M Moukari

S Picard

L Simon

F Jurie

DEEP MULTI-SCALE ARCHITECTURES FOR MONOCULAR DEPTH ESTIMATION

Keywords: monocular depth estimation, multi-scale features, CNN architecture

This paper aims at understanding the role of multi-scale information in the estimation of depth from monocular images. More precisely, the paper investigates four different deep CNN architectures, designed to explicitly make use of multi-scale features along the network, and compare them to a state-of-the-art single-scale approach. The paper also shows that involving multi-scale features in depth estimation not only improves the performance in terms of accuracy, but also gives qualitatively better depth maps. Experiments are done on the widely used NYU Depth dataset, on which the proposed method achieves state-of-the-art performance.

INTRODUCTION

The estimation of depth information from images has a very long history in the computer vision literature, for instance in stereo vision [START_REF] Scharstein | A taxonomy and evaluation of dense two-frame stereo correspondence algorithms[END_REF]. As mentioned by Michels et al. [START_REF] Michels | High speed obstacle avoidance using monocular vision and reinforcement learning[END_REF] many researchers have been inspired by the way humans use monocular cues (e.g.texture, perspective, defocus) for estimating depth information e.g., [START_REF] Wu | Perceiving distance accurately by a directional process of integrating ground information[END_REF][START_REF] Bulthoff | Top-down influences on stereoscopic depth-perception[END_REF].

One major breakthrough in this area arose from the use of Deep Convolutional Networks, in order to train filters capable of detecting optimal cues for depth estimation. Mohan [START_REF] Mohan | Deep Deconvolutional Networks for Scene Parsing[END_REF] and Eigen et al. [START_REF] Eigen | Depth map prediction from a single image using a multi-scale deep network[END_REF] were among the firsts to propose depth estimation algorithms using deep CNN. Such algorithms now constitute the mainstream approach for depth estimation. Numerous publications have extended this approach in several directions. However, none of them scrutinize the importance of multi-scale information, in particular regarding the performance of depth estimation. This paper aims at filling this gap by proposing and evaluating several architectures that are multi-scale by construction and evaluate them on a public dataset. Our experimentations allowed us to observe that if multi-scale architectures bring only mild improvements over careful designed single-scale architectures with small training sets, they lead to state-of-the-art results on larger ones.

The rest of the paper is structured as follows. Section 2 presents the related works, Section 3 the different multi-scale architectures while Section 4 experimentally compares them and draw some conclusions.

RELATED WORK

A number of deep learning methods have recently emerged to address the problem of monocular depth estimation. The seminal work of Eigen et al. [START_REF] Eigen | Depth map prediction from a single image using a multi-scale deep network[END_REF] introduced a powerful deep CNN approach allowing to estimate depth from monocular RGB images in a feed-forward way. The proposed architecture is composed of two stacked networks. The first one outputs a coarse depth map while the second one refines it by applying 3 successive convolutions, using the coarse output and the RGB image as inputs.

This work has been extended in [START_REF] Eigen | Predicting depth, surface normals and semantic labels with a common multi-scale convolutional architecture[END_REF] by adding a third scale and by combining several tasks (multi-task learning framework), predicting not only the depth but also the surface normals and the semantic labels of the input RGB images. Several authors (e.g., [START_REF] Wang | Towards unified depth and semantic prediction from a single image[END_REF]) have also tried to combine complementary tasks, such as semantic segmentation, to improve the estimation of the depth.

In practice, recent methods can be roughly categorized into two main families: global methods that estimate the depth map from the whole input image and local methods that work locally on image patches. For instance, Eigen et al. [START_REF] Eigen | Depth map prediction from a single image using a multi-scale deep network[END_REF][START_REF] Eigen | Predicting depth, surface normals and semantic labels with a common multi-scale convolutional architecture[END_REF] are both global methods. On the other hand, approaches such as [START_REF] Liu | Learning depth from single monocular images using deep convolutional neural fields[END_REF][START_REF] Liu | Deep convolutional neural fields for depth estimation from a single image[END_REF][START_REF] Li | Depth and surface normal estimation from monocular images using regression on deep features and hierarchical crfs[END_REF][START_REF] Roy | Monocular depth estimation using neural regression forest[END_REF] are fundamentally based on local predictions. Some of them [START_REF] Liu | Learning depth from single monocular images using deep convolutional neural fields[END_REF][START_REF] Liu | Deep convolutional neural fields for depth estimation from a single image[END_REF][START_REF] Li | Depth and surface normal estimation from monocular images using regression on deep features and hierarchical crfs[END_REF] enforce some global consistency between patches thanks to CRF, yet they still remain fundamentally local. Furthermore, to alleviate the cons of both categories, Wang et al. [START_REF] Wang | Towards unified depth and semantic prediction from a single image[END_REF] and Chakrabarti et al. [START_REF] Chakrabarti | Depth from a single image by harmonizing overcomplete local network predictions[END_REF] have designed hybrid approaches that fuse local and global predictions.

