Cyrille Migniot
email: cyrille.migniot@gipsa-lab.grenoble-inp.fr

Pascal Bertolino
email: pascal.bertolino@gipsa-lab.grenoble-inp.fr

Jean-Marc Chassery
email: jean-marc.chassery@gipsa-lab.grenoble-inp.fr

Contour segment analysis

come

Introduction

Simultaneous detection and segmentation of element of a known class , and in particular with the one of persons, is a much discussed problem. Due to the various colors and positions that a person can have, it is a challenging task. The aim is to avoid user supervision. It is a critical part in any applications such as video surveillance, driverassistance or video indexing. Dalal [START_REF] Dalal | Histograms of Oriented Gradients for Human Detection[END_REF] presented an efficient and reliable detection algorithm based on Histograms of Oriented Gradient (HOG) descriptors with Support Vector Machine classifier. Nevertheless, if person localization is performed, no information about his/her shape is provided and segmentation is not done. This paper describes a HOG local process. Indeed HOG global process in a detection window makes a decision about the presence of a person in the window. With HOG local process, information relied on the person silhouette is obtained. Our method uses this approach in order to determine the relevance as a person silhouette part for each contour segment (Figure 1). This data might allow segmentation, for example by looking for the shortest-path cycle in a graph. Here, contour segments are an interesting support. Given that only shape is discriminative of person class, it is logical to study contours. Furthermore, segments gather pixels that share the same location and orientation. Associating them with HOGs based results is therefore relevant. Our method studies a positive detection window (containing a person) and generates for each contour segment a human silhouette membership likelihood value. Learning and tests are made with the INRIA Static Person Data Set containing positive detection windows of Dalal's algorithm [START_REF] Dalal | Histograms of Oriented Gradients for Human Detection[END_REF].

Related Work

Most human detection works have used a "descriptor/classifier" framework (Figure 2a). The descriptor converts an image into a vector of discriminative features of the searched class. The classifier, from an image feature vector, determines whether a person is in the image. To describe a class, the classifier is made from feature vectors of positive (containing an instance of the target class) and negative detection windows. SVM [START_REF] Vapnik | The Nature of Statistical Learning Theory[END_REF], AdaBoost [START_REF] Freund | A Decision-Theoretic Generalization of On-line Learning and an Application to Boosting[END_REF] and neural networks [START_REF] Toth | Detection and Recognition of Moving Objects Using Statistical Motion Detection and Fourier Descriptors[END_REF] are the most used classifiers. Regarding descriptors, Haar wavelets [START_REF] Oren | Pedestrian Detection Using Wavelet Templates[END_REF] and HOGs are the most usual but Principal Component Analysis (PCA) [START_REF] Munder | Pedestrian Detection and Tracking Using a Mixture of View-Based Shape-Texture Models[END_REF], Riemannian manifold [START_REF] Tuzel | Human Detection via Classification on Riemannian Manifolds[END_REF] or Fourier descriptors [START_REF] Toth | Detection and Recognition of Moving Objects Using Statistical Motion Detection and Fourier Descriptors[END_REF] are used too. Detection can be done for the entire person or for body parts (limbs, torso, head, etc.) as in [START_REF] Alonso | Combination of Feature Extraction Methods for SVM Pedestrian Detection[END_REF]. Nevertheless, other works are based on other techniques as, for example, human movement recognition and particularly walking periodicity [START_REF] Ran | Pedestrian Classification from Moving Platforms Using Cyclic Motion Pattern[END_REF]. Simultaneous detection and segmentation need other processing. Using stereo [START_REF] Kang | Real-Time Pedestrian Detection using Support Vector Machines[END_REF] easily gives a segmentation of elements of different depths but requires particular equipment. Silhouette template matching [START_REF] Lin | Hierarchical Part-Template Matching for Human Detection and Segmentation[END_REF] [8] allows segmentation with the nearest known tem-plate. Contour segments analysis is used as well. [START_REF] Wu | Simultaneous Object Detection and Segmentation by Boosting Local Shape Feature Based Classifier[END_REF] captures most important edgelets with a cascade of classifiers but essentially for detection. [START_REF] Sharma | Integrating Appearance and Motion Cues for Simultaneous Detection and Segmentation of Pedestrians[END_REF] looks for the most likely contour segment cycles without any knowledge on the studied class. These cycles are attached to some person features, using a Markov random field. This is an approach inverse to ours.

HOG and SVM Combination

Our method is based on the combination used by [START_REF] Dalal | Histograms of Oriented Gradients for Human Detection[END_REF] for detection, as we remind in this section. The process consists in testing all possible detection windows for all locations and scales. A human presence measure is associated to each detection window. Descriptors are based on contour orientation. In order to obtain spatial informations, any studied detection window is divided into areas named blocks and cells.

