distillation through attention[END_REF]. After the data augmentation process consisting in cropping, rotating, and flipping input videos, the dataset becomes larger with 417024 video scenes (3 336 192 frames). Data augmentation has significantly enhanced the performances of our model.

C. Compared methods and performance metrics 1) Compared methods

Several state-of-the-art methods are used to evaluate the performances of our proposed approach for the video SCI reconstruction, including two iteration-based reconstruction algorithms:

• GAP-TV [START_REF] Yuan | Generalized alternating projection based total variation minimization for compressive sensing[END_REF] • DeSCI [START_REF] Liu | Rank minimization for snapshot compressive imaging[END_REF] and a recent deep learning based reconstruction algorithms:

• BIRNAT [4]
2) Performance metrics To quantify the performances of the evaluated algorithms, we use well known frames quality evaluation metrics: the Peak Signal-to-Noise Ratio (PSNR) and Structural Similarity Index(SSIM) [START_REF] Hore | Image quality metrics: PSNR vs. SSIM[END_REF].

D. Implementation details

The ViT-SCI algorithm has been implemented using Pytorch framework [START_REF] Paszke | Pytorch: An imperative style, high-performance deep learning library[END_REF]. We use the Mean Square Error (MSE) as a loss function in the main implementation. To minimize the MSE function, we use Adam optimizer [START_REF] Kingma | Adam: A method for stochastic optimization[END_REF] with an initial learning rate of 0.0003 (the learning rate is reduced by 5% every 5 epochs). The performance evaluation of the different approaches is done on an NVIDIA RTX 2080 GPU (8GB GDDR6). Our method is trained for 100 epochs and it took about 190 hours to train the entire ViT-SCI network.

E. Network architecture

In the ST-ConvMHA, we used three convolutional layers to learn the spatial information of each frame. The output of the ST-ConvMHA layer passes through a convolutional layer to generate the final feature map. To decrease the computational cost of our model, we use gray scale frames for the training process (c = 1). The low frequency feature extraction module has 5 residual blocks. The deep feature extraction module uses 4 transformer encoder layers. The final video reconstruction module has 30 residual blocks.

F. Ablation study

In this section, we study the core implementation of ViT-SCI through a profound experimental study to demonstrate the effectiveness of our model design choices.

1) Frame size

Table I reports that ViT-SCI has larger computational cost (Training Time) when having higher spatial resolution. Indeed, about 155 more hours is required to train our model on DAVIS dataset with a spatial resolution of 80 × 80 than on the same dataset with a spatial resolution of 64 × 64. This large computational cost can threaten the scalability of the model while maintaining efficiency. However, we notice from Figure 5 and 6 that training ViT-SCI on smaller images with smaller spatial dimensions does not affect quality performances as much. Therefore, we believe that our model can be extended to process larger training datasets.

2) Positional Embeddings

In Transformer based architectures, positional embeddings is of huge importance since all tokens are taken parallelly. The 3D positional encoding used in this implementation indicates the spatial and temporal positional embeddings which refers to the position in the video scene. To investigate the importance of our 3D positional embedding module, we conduct the following experiments (with 2 attention heads):

• No 3D positional embedding • 3D (Spatiotemporal) positional embedding The experiments, illustrated in Figure 7, shows that the model trained with 3D positional embeddings achieves better performances (+4.65% in terms of PSNR by passing from 30.1939 to 31.5969 and +1, 47% in terms of SSIM by passing from 0.9084 to 0.9218). This result proves that the positional information of every token is implicitly provided in the Transformer based architectures. However, it is important to enhance this positional information with explicitly implemented positional embeddings.

3

) Number of heads

We have carried out a series of experiments by training our model with one single attention head and with some independent attention layers applied in parallel to answer the famous question that has already been asked in [START_REF] Michel | Are sixteen heads really better than one?[END_REF]: "Are Sixteen Heads Really Better than One?" or "is more than one head even needed?". From the figures of Table II, it is clear that the training time slightly increases with the number of heads, while the performances do not follow a monotonic behavior and even don't show significant differences.

We can further notice from Figure 8 and 9 that the model is not sensitive to the number of attention heads. Therefore, one single attention head is sufficient, thus reducing the training computational cost. This may be explained by the fact that we have trained our approach on small sized video clips where the number of dynamic objects is limited and multiple heads are not needed to detect and learn syntactic relations between different objects. However, we believe that with higher temporal and spatial resolution datasets, pruning the attention heads in our model will result in significant performance degradation. Therefore, as suggested in [START_REF] Voita | Analyzing multi-head self-attention: Specialized heads do the heavy lifting, the rest can be pruned[END_REF], it is advisible to retain more than one attention head and enhance formula 13 to:

ST -ConvM HA = i Concat i (λ i head i )W O , (17)
where λ i is a learnable parameter offering the capability to the neural network to learn more effective interactions between attention heads.

4) Number of extraction blocks

The extraction network is important to extract the main features of input frames. Thus, we trained ViT-SCI with different numbers of extraction blocks to evaluate their impact on performances. As reported in Table III, the computational cost increases linearly with the number of extraction blocks. Indeed, about 11 more hours is needed when increasing the extraction blocks from 1 to 5. In addition, the average of the quality figures increases also when adding more extraction blocks (Figure 10). However, Figure 11 shows that the optimal number of extraction blocks can depend on the testing dataset. On the one hand, Aerial, Kobe and Traffic perform better with 5 extraction blocks. While on the other hand, Drop, Runner and Vehicle have better reconstruction quality with only 3 extraction blocks.

5) Number of reconstruction blocks

The reconstruction module can also significantly impact the quality performances of ViT-SCI. Therefore, finding an optimal trade-off between the computational cost and the number of reconstruction blocks can be very challenging. So, we trained our model on 3 different numbers of reconstruction blocks and we compared our model's quality performances. It is obvious that the training time increases when increasing the number of reconstruction blocks since about 1.000.000 more model parameters must be learned when adding 10 reconstruction blocks. The average PSNR and SSIM results on 6 test sets, presented in Figure 12, show ViT-SCI performs well with 30 reconstruction blocks. However, when we study each dataset separately in Figure 13, we notice that Aerial, Kobe and Traffic need 30 reconstruction blocks for better PSNR and SSIM performances while Drop, Runner and Vehicle are well reconstructed with only 20 reconstruction blocks.

6) Number of ST-ConvMHA Attention layers or depths

To explore the impact of the number of ST-ConvMHA attention layers on the performances of ViT-SCI, we trained our model with different attention layers or depths. Each layer has 8 attention heads. Table V shows that the number of learnable parameters increases linearly when increasing the number of the encoder layers and the training becomes computationally heavier. This ablation study aims to find the smallest number of attention layers that gives better quality performances to ensure the trade-off between the output quality and the computational cost. Figure 14 shows that 2 attention layers performs well, outperforming the same model configuration but with 1 and 4 attention layers on the average reconstruction quality on the 6 testing datasets. However, we notice from Figure 15 that these better quality performances are valid on Drop, Runner and Vehicle. For Aerial, Kobe and Traffic, 4 attention layers are needed for better reconstruction quality. These results prove that for some datasets deeper is better but it is not always the case for every dataset. We notice also, from Table V and Figure 14 and 15, that the difference in performance is very small because we train our model on very short videos of 8 frames so we believe that with larger datasets the difference can be more noticeable.