As we write, the state-of-the-art architecture for depth estimation is an encoder-decoder network with a powerful encoder typically pre-trained for semantic purposes. For example, Laina et al. [START_REF] Laina | Deeper depth prediction with fully convolutional residual networks[END_REF] use the ResNet architecture as a fea- Convolutional architectures that are designed to use multiscale features all along the network are now well studied in some research areas such as object detection, object recognition and semantic segmentation. The U-net [START_REF] Ronneberger | U-net: Convolutional networks for biomedical image segmentation[END_REF] is a popular encoder-decoder architecture where skip connections allow the decoder to benefit from the mirror encoder features, to help awareness of the finer details. Furthermore, Lin et al. [START_REF] Lin | Feature pyramid networks for object detection[END_REF] argue that, in addition to skip connections, it is useful to have some intermediate outputs at lower resolution in the decoder, constructing high-level semantic feature maps at different scales. They demonstrate that, at last, it allows to improve both object detection and segmentation tasks. Eventually, dilated convolutions have been introduced to avoid down sampling in the encoding part of the network while preserving the same receptive field. These layers have been exploited for semantic segmentation in a recent work [START_REF] Chen | Rethinking atrous convolution for semantic image segmentation[END_REF] to implement deep feature extraction in a so-called spatial pyramid.

MULTI-SCALE ARCHITECTURES FOR DEPTH ESTIMATION

Given the lack of literature on the proper handling of multiscale features for depth estimation, we propose here four architectures inspired from other research areas such as semantic segmentation. All four of them are built upon a highly optimized single-scale network that we present first. The five resulting network architectures are depicted in Fig. 3).

Single-scale network (SSN)

This architecture is similar to hourglass nets, combining an encoder and a decoder part. We use the ResNet200 network [START_REF] He | Deep residual learning for image recognition[END_REF], pre-trained on the Ima-geNet classification task, as a feature extractor. The last two layers, originally dedicated to classification purposes, are removed. Furthermore, we modify the vanilla convolutions of the last ResNet block to become 2-dilated convolutions. In this case, the output features have a higher spatial resolution, For the decoding part, we use a slightly different version of the up-projection module, originally proposed in the work of Laina et al. [START_REF] Laina | Deeper depth prediction with fully convolutional residual networks[END_REF]. In our version (Fig. 2), we choose to use the nearest neighbors up-sampling instead of the original unpooling layer because we found it faster, with no impact on the final performance. Furthermore, as we conduct experiments on few training images, we use randomized ReLUs as activation units instead of the original ReLUs. We found it to generalize better when we learn on small-scale training set (typically less than 800 images). Similar results are observed in [START_REF] Xu | Empirical evaluation of rectified activations in convolutional network[END_REF], reporting the effectiveness of randomized ReLUs to combat overfitting on small-scale datasets. At the far end of the network, we use a 3x3 convolution with 1 output channel, followed by a ReLU. The output of this last layer serves as our depth estimate.

Skip Connections (Skips) This first multi-scale variant is inspired by the U-Nets [START_REF] Ronneberger | U-net: Convolutional networks for biomedical image segmentation[END_REF]: we use skip connections between mirror layers (in terms of spatial resolution) to concatenate features from the encoder with their counterpart of the same size in the decoder (see Fig. 3 (b)). Multi-Scale Middle Layer (MSML) This architecture also uses an encoder based on the Dilated ResNet. However, the last two ResNet blocks are replaced by 2 and 4 dilated ResNet blocks. Thus, we keep a higher spatial resolution in the middle layer whose size becomes 1/8 of the input resolution. Then, we gather all same-sized multi-scale encoder features in the middle layer before building on the decoder. Early features that are not the same size are first forwarded through a max pooling layer to downsample them before concatenation (see Fig. 3 (d)).