Blocks and Cells

The detection window is divided by a rectangular grid. Cells are integrated into blocks. Each block contains a fixed number of cells. Each cell belongs to at least one block. Without block overlapping, a cell belongs to a single block. Division into blocks is also performed using a rectangular grid (Figure 2b). With block overlapping, a cell may belong to several blocks. First, we do not use block overlapping.

Histograms of Oriented Gradients For a given area, a HOG means the proportion of contour pixels for each orientation bin. Orientation interval is evenly divided in N bin orientation bins in ∆:

∆ = I k = 2k -1 2N bin π, 2k + 1 2N bin π , k ∈ [1, N bin ] (1)
Let ν cell I be the occurrence frequency of contour pixels whose orientation belongs to interval I in the cell cell. The cell HOG is given by:

HOG cell = ν cell I , I ∈ ∆ (2)
Descriptor HOGs are descriptors and thus are described by feature vectors. Clothe colors and texture variations are not discriminative. HOGs describe shape and are also relevant for human detection.

Classifier For a block block, the feature vector V block is made from all HOGs of its cells:

V block = {HOG cell , cell ∈ block} (3)
With the normed concatenated vector of the feature vectors of all blocks, the SVM classifier indicates whether a person is in the detection window. An overview of the method is shown in Figure 3a.

Combining HOG with SVM for each Block

In our approach, the classification process is not anymore performed globally but at the block level in order to classify any piece of contour has being or not a piece of the person's silhouette. First, a contour map of the detection window is computed using the Canny algorithm. Then a list of contour segment is created.

The window is divided into blocks and cells. In the following, we present the method without then with block overlapping.

Without Block Overlapping

The method shown previously provides a value for each window. Our aim is to have one for each contour segment.

Feature Vectors Each contour pixel with orientation θ ∈ k N bin π, k+1 N bin π updates two bins of the cell's HOG with weights ω (for bin ν I k ) and 1-ω (for bin ν I k+1 ) as follows:

ω = k+1 N bin π -θ π N bin (4)
HOG cell are still obtained with equation 2 and V block with equation 3. For segment likelihood calculation, HOG block is defined by:

HOG block = ν block I , I ∈ ∆ (5)
Normalization Illumination and contrast can greatly modify HOG values. This leads to a reduction of the efficiency in the learning and classification process. Thus a normalization of V block is performed. Let N be the vector size. Four different normalization schemes were tested:

-L1-norm: averaging by the norm L1 of the vector:

V L1 bloc (i) = V block (i) N k=1 V block (k) (6)
-L1sqrt norm: the square root of the L1-norm is used to express the feature vector as a probability distribution

V L1 sqrt bloc (i) = V block (i) N k=1 V block (k) (7)
-L2-norm: averaging by the norm L2 of the vector:

V L2 bloc (i) = V block (i) N k=1 V block (k) 2 (8)
-L2-Hyst norm: non linear illumination variations could make saturations in the acquisition and cause sharp magnitude variations. High magnitude gradient influence is here decreased by applying a threshold of 0.2 after L2-normalization. Finally L1-normalization is performed.

These norms effects are shown in section 3.

Likely Segments From the database examples, SVM constructs a hyperplane to separate positive and negative elements. Algebraic distance between an example and this hyperplane provides the classification and associates to each block a value S SV M block . An overview is shown in Figure 3b. For contour segment seg with orientation θ ∈ I, let t b k Fig. 3. An overview of person detection (a) and an overview of our approach (b). be this segment pixel count belonging to block b k . In a window containing N b blocks, the value associated to this segment is:

P seg = N b k=1 t b k HOG b k (I)S SV M b k N b k=1 t b k (9)
This value is the likelihood of a segment to be part of a human silhouette.

With Block Overlapping

An important number of cells in a block leads to an efficient descriptor. An important number of blocks gives more spatially accurate results. Nevertheless cells with too small size is not informative enough. The solution is block overlapping. A cell could belong to several blocks. Hence, cell number and size could be higher than previously.

In practice, in SVM output, a value S SV M block is always allocated to each block. But, since a single cell can belong to several blocks, a value S SV M cell is associated with each cell.

S SV M cell = mean bloc cell (S SV M block ) (10)
For a contour segment seg with orientation θ ∈ I, let t c k be segment pixel count belonging to cell c k . In a window containing N c cells, the value associated with this segment is:

P rec seg = Nc k=1 t c k HOG c k (I)S SV M c k Nc k=1 t c k (11)

Results

To evaluate the ability of our method to extract information about person silhouette, we tested it on INRIA Static Person Data Set consisting of images of person in various upright poses. Learning is realized with 200 binary silhouette images obtained from positive examples and 200 negative examples. SVMlight algorithm [START_REF] Joachims | Making Large-Scale SVM Learning Practical[END_REF] is used as classifier.