G. Main simulation results

After confirming our design choices by means of the ablation study, we perform experiments to compare our proposed ViT-SCI algorithm with the state-of-the-art approaches on video compressive sensing. The quantitative results are summarized in Table VI, from which we compared the reconstruction quality of different reconstruction models and their recovery speed. For PSNR and recovery speed measured, our ViT-SCI achieves the best results among the video reconstruction methods with good SSIM results. On Arial dataset, ViT-SCI slightly outperforms BIRNAT in terms of the reconstruction quality (+1.49% and +0.5% for PSNR and SSIM, respectively) and largely outperforms GAP-TV (+32.23% and +10.19% for PSNR and SSIM, respectively) and DeSCI (+28.33% and +8.37% for PSNR and SSIM, respectively) on the same metrics. On Kobe dataset, a limited improvement is noticed over DeSCI (+0.55%) in terms of PSNR. DeSCI performs better in terms of SSIM on Kobe dataset. The quantitative results prove the efficiency of our proposed approach, based on an attention mechanism, on complex background datasets. Table VI, also shows that DeSCI has better PSNR and SSIM performances on Drop, Runner and Traffic datasets over our Transformer based approach. These results can be justified because our training dataset rarely includes high speed motions. So, our model is not well trained (BIRNAT also) to reconstruct video scenes with very high speed motions. In figure 16, we show the qualitative results of our ViT-SCI compared with the-state-of-the-art. Our ViT-SCI could synthesize finer details and clearer edges on the six evaluation datasets which confirm the quantitative results and illustrate the effectiveness of the ST-ConvMHA module on the reconstruction process. Further, considering a real-time application, the most interesting performances of the experimental results remains those of the recovery time. Our algorithm is able to reconstruct a video scene of size 8×64×64 in about one centisecond which is faster than BIRNAT by 12 times and much faster than the leader of the optmization based methods DeSCI by 730×10 +3 times. Specifically, ViT-SCI can achieve good results in only 8ms. So, it is able to perform real-time reconstruction of up to 125 measurements per second. Both the quantitative and qualitative results prove the ViT-SCI can be used as a reconstruction model in a video compressive sensing framework in real-time applications because of the good quality performances and especially the excellent recovery time.

H. Discussion

As supported by our ablation study, we want to highlight that optimizing the hyperparameters of our proposed architecture is non trivial and strongly depends on the dynamics and information content of the input videos. Owing to limitations in computational ressources, we could not achieve a fully satifying optimization of the hyperparameters. Furthermore the videos size had to be restricted to afford the training process. Theses limitations prevented our algorithm from reaching its best potential. Future researches should tackle these limitations by devising a memory optimized architecture.

V. CONCLUSION

Designing efficient video compressive sensing reconstruction algorithms has been very challenging in inverse problems. Inspired by recent advances in Deep learning and motivated by the huge success of Transformer-based architectures in NLP, we propose the first video SCI reconstruction algorithm built upon Transformers. in this model, the recovery approach is viewed as an end-to-end decoding task. The proposed approach, trained on DAVIS dataset, achieves state-of-the-art quality performance on 6 different simulation datasets. Also, it is much faster than all existing approaches since it is able to perform real-time acquisition and reconstruction of up to 125 measurements per second. A complete ablation study is provided to justify the choice of some hyperparameters. We strongly believe that our algorithm can pave the way for more research work on video compressive sensing based on recent advances in Deep Learning. Also, we assume that ViT-SCI is now ready to be widely exploited in energy-efficient real-time applications.

  • ViT-SCI achieves strong results on DAVIS2017 training dataset in comparison with other video SCI reconstruction algorithms based on Deep Learning architectures and optimization methods. The remaining paper is organized as follows: Section II discusses related works in video compresive sensing and Transformer based architectures. In Section III, we present the main architecture behind ViT-SCI. In Section IV, we evaluate the performance of our proposed algorithm in a video SCI context with an extensive ablation study on different hyperparameters. Finally, Section V provides conclusion and the main perspectives of our research work.

Fig. 1 .

 1 Fig. 1. Schematic of the CACTI system

,

  pos is the position the corresponding dimension, d c represents the size of the channel dimension and k ∈ N s.t. k ∈ [0, d 6 ].

Fig. 2 .

 2 Fig. 2. The architecture of the proposed ViT-SCI for video reconstruction in Video Snapshot Compressive Imaging

Fig. 4 .

 4 Fig. 4. The Deep Feature Extraction Module

Fig. 5 .

 5 Fig. 5. Ablation study on the effect of the frame size in training video clips: the average quality performances (Left: in terms of PSNR; Right: in terms of SSIM) on 6 test datasets

Fig. 8 .

 8 Fig. 8. Ablation study on the effect of the number of attention heads: the average quality performance(Left: in terms of PSNR; Right: in terms of SSIM) on 6 test datasets

Fig. 16 .

 16 Fig. 16. Reconstructed frames of GAP-TV, DeSCI, E2E-CNN, BIRNAT and ViT-SCI on six simulated video SCI datasets

TABLE VI AVERAGE

 VI PSNR(DB), SSIM AND RUN TIME(IN SEC) PER MEASUREMENT FOR DIFFERENT APPROACHES ON 6 EVALUATION DATASETS. BEST RESULTS ARE IN BOLD, SECOND BEST RESULTS ARE IN GRAY

	Algorithms	Aerial	Drop	Kobe	Runner	Traffic	Vehicle Average
		22.09	27.73	25.74	31.29	24.17	24.72	25.95
	GAP-TV [20] 0.8719	0.9141	0.7909	0.9177	0.7515	0.8700	0.8526
		8.0	8.0	8.1	8.1	8.3	8.2	8.12
		22.76	36.51	31.08	38.48	31.59	26.05	31.07
	DeSCI [22]	0.8866	0.9840	0.9278	0.9609	0.9138	0.9140	0.9311
		6168.2	6336.9	6396.5	6331.5	6215.3	6258.8	6284.5
		28.74	32.77	28.96	35.41	26.49	28.23	30.10
	BIRNAT [4]	0.9560	0.9626	0.8594	0.9337	0.8199	0.9019	0.9056
		0.1050	0.1097	0.1056	0.1057	0.1087	0.1132	0.1079
		29.21	35.40	31.25	37.67	28.15	25.40	31.18
	Ours	0.9608	0.9759	0.9047	0.9509	0.8696	0.8161	0.9130
		0.0092	0.0090	0.0089	0.0079	0.0089	0.0080	0.0086


Khaled Belahcene

Jérome Gaigne

Sylvain Lagrue

Fragment of the necessary decisions explained by an argument scheme in a non compensatory sorting model

We focus on an argument scheme explaining a necessary decision with respect to a jurisprudence considering the non compensatory sorting model. In this model, a set of candidates is sorted in two ordered categories by a set of points of view expressing an individual preference over the set of candidates. Candidates are ranked in the upper category if they are accepted by a sufficient subset of points of view. We show that the argument scheme is an approximation of this model and share some counterintuitive results concerning the relative performance of some well-known solvers.

Introduction

The accountable approval sorting problem has been introduced in [START_REF] Belahcene | Accountable approval sorting[END_REF]. It corresponds to the situation where a jury, composed of multiple members, has to assign a set of candidates into two ordered categories. The authors used GOOD and BAD, it could also be ACCEPTED and REFUSED if the context is a review of job applicants. Each member of the jury has an opinion over all the candidates as a preference relation and approves a subset of them. Those approval sets are aggregated with respect to sufficient coalitions in order to assign each candidate to the right category. Therefore a candidate is assigned to the upper category (i.e. ACCEPTED) if this candidate is approved by a sufficient subset of the jury.

The authors named two situations of interest for accountability in an approval sorting setting :

• being able to justify that all the decisions taken by the jury follow the approval sorting procedure; • being able to justify a candidate's category.

Here, we focus on the second case where a candidate wishes a justification to his/her ranking. Again there are two situations of interest :

• the candidate could have been ranked in the other category;

• the candidate cannot be ranked in the other category.