Dilated Spatial Pyramid (DSP) It relies on the same Atrous Spatial Pyramid Pooling (ASPP) blocks as the ones introduced by Chen et al. [START_REF] Chen | Rethinking atrous convolution for semantic image segmentation[END_REF] (see Fig. 3 (e)). The decoder part is build on top of the ASPP block [START_REF] Chen | Rethinking atrous convolution for semantic image segmentation[END_REF]. We use a pyramid of 1, 3, 6 and 12-dilated convolutions to construct our ASPP block instead of the originals 1, 6, 12 and 18-dilations to adapt to the size of our features, as illustrated in Fig. 4.

Decoder designs Between the five architectures, the decoder part varies slightly in the number of up-projections involved to recover the same spatial size as the input, and also in the amount of input and output features in each module. Hence, we use the following syntax to shorten the notations:

[N i - N o ].

EXPERIMENTS

This section presents an experimental validation of the different architectures proposed in the previous section. These models are evaluated both quantitatively and qualitatively and compared with the recent state-of-the-art results.

The NYU Depth v2 dataset. All our experiments have been conducted on the popular and challenging NYU Depth benchmark [START_REF] Kohli | Indoor segmentation and support inference from rgbd images[END_REF]. This dataset is composed of 464 video sequences of RGB-D indoor images, acquired with a Microsoft Kinect camera. The spatial resolution is of 640x480 pixels. The depth range goes from 0 to 10 meters. The dataset is split into 249 training sequences and 215 test sequences, among which only 795 training images and 654 testing images are densely annotated with filled-in depth values [START_REF] Kohli | Indoor segmentation and support inference from rgbd images[END_REF]. For both training and testing, we down sample the images by a factor of 2, using the nearest-neighbor algorithm.

Extended NYU dataset. In the prospect of comparing our approaches with state-of-the-art result, we have extended the official NYU dataset. To do so, we randomly select equally spaced frames from the raw training scene of the NYU Depth dataset. We end up with about 16K unique images that we process using the depth colorization routine provided in the matlab NYU Depth toolbox. Similar unofficial training sets are exploited in recent published work such as [START_REF] Laina | Deeper depth prediction with fully convolutional residual networks[END_REF].

Details on the learning procedure To obtain as fair comparisons as possible, we train our different architectures with the same hyper parameters and training procedures. We always rely on L2 losses and Stochastic Gradient Descent with a batch of size 3 (except for MSML and the last 2 scales of FPO where, due to memory limitations, we have used batches of size 2). We train each network for at least 800 epochs and stop the training when the score on the validation set increases during 50 epochs. The learning rate is set to 5e-3, the weight decay to 5e-4 and momentum to 0.9. We also apply online data augmentation, following the procedure reported in [START_REF] Eigen | Depth map prediction from a single image using a multi-scale deep network[END_REF].

Evaluation criteria. We report the following four standard performance criteria ( ŷi and y i denote respectively the predicted and ground-truth depth at pixel i. N stands for the total number of pixels): Experiments on the Official NYU dataset. As a first analysis stage, we trained all our architectures on the small official NYU training set (795 images). The results are presented in Table 2. In regard to the multi-scale vs single-scale comparison, we notice that every multi-scale network performs similarly or better than SSN. Though the performance gap is not so pronounced, we can conclude that for such a small dataset a carefully designed architecture as SSN is quite competitive.

Considering now the multi-scale methods, DSP consistently outperforms the other challengers. As a result, DSP stands out as a promising design to achieve state-of-the-art results.

Experiment on the Extended NYU dataset. Given our previous analysis, we singled out the DSP method for training on the unofficial but much larger scale NYU dataset. The learning procedure is the same, except that we train for about 60 epochs. The results are reported in Table 3, along with recent state-of-the-art performance. We see that our proposed Large Scale DSP (LS-DSP) network achieves the best results for almost all the usual metrics. Please note that, in the table, the results of Laina et al. [START_REF] Laina | Deeper depth prediction with fully convolutional residual networks[END_REF] were deliberately restricted to the L2 loss while the authors recommend the berHu loss. This intentional choice was made to mitigate the effect of the loss in our comparison. Nonetheless, for the sake of fairness, we should mention that Laina et al. [START_REF] Laina | Deeper depth prediction with fully convolutional residual networks[END_REF] obtain slightly better results on rel and log errors, while we achieve better results on the 4 remaining metrics. We expect that in the case of DSP as well, the berHu loss could further improve the performance.