In Figure 4 and7, each block (or cell) is colored with a gray level proportional to its SVM output (then, each segment). For a positive detection window pdw, let M pdw pos be the mean of P values associated to segment's pixels belonging to human silhouette and M pdw neg the mean of P values associated to other segment's pixels. Let benchmark be a value relative to our method performance, computed from N w detection windows as follows:

benchmark = 1 N w Nw pdw=1 M pdw pos M pdw neg ( 12
)
Overlapping Effects Benchmark values are very similar with and without block overlapping. Nevertheless, behaviors are not the same. With block overlapping, the value associated to a cell depends on several blocks. Thus false segment suppression (in particular on window boundaries) is less efficient (Figure 4). However, block overlapping provides spatially finest study. In this way, less silhouette segments are omitted by detection.

Cell and Block Size Effects Cell size modifies SVM outputs spatial accuracy and studied area relevance. Moreover bigger block size leads to more efficient descriptors but spatially less accurate results.

In Figure 5a, the benchmark is computed with the L2-norm. Blocks of 4 and 16 cells make block overlapping of 1 2 , while blocks of 9 cells make block overlapping of 1 3 . Blocks of 4 cells give the best results. Results for various cell sizes are stable, but, for next tests, cells of 25 pixels are used because they give the best benchmark. Normalization Effects Feature vector normalization prevents illumination variation effects. The goal of this normalization is to increase the similarity between images of the same class. Results of Figure 5b stems from tests with blocks of 4 cells of 25 pixels and block overlapping. Mean of M pdw pos , mean of M pdw neg and benchmark are shown. Without normalization, variations between elements of the same class are higher. Thus, descriptor is less discriminative and benchmark response is less good. The benchmark promotes L1 and L2-Hyst norms. However, for these normalizations, mean of M pdw pos is low. That means that, with these norms, our method is accurate but not efficient. L2-norm may also be the best choice.

Particular Cases Segmenting a person is harder if portions of the silhouette contours are missing (Figure 6a). Indeed, the edge detector can miss parts of the silhouette if the transition is not sharp enough. This has no influence in computing, but perturbs segment gathering.

A typical detection and segmentation difficulty is the occlusion (Figure 6b). If a part of the studied person is hidden by an object, detection is a delicate issue and segmentation suffers from contour discontinuities. However, with our local study, each area is independently studied. Thus, each segment can be locally likely, even though it is not linked to other likely segments.

If the studied image has too cluttered background (Figure 6c), edges are numerous. HOGs are so locally perturbed and do not permit recognition.

Illumination can pose problems. In example (d) of Figure 6, only a part of the leg is lighted. An edge matches the leg and the background boundary while another one matches the lighted and dark area boundary. These two edges are close and parallel. Our method can not recognize which one belongs to the silhouette.

Conclusion

This paper presented a method for quantifying the likelihood of contour segment as being part of a human silhouette. To do that, in a positive detection window, the HOGs that were previously computed for people detection give a local description of his/her silhouette. The main perspective of this work is to use this local description to perform accurate segmentation. An oriented graph may be created with all the contour segments. The graph edges will be weighted with values found by our method. A shortest-path algorithm, such as Dijkstra's algorithm, will find the most likely contour segment cycle representing the person's silhouette.

Fig. 1 .

 1 Fig. 1. Input image (a), contour image computed with Canny algorithm (b), contour pixels gathered in contour segments (c) and likely segments (d).

Fig. 2 .

 2 Fig. 2. The "descriptor/classifier" framework principle (a) and image partitioned in blocks (in red) and cells (in blue) with no block overlapping (b).

Fig. 4 .

 4 Fig. 4. Effects of block overlapping: SVM outputs and likely segments with block overlapping (a) and same results without block overlapping (b).

Fig. 5 .

 5 Fig. 5. Cell and block size effects on the algorithm performance (a) and normalization effects on the algorithm performance (b).

Fig. 6 .

 6 Fig. 6. If a silhouette part does not belong to the computed edges (right arm), segmentation is difficult (a). With occlusion, the calfs are not detected but it does not perturb the processing of feet (b). Cluttered background increases difficulty (c). Illumination variations provide two parallel edges in the leg (d).

Fig. 7 .

 7 Fig. 7. First column: INRIA's database images. Second column: SVM outputs. Third column: likely segments obtained by our method with block overlapping, L2-normalization and blocks of 4 cells of 25 pixels.