In order to be accountable in this last situation, it has to be shown that the decision taken for this candidate was necessary, namely if the candidate is ranked in the other category, the decision would not respect the approval sorting procedure. The authors proposed an argument scheme, whose principle is presented by the authors of [START_REF] Walton | Argumentation schemes[END_REF], to get a justification of the necessity of the decision which is humanly readable. This scheme has been proven to be a sufficient condition to determine the necessity of a decision, but, to our knowledge it has not been proven to be a necessary condition. Furthermore, to diminish the amount of private information about the juries' decision, a jurisprudence can be used as a reference case to build the argument scheme.

In Section 2, we introduce or recall important definitions about the non compensatory sorting model underlying the approval sorting, and about its inverse problem. Then, Section 3 introduces the argument scheme justifying the necessity of a decision that has been proposed by the authors of [START_REF] Belahcene | Accountable approval sorting[END_REF]. In this section, we study the necessity of the scheme and we prove it only covers a fragment of all the necessary decisions, i.e. the scheme is not necessary when there is a necessary decision. For obtaining these results, we used multiple solvers and models to compute the scheme, so we provide a study of their relative performances in Section 4. Lastly, Section 5 recalls our results and provides new questions.

Non compensatory sorting model and inverse NCS problem

In this section, we introduce the non compensatory sorting model with two categories together with some of its properties. We recall its inverse problem along with an important representation theorem brought by the authors of [START_REF] Belahcene | Accountable approval sorting[END_REF].

Non compensatory sorting model

The approval sorting model is also called non compensatory sorting model, NCS for short.

In order to ease the definition of the non compensatory sorting model, we recall the definition of an upset. Definition 1 (upset). Let A be a set and R be a binary relation over A. The subset

B ⊆ A is an upset for (A, R) iff ∀a ∈ A, ∀b ∈ B, if bRa, then a ∈ B.
Definition 2 (Non compensatory sorting model, [START_REF] Bouyssou | An axiomatic approach to noncompensatory sorting methods in MCDM, I: the case of two categories[END_REF]). Let X be a set of alternatives and N be a set of points of view. Let ACCEPTED and REFUSED be two ordered categories. Each point of view i ∈ N has a preference over the whole set of alternatives as a complete preorder written ≿i. To stay syntactically and semantically coherent, we may use the preorder written ≾i defined as ∀a, b ∈ X, a ≾i b iff b ≿i a. Each point of view i ∈ N approves a subset of alternatives written Ai such as Ai is an upset for (X, ≾i). Lastly, let S be the set of all the sufficient coalitions of points of view such as S is an upset for 2 N , ⊆ . A sorting process respects the NCS model if and only if each ACCEPTED alternative has been approved by a sufficient coalition of points of view, i.e. let α : X → {ACCEPTED, REFUSED} be the function characterizing the decision process, α corresponds to a non compensatory sorting if and only if

∀x ∈ X, α (x) = ACCEPTED, if {i ∈ N : x ∈ Ai} ∈ S REFUSED, otherwise
In short, a sorting process respecting the NCS model takes as inputs a set of alternatives X, a set of points of view N , the preferences of each point of view over the alternatives ≿i, ∀i ∈ N , the approval set of each point of view Ai, ∀i ∈ N and the set of all sufficient coalitions of points of view S. Its output is the sorting of the alternatives into the two categories.

From this definition, we can pull two rationality properties that this process satisfies. Firstly, the individual rationality which states that there is no alternative disapproved by i ∈ N that is preferred to an alternative accepted by i, i.e. Ai is an upset for (X, ≾i). Secondly, the collective rationality states that if a subset of points of view is a sufficient coalition, then any super set of it is also sufficient, i.e. S is an upset for 2 N , ⊆ .

Example 1. Let X = {a, b, c, d, e} be a set of alternatives, N = {1, 2, 3} be a set of points of view. Individual preferences are as follows:

e ≻1 a ≻1 c ≻1 b ≻1 d b ≻2 a ≻2 c ≻2 d ≻2 e c ≻3 b ≻3 a ≻3 d ≻3 e
Here are the approval sets of each jury. They approve their two most preferred ones:

A1 = {a, e} A2 = {a, b} A3 = {b, c}
Lastly, the sufficient coalitions are as follows:

S = {{1, 2} , {2, 3} , {1, 3} , {1, 2, 3}}
This setting results in the assignment: α = {(a, ACCEPTED) , (b, ACCEPTED) , (c, REFUSED) , (d, REFUSED) , (e, REFUSED)}.

Inverse NCS problem

If the process is to be inquired, it is necessary to be able from the output to find input parameters that could have led to obtaining α through a non compensatory sorting process. To do so, we need the output of the sorting process, α, and some other data, the individual preferences of the points of view. With those, we verify that it is possible to find a set of approval sets and a set of sufficient coalitions that allow obtaining α through a non compensatory sorting process. In the inverse NCS problem (also written inv-NCS), we only focus on finding the existence or the nonexistence of those parameters (i.e. it is unnecessary to reveal their actual values).

The representation theorem is presented in [START_REF] Belahcene | Accountable approval sorting[END_REF]. This theorem is important because it allows one to answer inv-NCS without having to deal with the notion of sufficient coalition. By this theorem, the sufficient coalitions do not have to be revealed.

Theorem 1 (NCS pairwise formulation, [START_REF] Belahcene | Accountable approval sorting[END_REF]). A function α mapping the alternatives to the categories can be represented in NCS if and only if there exists ⟨Ai⟩ i∈N ⊆ (2 X ) N such as:

1. ∀i ∈ N , Ai is an upset for (X, ≾i). 2. ∀ ⟨g, b⟩ ∈ α -1 (ACCEPTED) × α -1 (REFUSED), ∃i ∈ N such as g ∈ Ai and b / ∈ Ai.

Theorem 1 allows one to reduce inv-NCS to SAT ([3], Section 3.4). In [START_REF] Tlili | Learning non-compensatory sorting models using efficient sat/maxsat formulations[END_REF], this formulation has been extended in case where one has more than 2 categories, and relaxed in a MaxSAT formulation to be able to take into account noisy data.

Argument scheme's study

The authors of [START_REF] Belahcene | Accountable approval sorting[END_REF] proposed an argument scheme allowing one to build a humanly readable justification of the necessity of a decision about the sorting of an alternative. In this section, we recall this scheme. Then we answer the question about the necessity of this scheme to assess the necessity of a decision. We found out various counter examples showing that the scheme isn't a necessary condition.

Argument scheme justifying a necessary decision with respect to a jurisprudence

First of all, we recall the definition of a necessary decision. Let us say that we have a set of reference cases (a jurisprudence) α ⋆ : X ⋆ → {ACCEPTED, REFUSED} with X ⋆ ⊆ X. To show that a decision about an alternative sorting is necessary, we can use the fact the inv-NCS problem tells us if a decision process is a possible outcome of a NCS process. If we prove that the other decision concerning the alternative would not be possible (i.e. this new instance is then negative of the inv-NCS) it results that the original decision was necessary.

Definition 3 (Necessary decision with respect to a jurisprudence, [START_REF] Belahcene | Accountable approval sorting[END_REF]). Let α ⋆ be a positive instance of the inv-NCS problem, we say that x ∈ X is necessarily assigned to C ∈ {ACCEPTED, REFUSED} with respect to α ⋆ if α ⋆ ∪ x, C is a negative instance of the inv-NCS problem, with C the category opposed to C.