Qualitative evaluation. Fig. 5 allows some qualitative comparisons of the single-scale network and two multi-scale ones (Skips and DSP). We observed that the methods using skip connections often produce sharper outputs, which is visible on the edge of the furniture (2nd column), or the ridges on the ceiling (1st column of the RGB images transferred to the depth map), visible on the paintings (2nd column, 4th row). We made similar observations for FPO whose architecture is close to Skips, and, to lesser extent as the skips are in the middle layer of the network, with MSML. We also show our best results using the DSP trained on the extended NYU dataset. We see that the network generalizes well and produces better depth map than SSN. We can see, for instance, better inferences of farther distances (3rd columns) and better defined shapes, e.g.the head of the person (4th column).

CONCLUSIONS

This paper investigates the use of multi-scale information for depth inference, a question that has not been discussed so far in the literature. The presented experiments show that, on the official NYU dataset with 795 training images, the proposed multi-scale architectures consistently outperform the counterpart single scale architecture. When trained on a larger training set, of a size comparable to the latest published methods, the proposed DSP method gives impressive state-of-the art results while, in addition, the inferred depth maps look qualitatively better.

Fig. 1 .

 1 Fig. 1. An example of our state-of-the-art monocular depth estimation trained with a deep multi-scale architecture.

Fig. 2 .

 2 Fig. 2. Our modified up-projection module. Unlike the original proposed in [14], we perform nearest neighbors upsampling x2 and we use randomized ReLU. ture extractor, after cutting off the last classification layers and training a decoder on top of it to achieve the regression task of depth prediction. They propose up-sampling layers based on the principle of ResNet with skip connections and also show that training with the BerHu loss can further improve the final error. They achieve state-of-the-art performance on depth estimation from single monocular images. Although their work yields impressive results, they do not integrate multi-scale analysis along the network to enhance their prediction.Convolutional architectures that are designed to use multiscale features all along the network are now well studied in some research areas such as object detection, object recognition and semantic segmentation. The U-net[START_REF] Ronneberger | U-net: Convolutional networks for biomedical image segmentation[END_REF] is a popular encoder-decoder architecture where skip connections allow the decoder to benefit from the mirror encoder features, to help awareness of the finer details. Furthermore, Lin et al.[START_REF] Lin | Feature pyramid networks for object detection[END_REF] argue that, in addition to skip connections, it is useful to have some intermediate outputs at lower resolution in the decoder, constructing high-level semantic feature maps at different scales. They demonstrate that, at last, it allows to improve both object detection and segmentation tasks. Eventually, dilated convolutions have been introduced to avoid down sampling in the encoding part of the network while preserving the same receptive field. These layers have been exploited for semantic segmentation in a recent work[START_REF] Chen | Rethinking atrous convolution for semantic image segmentation[END_REF] to implement deep feature extraction in a so-called spatial pyramid.

Fig. 3 .

 3 Fig. 3. Schematics of the different compared methods.

Fig. 4 .

 4 Fig.4. ASPP block (adapted from[START_REF] Chen | Rethinking atrous convolution for semantic image segmentation[END_REF]), with a pyramid of 1, 3, 6 and 12-dilated convolutions. BN stands for Batch Normalization.Feature Pyramid Outputs (FPO) This architecture is inspired by the work of Lin et al.[START_REF] Lin | Feature pyramid networks for object detection[END_REF]. It is the same architecture as the one used for Skips but with additional extra intermediate outputs after the up-projection modules. These extra outputs are designed as a fork of the network and each contains one 3x3 convolution followed by a ReLU, allowing depth estimation at lower scales (see Fig.3 (c)). We experimentally find more efficient to train each output sequentially, starting with the lower resolution and fixing the previously trained part of the network to learn a new output resolution.

  (a) Mean relative error (rel): 1 N N i=1 | ŷi -y i | /y i (b) Mean log 10 error (log): 1 N N i=1 |log 10 ŷi -log 10 y i | (c) RMSE (rms): 1 N N i=1 ( ŷi -y i ) 2 (d) Threshold: % of y i s.t. max ŷi yi , yi ŷi = δ < thr. In our evaluations, ∀k ∈ {1, 2, 3}, the δ k stands for the threshold metric δ < 1.25 k .

Fig. 5 .

 5 Fig. 5. From top to bottom: RGB test image, depth groundtruth, SSN, Skips and LS-DSP outputs. Best viewed in color.

Table 1 .