The argument scheme below is a sufficient condition to show the necessity of a decision. An open question is to know if it is also a necessary condition to it. Definition 4 (Necessary decision argument scheme with respect to a jurisprudence, [START_REF] Belahcene | Accountable approval sorting[END_REF]). We say that a set {⟨g1, b1⟩ , ..., ⟨g k , b k ⟩} instantiates the argument scheme rejecting the function α with respect to α ⋆ , with ∀j = 1, ..., α -1 (ACCEPTED) , α (gj) = ACCEPTED and ∀j = 1, ..., α -1 (REFUSED) , α (bj) = REFUSED and ∀x ∈ X ⋆ , α(x) = α ⋆ (x), if : This definition relies on a subset B of points of view of cardinality lesser than the number of pairs of alternatives that we want to split (cf. theorem 1 about the pairwise model). Condition 1 states that none of the points of view outside of B is able to split any pair. Condition 2 says that only one pair at most at a time can be separated by each point of view inside B.

∃B ⊆ N , |B| ≤ min (|N | , k -1) , ∀i / ∈ B,
Proof. (Sufficient condition). The fact that this scheme is a sufficient condition to assess that an instance is negative to the inv-NCS problem is straightforward since k, the number of pairs to split, is strictly greater that the cardinality of B that is at most k -1. Knowing that, it says that at most k -1 pairs can be separated over the k ones thanks to condition 2 and 0 pairs can be separated by condition 1, so one remains non-split. Thus, by theorem 1, proving that the instance is indeed negative for inv-NCS. This scheme allows one to keep the individual approval sets private while justifying the decision in a humanly readable way. Now, let us say that we want a justification of b's sorting. If we switch b to the other category, i.e. α ′ = α\ {(b, REFUSED)} ∪ {(b, ACCEPTED)}, the new function α ′ is a negative instance of inv-NCS. Searching for an argument scheme in this configuration, we can for instance find the set B = {1} with the pairs {⟨a, d⟩ , ⟨b, e⟩}. In other words, the point of view 1 can only split one pair at a time, while the point of view 2 cannot split any of them since for each interesting pair, the REFUSED alternative is preferred to the ACCEPTED one according to ≻2.

This scheme is not unique, we can also take B = {1, 2} with {⟨a, d⟩ , ⟨b, d⟩ , ⟨b, e⟩} for instance.

Argument scheme as a necessary condition for necessary decision

To verify that the scheme is a necessary condition for identifying inv-NCS negative instances, we decided to try to compute a counterexample. To do this, we have to look for an instance satisfying the two following conditions:

1. the instance is negative for the inv-NCS problem; and 2. there's no scheme to be found in it.

We check the first condition using the SAT formulation of the inv-NCS problem proposed in [START_REF] Belahcene | Accountable approval sorting[END_REF]. We then check the second condition using a novel SAT formulation Φ scheme described in the next section.

Argument scheme's SAT formulation

To ease the search, we restrict the search space to preference profiles in the form of strict total orders. In this case, the argument scheme can be written as follows:

Definition 5 (Necessary decision argument scheme with respect to a jurisprudence composed of linear orders). We say that the set {⟨g1, b1⟩ , ..., ⟨g k , b k ⟩} instantiates the argument scheme to reject the function α with respect to α ⋆ , with ∀j = 1, ..., α -1 (ACCEPTED) , α (gj) = ACCEPTED and ∀j = 1, ..., α -1 (REFUSED) , α (bj) = REFUSED and ∀x ∈ X ⋆ , α(x) = α ⋆ (x), if :

∃B ⊆ N , |B| ≤ min (|N | , k -1) , ∀i / ∈ B, ∀u ∈ {1, ..., k} , bu ≻i gu (3)
and ∀j ∈ B, ∀p, q ∈ {1, ..., k} , if p ̸ = q then ∃ ⟨g, b⟩ ∈ {gp, gq} × {bp, bq} , b ≻j g (4)

In this section, we denote: m := α -1 (ACCEPTED) , n := α -1 (REFUSED) and A := α -1 (ACCEPTED) × α -1 (REFUSED).

We use the following propositional variables:

βi := i ∈ B λ gb := ⟨g, b⟩ ∈ {⟨g1, b1⟩ , ..., ⟨g k , b k ⟩}
We seek to express the following cardinality constraint:

|{i ∈ N : βi is true }| < |{⟨g, b⟩ ∈ A : λ gb is true }| (5)
To write cardinality constraints in SAT, we use Bailleux and Boufkhad's encoding introduced in [START_REF] Bailleux | Efficient cnf encoding of boolean cardinality constraints[END_REF] modified so as to allow us to directly compare the cardinalities of two sets. We keep the totalizer as is (which computes the cardinality of each set) but we modify the comparator (which, in its original version, sets a lower and an upper bound to the set cardinality). The idea is to create a comparator between the two cardinalities. To do so, we add the following rule between the variables of both totalizers (which are a unary representation of each set cardinality):

∃i ∈ {1, ..., n} , S λ i ∧ ¬S β i
This rule simply says that, there is at least one less true variable in the unary representation of the cardinality set {(g, b) ∈ A : λ gb is true } than in the one of the set {i ∈ N : βi is true }. In other words, the cardinality of the set {⟨g, b⟩ ∈ A : λ gb is true } is at least equal to |{i ∈ N : βi is true }|+ 1.

We denote by n λ := α -1 (ACCEPTED) × α -1 (REFUSED) the upper bound of the cardinality of the set {⟨g, b⟩ ∈ A : λ gb is true }, and by n β := |N | the upper bound of the cardinality of the set {i ∈ N : βi is true }.

In order to represent this rule, we can create l := max (n β , n λ ) new variables noted Wi, i ∈ {1, ..., l} which represent the previous rule as it follows :

S λ i ∧ ¬S β i ↔ Wi i=1...l

Wi

The unary representation with the least number of variables has to be completed to get to the same amount as the largest one, i.e. l. All these new variables have to be set to false. In other words,

S ⋆ with ⋆ = argmin x∈{λ,β} (nx) : Ext : i=n⋆+1...l ¬S ⋆ i
Thus the cardinality representation is as follows :

• Totalizers: Totalizer (λ gb ) ∧ Totalizer (βi) • Cardinalities' comparator : Comp 1 : i=1...l ¬S λ i ∨ S β i ∨ Wi Comp 2 : i=1...l S λ i ∨ ¬Wi Comp 3 : i=1...l ¬S β i ∨ ¬Wi Comp 4 : i=1...l Wi
The other constraints of the scheme are almost straightforwardly translated to SAT as follow:

C1 : ∀i∈N , ∀⟨g,b⟩∈A, g≻ i b βi ∨ ¬λ gb C2 :
∀i∈N , ∀p̸ =q=1,...,mn, ∀⟨g,b⟩∈{bp,bq }×{gp,gq},

g≻ i b ¬βi ∨ ¬λ gpbp ∨ ¬λ gq bq
This cardinality comparison encoding is useful so that we can call the solver only once to solve the problem and not once per each cardinality value. The total formula is then:

ϕ scheme : Ext ∧ Totalizers ∧ 4 i=1 Comp i ∧ C1 ∧ C2
Theorem 2. Let a preference profile be composed of linear orders and α a negative instance of inv-NCS. There is an argument scheme explaining why α is rejected iff ϕ scheme is satisfiable.

Counterexample search

Using this SAT representation and the inv-NCS SAT representation, we found out that there are negative inv-NCS instances which do not have argument schemes. To do so, we exhaustively search all instances for various parameters' settings: number of alternatives, number of points of view and number of ACCEPTED (thus REFUSED) alternatives.

To reduce the search space and avoid a rather good number of symmetries, we used the following procedure to generate all instances:

• Set the following parameters : number of alternatives (nalternatives), number of points of view (nPOVs), number of ACCEPTED alternatives (nACCEPTED and thus nREFUSED = nalternatives -nACCEPTED). • Generate a preference profile:

-For the first point of view, compute a permutation of the multiset containing two types of items, g, the ACCEPTED alternatives, and b, the REFUSED alternatives. g has a multiplicity of nACCEPTED and b has a multiplicity of nREFUSED. This point of view sets the name of each alternative, i.e. for the AC-CEPTED alternatives: g1, ..., gn ACCEPTED by adding an index to the g items in the obtained permutation starting from the most preferred to the least preferred and for the REFUSED alternatives: b1, ..., bn REFUSED in the same manner.

-For every other points of view, compute a permutation over the alternatives.

• The function α is easily generated by taking α -1 (ACCEPTED) = {g1, ..., gn ACCEPTED } and α -1 (REFUSED) = {b1, ..., bn REFUSED }. • Compute a hash for each point of view.

• Sort this hash in the increasing order and store this sorted tuple in a table. • If this tuple is already in the table then generate a new profile.

• Otherwise, solve the inv-NCS problem using the SAT formulation from Section 3.4 of [START_REF] Belahcene | Accountable approval sorting[END_REF]. • If it is unsatisfiable, then use ϕ scheme presented in Section 3.2.1 to search for any scheme. • If there is none, then store this instance as a counterexample.

In both inv-NCS and the argument scheme, we only care about the (ACCEPTED, REFUSED) pairs of alternatives and the positions of the alternatives in each preference. Thus we can avoid the symmetries consisting of swapping gi and gj by generating the first point of view's preference using the permutation of the multiset containing nACCEPTED times a g item and nREFUSED times a b item. Using this point of view to set the name of the alternatives allows us to remove this type of symmetry.

Furthermore, to avoid the symmetries coming from the permutations of the points of view, we store a sorted tuple of hash (one hash per point of view computed with the point of view's preference) in a table that is checked every time a new profile is created.

There is at least one other type of symmetry that we ignore. For instance, the one consisting in swapping the categories of all alternatives. Despite this, we were able to exhaustively search the space for several parameters values. The results can be found in Table 1 The symmetrical aspect of Table 2 can be explained by the remaining symmetry between the role of ACCEPTED and REFUSED alternatives.

We found out that there are counterexamples, however, it is important to note that the ratio of counterexamples over the total number of instances or even the number of negative instances is really small knowing there are 6 alternatives and 3 points of view and all preferences are strict total orders.

Here is one of the counterexamples found by this process with 6 alternatives and 3 points of view: This configuration does not satisfy the inv-NCS problem neither does it contain an argument scheme (i.e. there is no way of selecting pairs of alternatives with one ACCEPTED and one RE-FUSED, so that we could create a subset B of points of view respecting the conditions of the argument scheme). However, if we consider α ′ = {(a, ACCEPTED) , (b, ACCEPTED) , (x, REFUSED) , (c, REFUSED) , (d, REFUSED) , (e, REFUSED)} where x is now in the REFUSED category, then it is a positive instance for the inv-NCS problem because:

A1 = {a, b, c, x} A2 = {a} A3 = {b, e} and S = {{2} , {1, 3} , {1, 2} , {2, 3} , {1, 2, 3}}
leads to α ′ . Thus there is a necessary condition not covered by the argument scheme.

Other formulations

The definition of an argument scheme naturally translates into a pseudo boolean representation, because it mixes logical constraints and cardinality constraints. The reader can note that this formulation is not a perfect translation of the scheme's definition since, here, the number of pairs and the number of points of view in B are constants. However, we are able to use this formulation because it does not impact the decision problem result, i.e. "is there an argument scheme?". Indeed, if we consider a scheme with k + n pairs to be split, with n > 0, and only k -1 points of view able to split at most one pair at a time, removing n pairs will not change the result since we still have more pairs to split than points of view to split them. This is possible because as it is defined, all pairs are two-by-two disjoints on all B's points of view meaning that for each pair of pairs of alternatives there is at least one of the REFUSED alternatives that is preferred to an ACCEPTED one, implying that for each pair of pairs, only one can be split at a time by a point of view in B. By removing n pairs, we still have k pairs two-by-two disjoint on each of the k -1 points of view, resulting in having k pairs to be split and only k -1 points of view capable to split at most one pair at a time. On all other points of view, the pairs are still non-splittable. Thus, we still have our deadlock leading to have one pair that is not splittable. The decision problem is then solved by this formulation but it cannot be used to enumerate all the schemes a configuration can contain. In other words, every scheme with a lot more pairs than points of view in B is subsumed by a scheme with only k pairs and k -1 points of view.

There exists a SAT counterpart of this formulation:

Card1SAT : |{⟨g, b⟩ ∈ A : λ gb is true }| = k Card2SAT : |{i ∈ N : βi is true }| = k -1 C1SAT : ∀i∈N ∀⟨g,b⟩∈A, g≻ i b βi ∨ ¬λ gb C2SAT :
∀i∈N ∀p̸ =q=1,...,mn ∀⟨g,b⟩∈{gp,gq }×{bp,bq},

g≻ i b ¬βi ∨ ¬λ gpbp ∨ ¬λ gq bq
Resulting to the formula:

Φ2 : Card1SAT ∧ Card2SAT ∧ C1SAT ∧ C2SAT

Explainable fragments

The fact that we found instances that both are negative to the inv-NCS problem and do not contain any argument scheme means that the scheme is not a complete representation of inv-NCS. However, it is too early to say that it lost all its value because its goal is to build a humanly readable explanation of a complex combinatorial problem, and so it does. Thus the question of how well it approximates the inv-NCS problem opens up.

To further investigate this question, we can see what happens if we fix the scheme's maximum length (i.e. the number of points of view in the set B). Its goal, as said before, is to build a humanly readable justification for a necessary decision and thus fixing the maximum size of the scheme can be interesting since the smaller, the clearer it is. This new model allows us to create nested explainable fragments that grow with the upper bound we place on the length. For instance, if we only consider schemes of size 1, we obtain a Pareto dominance characterization. If we increase the maximum size, we keep the fragment but add new deadlock characterizations with it. This is interesting because:

• The search for schemes of a given length can be implemented in polynomial time w.r.t the number of pairs in the assignment. This opens the question of the practical efficiency of this algorithm, compared to the SAT-based modeling and solving approach we used. • Increasing the size of the scheme may push the border of the uncovered necessary decision meanwhile it also reduces the readability of the explanation. Thus the purpose of simplifying a complicated combinatorial model is less and less respected, the longer the scheme is. The question of setting the value of the upper bound so as to achieve a good trade-off between the expressiveness of the scheme and its intelligibility is an interesting question, lying at the edge between decision theory and cognitive psychology.

This nesting opens up several research questions like quantifying the proportion that each maximum size model can explain.

Knowing that the original scheme is not complete, it can be interesting to see if there are remarkable structures that can be added into it that do not increase the complexity of the explanation. See for example the structure of example 3 where we added one point of view and one alternative to example 2 resulting into a counterexample.

Searching for counterexamples, we compare various argument scheme's formulations' and multiple solvers' efficiency. To implement these experimentations we used an ASUS G551J computer with an Intel® Core™ i7-4750HQ CPU @ 2.00GHz × 8 and 8 GiB of DDR3 clocked at 1600 MHz RAM. All execution times have been recorded using the user time of /usr/bin/time command on an Ubuntu 20.04.4 LTS OS.

Considering that the argument scheme contains cardinality constraints, we tried to use a pseudo boolean formulation. To solve both the SAT and pseudo boolean formulations we used Gophersat version 1.2 [5] which can solve both types of problems and has a useful API that helps us perform quick experiments. We compared the SAT resolution time by using Glucose version 4.0 [1] which is based on Minisat [START_REF] Eén | Minisat[END_REF].