 1 Architecture of the different up-projections modules for each experiment.

	Decoders up-projections (UpProj) architectures
	Method	UpProj1		UpProj2	UpProj3	UpProj4
	SSN	[2048-1024] [1024-512] [512-256] [256-64]
	Skips	[3072-1024] [1536-512] [768-256] [320-64]
	FPO	[3072-1024] [1536-512] [768-256] [320-64]
	MSML	[3904-512]		[512-256]	[256-64]		-
	DSP	[2560-1024] [1024-512] [512-256] [256-64]
		lower is better	higher is better
	Method	rel	log	rms	δ 1	δ 2	δ 3
	SSN	0.167 0.066 0.603 77.9 94.9 98.9
	Skips	0.163 0.065 0.599 79.2 95.1 98.9
	MSML 0.160 0.066 0.601 78.1 95.4 99.0
	FPO	0.161 0.065 0.599 78.5 95.3 98.9
	DSP	0.159 0.064 0.592 79.3 95.4 98.9

Each bracket represents one up-projection module with N i input features and N o output features. All decoders upprojections architectures are summarized in Table

1

.

Table 2 .

 2 Error comparison of the multi-scale methods on the NYU Depth v2 test set. We only use the official subset of 795 training images for the learning stage.

Table 3 .

 3 Comparison of our LS-DSP network trained on the extended dataset, against state-of-the-art results on NYU.

		lower is better	higher is better
	Method	rel	log	rms	δ 1	δ 2	δ 3
	Eigen [7]	0.158	-	0.641 76.9 95.0 98.8
	Li [22]	0.143 0.063 0.635 78.8 95.8 99.1
	Laina [14] 0.138 0.060 0.592 78.5 95.2 98.7
	Cao [23]	0.141 0.060 0.540 81.9 96.5 99.2
	LS-DSP	0.133 0.057 0.569 83.0 96.6 99.3

). Skip connections at the end of the network give sharpest inferences, at the price of artifacts (edges


Rawdha Dekhili

Thomas H Kauffmann

Hassen Aroui

Marc D Fontana

Phase transformations in LiH 2 PO 4 (LDP) revealed by Raman spectroscopy

Keywords: Raman spectroscopy, vibrational modes, phosphate compounds, phase transformations

We investigate Raman scattering measurements on polycrystalline LiH 2 PO 4 (LDP) between 30 and 240°C, in order to detect phase transformations. The thermal behavior of main lines in different wavenumber ranges is discussed and analyzed. Raman spectra show two anomalies at 176°C and 210°C, reflecting changes in chemical bonds, and are attributed to progressive polymerization, in agreement with earlier electrical measurements.

Introduction

The family of phosphate compounds with the general formula MX 2 PO 4 (M = K, Rb, NH 4 , Cs, Tl, Li, X = H, D) is the object of many investigations thanks to several physical and chemical properties interesting for basic research and practical applications [START_REF] Shchur | Lattice dynamics study and specific heat of CsH 2 PO 4 and CsD 2 PO 4[END_REF][START_REF] Mendes | Electrical Conductivity in the KDP, ADP, and K 1-x (NH 4 ) x H 2 PO 4 Crystals[END_REF]. Among these compounds, the most widely studied is potassium dihydrogen phosphate (KDP) for ferroelectric, piezoelectric, and electro-optic properties [START_REF] Deng | A study on dual laser beam separation technology of KDP crystal[END_REF][START_REF] Goma | Dielectric parameters of KDP single crystals added with urea[END_REF]. Much less investigated, LiH 2 PO 4 (LDP) has a particular interest for its relatively high protonic conductivity [START_REF] Lee | Crystal Growth and morphology of LiH 2 PO 4[END_REF][START_REF] Jung Kweon | Impedance spectroscopy of the superprotonic conduction in LiH 2 PO 4[END_REF][START_REF] Jung Kweon | Rotating-frame nuclear magnetic resonance study of the superprotonic conduction in LiH 2 PO 4[END_REF]. The structure of LDP consists into tetrahedral groups PO [START_REF] Goma | Dielectric parameters of KDP single crystals added with urea[END_REF] and LiO 4 which are bonded together by oxygen ions. Lee et al [START_REF] Lee | Raman spectroscopic study of LiH 2 PO 4[END_REF] have recorded Raman data on LDP between 70 and 300 K but did not observe any change in spectra at low temperature. They did not investigate high temperature range but by means of electrical conductivity and dielectric constant measurements they found two anomalies at 178°C (Tp 1 ) and 196°C (Tp 2 ) [START_REF] Lee | High-temperature phase transformations in LiH2PO4 and possible solid-state polymerization[END_REF]. They interpreted them as due to super-protonic phase transformations induced by polymerization at Tp 1 and Tp 2 . Therefore we have performed Raman spectroscopy measurements at high temperature with a special attention to the temperature range 170-220°C.