We first introduce the process we followed to compare the performances between formulations and solvers. We compared a SAT and an equivalent pseudo boolean formulation using Gophersat solver. Finally, we compared the efficiency of Gophersat and Glucose on finding an argument scheme.

On this last part, we expected to have at least better performances with Glucose. However this assumption has been proven wrong on most of the instances. We have to note that we used the default parameters of each solver, without trying to fit them to the task at hand, which might seem more penalizing for Glucose, which offers many options.

Process

In this subsection we introduce the process used to compare the influence of various parameters on the solving time. We still focus on the case where all preferences are strict total orders.

The parameters we studied are the following:

• The number of alternatives: n • The number of points of view: m • The rate of ACCEPTED alternatives: p • The minimal scheme's size (the number of pairs concerned): s

We generated the test samples using the algorithm of Listing 1 To test each parameter's influence, we generated the following set of configurations.

The default value for each configuration is:

• number of alternatives: 20

• number of points of view: 5

• rate of ACCEPTED alternatives: 35%

• minimal scheme's size: 2

Here are the various values tested for each parameter (only one parameter is tested at a time, the others take there default value):

• number of alternatives : 10, 20, 30, 40, 50 • number of points of view : 3, 4, 5, 7, 10 • rate of ACCEPTED alternatives : 10%, 35%, 50%, 75%, 90% • minimal scheme's size : -1 (no scheme), 1, 2, 3, 4, 5, 6 100 instances were generated for each setting.

Formulation comparison (pseudo boolean versus SAT)

In order to perform a comparison between a pseudo-boolean formulation and a SAT formulation, we cannot use the SAT formulation Listing 1: Sample generation process. shuffle() use the Fisher-Yates shuffle to ensure equiprobability to all permutations. formulate() use the SAT formulation presented in Section 3.2.3. resolveInc(formula) is a function that incrementally solves the formula using Gophersat by increasing the size of the scheme on each iteration. presented in Section 3.2.1. The reason is that this formulation allows us to call only once on the solver whereas the pseudo-boolean necessitate multiple calls (one for each possible value of k). Thus, we will compare the pseudo-boolean formulation and its SAT counterpart proposed in Section 3.2.3 Using Gophersat to solve equivalent SAT and pseudo-boolean equivalent instances with all generated instances, we obtain the scatter plot of Figure 1. This figure focuses only on the reading and solving time of either a SAT or pseudo-boolean instance. In this setting, one can see that the SAT solving time is most of the time faster than its equivalent pseudo-boolean instance. nPOV line shows the results with 1: 3, 2: 4, 3: 5, 4: 7, 5: 10 points of view. %Acc line shows the results with 1: 10%, 2: 35%, 3: 50%, 4: 75%, 5: 90% of ACCEPTED alternatives. min sol line shows the results with scheme of minimal size equal to 1: -1, 2: 1, 3: 2, 4: 3, 5: 4, 6: 5, 7: 6, with -1 meaning that there is no scheme in the instance.

Table 3 shows the relative performances of pseudo-boolean and SAT instances with respect to each parameter. We can easily see that the more alternatives there are, the more SAT solving outperforms pseudo boolean solving. If we focus on the impact of the number of points of view, we can see that there seems to have two effects. Firstly, we can note that the more points of view there are, the less pseudo-boolean instances are faster than the equivalent SAT instances. Secondly, we can see that the parity of the number of points of view impacts the number of SAT instances that are faster that the equivalent pseudo-boolean instances. The rate of ACCEPTED alternatives seems to slow down the SAT solving more than the pseudo-boolean solving when this rate tends to extreme values. Finally, we can see that the size of the smallest scheme present in the instance has a strong influence on the relative performances. In particular, we can note that the greater the smallest size is, the more the SAT solving outperforms the pseudoboolean solving. Considering the extreme case where there are no schemes, SAT is faster on every instance. of each SAT instance to its equivalent pseudo boolean instance. Line 2 : Scatter plot comparing the formulation+solving time (in seconds) with Gophersat of each SAT instance to its equivalent pseudo boolean instance.

We saw that solving SAT instances is generally faster with Gophersat than its pseudo boolean equivalent when considering our problem. This holds when we focus on solving time, but if we take into account the formulation time, which is mostly faster with pseudo boolean output, the pseudo boolean instances become faster to solve as Figure 2 show us. We can note that this difference is less important than the one when considering only the solving time.

Solver comparison (Gophersat versus Glucose)

This subsection shares some results concerning the relative performances of Gophersat 1.2 and Glucose 4.0 SAT solvers on finding argument scheme. It is important to note that no tweaking has been done on the solvers' settings.

We compared the efficiency of the two previously presented SAT formulations. We will note SAT the one using a constant k explained in Section 3.2.3 and SATNoK the one without any constant introduced in Section 3.2.1.

We expected to find out that Glucose is more efficient than Gophersat, however Figure 3 shows us that for both formulations (SAT and SATNoK), Gophersat is at best equivalent or at worst outperforms Glucose. To seek more details about those differences, Table 4 and Table 5 report the details about the parameters' influence.

We can see that the higher the number of alternatives is, the more Gophersat outperforms Glucose. However, the high number of points of view seems to be beneficial to Glucose. The ACCEPTED alternative rate is favorable to Glucose when it is set to an extreme value and to Gophersat when it is near 50%. The scheme size has a different influence with each of the formulations. With SAT the greater it is, the more Gophersat outperforms Glucose whereas it is the contrary if we use SATNoK.

We did not dig deeper on those results. We think that even if Gophersat is based on Glucose, the fact that we did not tweak any of the settings on either of them may influence the solving process to this point.

Conclusion

We considered the problem of explaining the results of a non compensatory model that are necessary w.r.t. a given jurisprudence, using Glucose and Gophersat with respect to all parameters. Each entry shows the number of instances where Glucose is faster/the number of instances where Gophersat is faster. nAlt line shows the results with 1: 10, 2: 20, 3: 30, 4: 40, 5: 50 alternatives. nPOV line shows the results with 1: 3, 2: 4, 3: 5, 4: 7, 5: 10 points of view. %Acc line shows the results with 1: 10%, 2: 35%, 3: 50%, 4: 75%, 5: 90% of ACCEPTED alternatives. min sol line shows the results with scheme of minimal size equal to 1: -1, 2: 1, 3: 2, 4: 3, 5: 4, 6: 5, 7: 6, with -1 meaning that there is no scheme in the instance. Table 5. Comparing SATNoK instances relative performances when solved on Glucose and Gophersat with respect to all parameters. Each entry shows the number of instances where Glucose is faster/the number of instances where Gophersat is faster. nAlt line shows the results with 1: 10, 2: 20, 3: 30, 4: 40, 5: 50 alternatives. nPOV line shows the results with 1: 3, 2: 4, 3: 5, 4: 7, 5: 10 points of view. %Acc line shows the results with 1: 10%, 2: 35%, 3: 50%, 4: 75%, 5: 90% of ACCEPTED alternatives. min sol line shows the results with scheme of minimal size equal to 1: -1, 2: 1, 3: 2, 4: 3, 5: 4, 6: 5, 7: 6, with -1 meaning that there is no scheme in the instance.

an argument scheme found in the literature. Contrarily to what has been conjectured, we show this scheme does not allow to explain all necessary decisions. This opens the question of assessing the size of the explainable fragment. During our experimentation, we found out that with preferences modeled as strict total orders and 6 alternatives and 3 points of view, the unexplainable fragment is extremely small. Furthermore, since the purpose of the scheme is to build a humanly readable explanation, it can be interesting to study a bounded version of the scheme where we fix an upper bound for the number of pairs or the number of points of view that are not trivially irrelevant. We found an interesting structure to extend the scheme to catch a part of the decision not covered by the scheme. The question of characterizing the decisions not covered by the scheme is left open.