Experimental

LDP polycrystals were prepared by employing a slow evaporation method using Li 2 CO 3 (purity of 99.99%), phosphoric acid (purity of 95%) and distilled water as starting materials according to the reaction Li 2 CO 3 (solid) + 2 H 3 PO 4 (liquid) <=> 2 Li(H 2 PO 4 ) 2 (solid) + CO 2 (gas) + 2 H 2 O(liquid) The resulting solution was stirred at room temperature. A quantity of the obtained homogeneous solution was kept in Petri dishes at ambient temperature during a few days; the final product consists into transparent cm-size samples. Raman spectra were recorded on a polycrystalline sample using confocal Raman microscope LabRAM HR Evolution (Horiba) with an exciting laser line of 532 nm. The system leads to a

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT

2 spectral resolution of 1 cm -1 using a 1800 grooves per millimeter grating and is equipped with an ultra-low frequency module to detect wavenumbers down to 5 cm -1 . The laser light was focused through a 50x objective (Olympus) at a distance above 1 cm from the sample placed into a temperature stage (Linkam). Raman spectra of LDP were recorded in the spectral range of 5-3500 cm -1 as a function of temperature from 30°C up to 240°C. The Raman spectrum recorded at room temperature is shown in figure 1 and is found to be relatively similar to spectra reported earlier [START_REF] Lee | Raman spectroscopic study of LiH 2 PO 4[END_REF][START_REF] Rhimi | Structural and vibrational study a new potassium lithium dihydrogenphosphate KLi(H 2 PO 4 ) 2[END_REF]. We distinguish different frequency ranges: below 300 cm -1 are lying the bands generally associated to external modes; between 300 and 1100 cm -1 the lines generally attributed to LiO 4 and PO 4 internal vibrations and above 1100 cm -1 very weak structures corresponding to OH vibrations.

Results and analysis

We report and discuss the temperature dependence of Raman spectra especially between 170 and 220°C. For this we describe and analyze the temperature behavior of the most intense lines lying in 4 different wavenumber ranges. These peaks are indicated by arrows in figure 1. In figures 2 and 3 are shown the Raman spectra recorded at different temperatures, in the ranges corresponding to external modes and high-frequency internal modes respectively. We note a continuous change of the profile when heating between 30 and 170°C, with a clear down-shift of the maxima positions, and a decrease of intensity of all considered lines. The diminishing of the maximum position with increasing temperature is expected regarding the thermal dilatation. The chemical bonds are expected to increase on heating, so that the force constant diminishes and subsequently the frequency decreases. Above 170°C the maximum position of main lines displays two breakdowns in its monotonous decrease on heating (see insert of figure 2). Likewise, regarding the intensity we remark that for larger temperature, all lines (see figures 2 and 3) exhibit a large discontinuity in their behavior, with an abrupt decrease of intensity, between 210 and 220°C. In particular, we can point out the large diminishing of the line at about 85 cm -1 which is very intense at room temperature. To attempt to rely more precisely the Raman data with the phase transformations, we report in figure 4 the temperature dependence of the maximum intensity of the main lines.

M A N U S C R I P T A C C E P T E D

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The intensity of all peaks shows firstly a monotonous decrease on heating, but the line at 85 cm -1 which is nearly constant, and then two clear breakdowns in their dependence. These two clear anomalies appear around 176°C and 210°C, close -but slightly shifted -to Tp 1 and Tp 2 reported by Lee by means of electrical measurements [START_REF] Lee | High-temperature phase transformations in LiH2PO4 and possible solid-state polymerization[END_REF]. It is to be noted that, on the one hand the external modes, and on the other hand the internal modes sensitive to P-O and Li-O bonds, are all affected by a structural rearrangement. These features reflect a change of the whole structure of the material, and constitute therefore an evidence of phase transformations.