Example 2 .

 2 Let X = {a, b, c, d, e}, N = {1, 2} and α = {(a, ACCEPTED) , (b, REFUSED) , (c, REFUSED) , (d, REFUSED) , (e, REFUSED)}. a ≻1 c ≻1 d ≻1 b ≻1 e e ≻2 b ≻2 d ≻2 a ≻2 c This configuration is positive for inv-NCS. If we choose A1 = {a} and S = {{1} , {1, 2}}, we produce α.

Example 3 .

 3 This example is a continuation of Example 2 adding one point of view and one alternative. Let us take X = {a, b, c, d, e, x}, N = {1, 2, 3} and α = {(a, ACCEPTED) , (b, ACCEPTED) , (x, ACCEPTED) , (c, REFUSED) , (d, REFUSED) , (e, REFUSED)} x ≻1 c ≻1 a ≻1 b ≻1 d ≻1 e a ≻2 c ≻2 x ≻2 d ≻2 b ≻2 e e ≻3 b ≻3 d ≻3 a ≻3 c ≻3 x

  b)∈A λ gb = k Card2PB : i∈N βi = k -1 C1PB : ∀i ∈ N , ∀ ⟨g, b⟩ ∈ A, g ≻i b, βi + ¬λ gb ≥ 1 C2PB : ∀i ∈ N , ∀p ̸ = q = 1, ..., mn, ∀ ⟨g, b⟩ ∈ {gp, gq} × {bp, bq} , g ≻i b, ¬βi + ¬λ gpbp + ¬λ gq bq ≥ 1

  GENERATE(n,m,p,s): alternatives := [1..n] accepted := [1..floor(n * p)] refused := [floor(n * p)+1..n] do preferences := [] for i from 0 to m-1 preferences[i] := copy(alternatives) preferences[i] = shuffle(preferences[i]) done formula := formulate(preferences, accepted, refused) min_size := resolveInc(formula) while min_size != s return (preferences, (accepted, refused))

Figure 1 .

 1 Figure 1. Scatter plot comparing the solving time (in seconds) with Gophersat of each instance in SAT to its equivalent pseudo boolean instance

Figure 2 .

 2 Figure 2. Line 1 : Scatter plot comparing the formulation time (in seconds)of each SAT instance to its equivalent pseudo boolean instance. Line 2 : Scatter plot comparing the formulation+solving time (in seconds) with Gophersat of each SAT instance to its equivalent pseudo boolean instance.

Figure 3 .

 3 Figure 3. Line 1 : Scatter plot comparing the resolution time of a SAT instance on Gophersat and on Glucose. Line 2 : Scatter plot comparing the resolution time of a SATNoK instance on Gophersat and on Glucose.

Table 2 .

 2 and

	# alternatives # points of view # counterexamples
	3	2	0
	4	2	0
	4	3	0
	5	2	0
	5	3	0
	5	4	0
	6	2	0
	6	3	898

Table 1 .

 1 Number of counterexamples with respect to the number of alternatives and the number of points of viewThe number of counterexamples for 6 alternatives and 3 points of view is very small considering the number of tested configurations. Without the hash table, we have 5 k=1 C k 6 × 6! × 6! = 32 140 800 configurations, which can be shrunk to 15 785 210 by using the hash table. In this, 8 387 380 are negative for the inv-NCS problem. Thus leading us to a frequency of

		898 15 785 210 ≃ 5.69 × 10 -5 over all config-
	urations (positive and negative ones) and a ratio of negative instances
	not covered by the scheme of 898 8 387 380 ≃ 1.07 × 10 -4 .
	# ACCEPTED alternatives # counterexamples
	1	0
	2	12
	3	874
	4	12
	5	0

Table 2 .

 2 Number of counterexamples with respect to the number of ACCEPTED alternatives when there are 6 alternatives and 3 points of view

Table 3 .

 3 Comparing pseudo boolean (PB) and SAT relative performances on Gophersat with respect to all parameters. Each entry shows the number of instances where PB is faster/the number of instances where SAT is faster. nAlt line shows the results with 1: 10, 2: 20, 3: 30, 4: 40, 5: 50 alternatives.

	Name	1	2	3	4	5	6	7
	nAlt	2/1	14/61	0/95	0/100	0/99	x	x
	nPOV	10/61	8/52	8/62	8/53	5/62	x	x
	%Acc	7/29	13/50 14/52	6/54	10/18	x	x
	min sol 0/100 16/42 10/50 12/67	9/69	15/74 6/92

Table 4 .

 4 Comparing SAT instances relative performances when solved on

	Name	1	2	3	4	5	6	7
	nAlt	2/0	5/62 0/100 0/100 0/100	x	x
	nPOV	0/74	1/73	7/66	4/63	6/57	x	x
	%Acc	10/0	5/70	1/60	2/60	12/0	x	x
	min sol 10/82 4/64	4/69	1/93	0/97	3/90 11/79

Heudiasyc, Université de Technologie de Compiègne et CNRS, France, email: khaled.belahcene@hds.utc.fr

Heudiasyc, Université de Technologie de Compiègne et CNRS, France, email: jerome.gaigne@hds.utc.fr

Heudiasyc, Université de Technologie de Compiègne et CNRS, France, email: sylvain.lagrue@hds.utc.fr


Chen Xia
email: chenxia@mail.crsri.cn

Yan Jianjun
email: byanjj@mail.crsri.cnl

Yang Huaquan

Influence of Aggregates on Cracking Sensitivity of Concrete

Keywords: Concrete, Aggregate, Strength, Sensitivity, Cracking temperature

Concrete cracking sensitivity made from different aggregates was compared with employment of advanced temperature and stress test machine by tracking thermal and stress development of concrete with age and elements distribution in interfacial transition zone of each concrete was explored by SEM and EDS. Test results indicated that compressive strength of concrete made from basalt and limestone was greater than diorite and natural aggregate and crack resistance of concrete based on cracking sensitivity comparison in terms of cracking temperature varying from minus 0.3°C to minus 2.5°C as diorite, basalt, limestone and natural aggregate. Elements Ca and Mg were apparently accumulated in vicinity of ITZ and some reaction between limestone and cement matrix might have taken place which blurred the boundary between aggregates and cement matrix.

Introduction

Aggregates, accounting for over 75 per cent in concrete by volume, have prominent influence on concrete performances [START_REF] Huaquan | Research and application of hydraulic concrete[END_REF] and selection of proper aggregates in dam construction was one of the most significant steps with respect to dam construction in which various factors including lithology, grade, processability, toughness and et al should be taken into account. In order to obtain general law of aggregates influence on concrete, numerous researches [START_REF] Huang Xiushan | characteristics of concrete affected by grain group of aggregate[END_REF][START_REF] Donza | High strength concrete with different fine aggregates[END_REF][START_REF] Sadowskia | Effect of aggregate kind and graining on modelling of plain concrete under compression[END_REF] have been conducted and only a few was concerned about shrinkage property, closely related to crack resistance of mass concrete. G.E.Troxell [START_REF] Troxell | Long-time creep and shrinkage tests of plain and reinforced concrete[END_REF] revealed that shrinkage difference among concrete made from aggregates would reach as high as 800×10 -6 given concrete was located under condition of 21°C and 50 per cent RH for three decades.

Explanations in terms of physical features, chemical and particle characteristics as well as thermal characteristics have been provided. Ref. [6] presumed that sandstone aggregate was of sedimentary rock consisted of original biological detritus while griotte was of metamorphic rock stem from limestone recrystallization characterized by condense blastic texture, with linear expansion coefficient of 12×10 -6 /°C and 6×10 -6 /°C respectively and 24h water absorption of 1.4 and 1.1 per cent for each, and the reduction in drying shrinkage and self-shrinkage of concrete at 180d was 33.6 per cent and 49.4 per cent when combined aggregates superseded sandstone aggregate. Whereas, others retorted [START_REF] Fenglan | Self-shrinkage test of high-strength concrete with different aggregates[END_REF] that shrinkage of concrete was primarily dependent on coarse aggregate and its long term water absorption property, indirect reflection of parent rock on pore structure, and self and drying shrinkage was stem from constant water absorbing by capillary and corresponding water loss from paste. So far no general rule has been obtained considering the fact that various factors were involved with regard to influence of aggregates on concrete performances and most significantly the majority of investigations was conducted by conventional methods, a bottleneck with regard to aggregates research, and hereafter with introduction of advanced temperature and stress machine, cracking sensitivity of concrete made from basalt, diorite, limestone and natural aggregate was compared and aggregates interfacial transition zone features for each, the key to unknot the puzzle, was disclosed with employment of element map.

Materials and Methods

The chemical compositions of 42.5 medium-heated Portland cement and chemical components of class I fly ash by mass volume of 35 per cent used for test were listed in Table1.Crushed basalt fine and four coarse aggregates correlated with DL/T 5144-2001, including artificial basalt, diorite, limestone and natural aggregates as shown in Fig. 1 and table 2, were used to make concrete. Bote ZB-1A water reducing agent, GYQ air entertainer were incorporated. Water/binder ratio was 0.41, sand/ and 35 per cent fly ash. The mix proportion of each batch was shown in Table 3. All ingredients were batched and placed in the formwork directly. The temperature of the fresh concrete should be 20°C±1°C. The thermally-insulated upper formwork was fixed onto the specimen. When the power was turned on, synchronous motor was started to ensure 100 per cent restraint implemented on specimen and core temperature development was recorded. At 7d, 28d and 90d small pieces containing ITZ were extracted from specimen and used for microscopic tests after treatment.

Results and Discussions

Mechanical property: Compressive and splitting tensile strength of concretes was listed respectively in table 3. As could be seen from table 3 that concrete strength exhibited rising tendency with age and by comparison the compressive strength of concrete made from different aggregates was decreased in sequence as basalt, limestone, diorite and natural aggregate at 90d and the splitting tensile strength as limestone, diorite, basalt and natural aggregate. Cracking sensitivity: Characteristic parameters during thermal and stress development of concrete were summarized in table 4. As could be seen that the test could be finished within 10 days, in comparison with conventional deformation testing method, the beneficial effects of TSTM were clearly demonstrated. When tensile stress sprang up during cooling process surpassed ultimate tensile strength, cracks was produced and corresponding cracking temperature could characterize the cracking tendency of concrete which was adopted by RILEM [START_REF] De | Thermal properties [R[END_REF] as indicator of cracking sensitivity and it did specify that concrete with cracking temperature below 10 degree was rated as good one.

As demonstrated in table 4 that cracking stress varied with type of aggregates used. Concrete made from basalt was rated the best in terms of cracking stress, followed by diorite and limestone, natural aggregate the least. In the light of cracking temperature, it was apparent that cracking temperature varied from minus 0.3°C to minus 2.5°C, of which concrete made from diorite was rated the best and natural aggregate the least. Elements distribution in ITZ of concrete: Elements distribution on ITZ of concrete made from different aggregates was demonstrated in Fig. 2 and colors on left side stand for enrichment degree of each element, and its decreasing manner was accorded with color darkening from white to black. Interfacial transition zone between aggregate and cement matrix could be discerned by enrichment of elements Al, Si and Fe, major components in aggregates. As shown in Fig. 2, boundary was obviously observed in Fig. 2 (a) and (b) while no explicit boundary could be identified in Fig. 2 (c) and (d) and elements in the latter were more uniformly distributed than in the former. Most significantly, elements Ca and Mg were apparently accumulated in vicinity of ITZ (white and pink color) and their enrichment on ITZ of diorite and basalt concrete was more evident than limestone and natural aggregate concrete. It was presumed that some reaction between limestone and cement matrix, might have taken place as revealed in Reference [9] that limestone surface was dissolved and C 4 AH n •1/2CO 2 was produced on ITZ, resulted in micro structure alteration of ITZ, pore structure and hydration products phase in particular, and directly affect mechanical and deformation properties of concrete as displayed in table 4. Besides, different ions dissolved from aggregates would directly lead to different products of hydration formation.

Fig. 2

 2 Fig.2 Elements distribution on interfacial transition zone of concrete

Table 1

 1 Chemical and mineralogical compositions of cement and fly ash

	Type			Chemical compositions(%)
		SiO 2	Fe 2 O 3	Al 2 O 3	CaO	MgO	R 2 O	SO 3	loss
	Cement	21.7	5.0	4.1	62.1	4.8	0.4	0.8	0.2
	Fly ash	52.53	11.11	24.51	3.87	0.87	1.21	0.69	2.15
			Table 2 Quality results of coarse aggregates
		Apparent density	Water absorption	Incorruptibility	Crushing index
	Type	(kg/m 3 )			(%)			(%)	(%)
	20~40mm	5~20mm	20~40mm	5~20mm		5~20mm
	Basalt	2940	2930	0.29		0.38		0.55	3.7
	Diorite	2890	2850	0.31		0.64		0.09	6.8
	Limestone	2770	2760	0.22		0.30		1.7	11.3
	Natural	2680	2670	0.60		1.00		1.9	4.0
	DL/T 5144-2001	≥2550			≤2.5			≤5	≤13

Fig.1 Surface texture of coarse aggregats

Table 3

 3 Compressive and splitting tensile strength of concrete

	No.	Aggregate	Compressive strength (MPa) 7d 28d 90d	Splitting tensile strength (MPa) 7d 28d 90d
	AG1	Basalt	18.9	29.0	43.2	1.4	1.9	3.1
	AG2	Diorite	16.5	27.0	38.1	1.6	2.1	3.3
	AG3	Limestone	20.3	31.4	41.6	1.8	2.8	3.8
	AG4	Natural	16.2	25.1	35.5	1.2	1.8	2.7

Table 4

 4 Characteristic parameters of concrete made from different aggregates

	No.	Aggregate	Maximum compressive stress (MPa)	Second zero-stress temperature (°C)	Maximum temperature (°C)	Cracking stress (MPa)	Cracking temperature (°C)
	AG1	Basalt	1.00	58.3	65.8	1.15	-2.3
	AG2	Diorite	1.27	58.3	66.3	0.80	-2.5
	AG3	Limestone	1.10	59.3	64.1	0.62	-1.7
	AG4	Natural	1.15	53.6	63.1	0.38	-0.3

Acknowledgements

This work was supported in part by NSFC under Grant No. 51139001 and Central non-profit Research Funds for Institutes No.CKSF2012035/CL.

Summary

Based on what discussed above, it was safe to conclude that natural and basalt aggregate were featured by high porosity and compressive strength of concrete made from different aggregates was decreased in sequence as basalt, limestone, diorite and natural aggregate. Cracking sensitivity tests implicated that concrete made from diorite rated the best in crack resistance, followed by basalt and limestone, and concrete made from natural aggregate was the last on lap. Micro structure analysis disclosed that superficial features and chemical activity of aggregates were the primary factors that influenced ITZ structure and concrete performances.