An additional remarkable feature is given in figure 5 by the temperature dependence of the quasielastic scattering, i.e. the signal beside the Rayleigh and external mode bands (see the insert). In this figure is reported the integrated intensity between 5 and 30 cm -1 as function of temperature. This plot shows a large peak around 176°C and a second small maximum around 210°C. The spectral range below 20 cm -1 covers (even partly) the collective protonic and relaxational motions of PO 4 and LiO 4 .

Figure 5. Temperature dependence of the integrated intensity between 5 and 30 cm -1 of the quasielastic scattering reported in the insert.

The quasi-elastic scattering can be related to the dielectric response in the teraHertz or sub-teraHertz frequency range. Thus, as the Stokes Raman intensity is proportional to the imaginary part of the dielectric susceptibility χ"(ω) [START_REF] Tominaga | Central component of y(xy)x Raman spectra in paraelectric KH 2 PO 4[END_REF], the integration of the Raman intensity over the frequency range 0-30 cm -1 provides the real part of the susceptibility χ'(ω) at 1 THz. This last feature can be related to dielectric permittivity measurements of Lee et al [START_REF] Lee | High-temperature phase transformations in LiH2PO4 and possible solid-state polymerization[END_REF] and thus shows some consistencies with them. This demonstrates that the whole Raman spectrum is found to be affected by structural transformations. As shown in figure 6 the thermal change affects also the very high wavenumber range, corresponding to the OH bands. At first we note a shift on heating of the whole spectrum, which is

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mainly caused by the increase of the background (see level around 1400 cm -1 , i.e. far from any Raman band). This can be related to the loss of transparency of the sample which becomes milky above 200°C. The most remarkable change occurs at 240°C. The narrow lines around 1200 cm -1 , and the broad stretching bands between 2600 and 3600 cm -1 disappear, and are replaced by a very intense and wide signal. This huge profile change occurs simultaneously with the appearance of water on the surface of the sample. In fact the sample is notably damaged by heating process. This is confirmed by the spectrum obtained after cooling which completely differs from this previously recorded on heating (see figure 6). Therefore the transformations invoked above are clearly irreversible. A thermal decomposition does exist at high temperature with water releasing. The progressive let of water is consistent with the chemical reactions proposed by Lee et al. [START_REF] Lee | High-temperature phase transformations in LiH2PO4 and possible solid-state polymerization[END_REF] Figure 6. High wavenumber Raman spectra of LiH 2 PO 4 recorded during a heating-cooling process. Spectra at 240°C (heating) and 210°C (cooling) are vertically shifted for clearness.

The thermal behavior of the largest peak around 888 cm -1 (position at room temperature) reported in figure 4 corroborates this analysis. This line which shows the strongest decrease of intensity on heating, corresponds to the internal vibration ν 1 (P-OH). This bond is therefore affected by the continuous releasing of water, explaining more abrupt changes at Tp 1 and Tp 2 , suggested to be the onset of dimerization and oligomerization respectively.

Conclusion

On heating up to 170°C, Raman lines show an expected decrease of their maximum position. At higher temperature their intensity shows a breakdown in their monotonous behavior with temperature. Two anomalies were detected in the temperature dependence around 176°C and 210°C, for all main Raman lines. Consequently these features reflect the large modifications of the chemical bonds at high temperature, and therefore transformations of the whole structure. These

Figure 1 .

 1 Figure 1. Raman spectrum of LiH 2 PO 4 recorded at room temperature in the region from 5 to 1400 cm -1 . Arrows indicate the lines which are particularly analyzed.

Figure 2 .

 2 Figure 2. Raman spectra of LiH 2 PO 4 recorded at different temperatures in the region from 50 to 95 cm -1 . In the insert is reported the temperature dependence of the position of the line lying at 85 cm -1 at room temperature.

4 Figure 3 .

 43 Figure 3. Raman spectra of LiH 2 PO 4 recorded at four different temperatures in the region from 860 to 960 cm -1 .

Figure 4 .

 4 Figure 4. Temperature dependence of the maximum intensity of four different Raman peaks of LiH 2 PO 4 (the lines are labeled according to their position at room temperature).

•

  with electrical data reported by Lee et al[START_REF] Lee | High-temperature phase transformations in LiH2PO4 and possible solid-state polymerization[END_REF] and corroborate their interpretation. Use of Raman spectroscopy to detect phase transformations• Phase transformations in a new material (LDP) among KDP family compounds • Link between Raman data and dielectric measurements