Panikos Heracleous
email: panikos.heracleous@gipsa-lab.grenobe-inp.fr

Denis Beautemps
email: denis.beautemps@gipsa-lab.grenobe-inp.fr

Noureddine Aboutabit
email: noureddine.aboutabit@gipsa-lab.grenobe-inp.fr

Cued Speech Recognition for Augmentative Communication in Normal-hearing and Hearing-impaired Subjects

Keywords: Cued Speech, hidden Markov models, automatic recognition

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

To date, visual information is widely used to improve speech perception or automatic speech recognition (lipreading). With lipreading technique, speech can be understood by interpreting movements of lips, face and tongue. In spoken languages, a particular facial and lip shape corresponds to each sound (phoneme). However, this relationship is not one-to-one, and many phonemes share the same facial and lip shape (visemes). It is impossible, therefore to distinguish phonemes using visual information alone.

Even with high lipreading performances, speech cannot be thoroughly perceived without knowledge about the semantic context. To date, the best lip readers are far way of reaching perfection. On average, only 40 to 60% of the vowels of a given language (American English) are recognized by lipreading [START_REF] Montgomery | Physical characteristics of the lips underlying vowel lipreading performance[END_REF], and 32% when relating to low predicted words [START_REF] Nicholls | Cued speech and the reception of spoken language[END_REF]. The best result obtained amongst deaf participants was 43.6% for the average accuracy [START_REF] Auer | Enhanced visual speech perception in individuals with early-onset hearing impairment[END_REF][START_REF] Bernstein | Lipreading, the lexicon, and cued speech[END_REF]. The main reason for this lies in the ambiguity of the visual pattern. However, as far as the orally educated deaf people are concerned, the act of lipreading remains the main modality of perceiving speech.

To overcome the problems of lipreading and to improve the reading abilities of profoundly deaf children, in 1967 Cornett [START_REF] Cornett | Cued speech[END_REF] developed the Cued Speech system to complement the lip information and make all phonemes of a spoken language clearly visible. As many sounds look identical on lips (e.g., /p/, /b/, and /m/), using hand information those sounds can be distin-Figure 1: Hand positions for vowels (top) and handshapes for consonants (bottom) in Cued Speech for French guished, and thus make possible for deaf people to completely understand a spoken language using visual information only.

Cued Speech (also referred to as Cued Language [START_REF] Fleetwood | Cued language structure: An analysis of cued american english based on linguistic principles[END_REF]) uses handshapes placed in different positions near the face in combination with natural speech lipreading to enhance speech perception from visual input. This is a system where the speaker faces the perceiver and moves his hand in close relation with speech. The hand, held flat and oriented so that the back of the hand faces the perceiver, is a cue that corresponds to a unique phoneme when associated with a particular lip shape. A manual cue in this system contains two components: the handshape and the hand position relative to the face. Handshapes distinguish among consonant phonemes whereas hand positions distinguish among vowel phonemes. A handshape together with a hand position cue a syllable.

Cued Speech improves speech perception for deaf people [START_REF] Nicholls | Cued speech and the reception of spoken language[END_REF][START_REF] Uchanski | Automatic speech recognition to aid the hearing impaired: Prospects for the automatic generation of cued speech[END_REF]. Moreover, for those who have been exposed to this method since their youth, offers a thorough representation of the phonological system, and therefore it has a positive impact on language development [START_REF] Leybaert | Phonology acquired through the eyes and spelling in deaf children[END_REF]. Fig. 1 describes the complete system for French. In Cued Speech for French, eight handshapes in five positions are used. The system was adapted from American English to French in 1977. To date, Cued Speech is adapted to more than 60 languages.

Another widely used communication method for deaf individuals is the Sign Language [START_REF] Dreuw | Speech recognition techniques for a sign language recognition system[END_REF][START_REF] Ong | Automatic sign language analysis: A survey and the future beyond lexical meaning[END_REF]. Sign Language is a language with its own grammar, syntax and community; however, Cued Speech is a visual representation of a spoken language, and it was developed to help raise the literacy levels of deaf individuals. Cued Speech was not developed to replace Sign Language. In fact, Sign Language will be always a part for deaf community. On the other hand, Cued Speech is an alternative communication method for deaf individuals. By cueing, children who are deaf would have a way to easily acquire the native home language, read and write proficiently, and more easily communicate with hearing family members who cue.

Access to communication technologies has become essential for handicapped people. The current study is a part of the TELMA project (Phone for deaf people) aiming at developing an automatic translation system of acoustic speech into visual speech completed with Cued Speech and vice versa, i.e. from Cued Speech components into auditory speech [START_REF] Beautemps | Telma: Telephony for the hearing-impaired people. from models to user tests[END_REF]. This project would enable deaf users to communicate with each others and with normal-hearing people through the help of the autonomous terminal TELMA. In this context, the automatic translation of Cued Speech components into a phonetic chain is a key issue. The Cued Speech system allows both hand and lip flows to convey a part of the phonetic information. Therefore, in order to recover the complete phonetic and lexical information, lip and hand components should be used jointly.

Previously, the authors reported experimental results on vowel-and consonant recognition in Cued Speech for French based on HMMs [START_REF] Heracleous | Lip shape and hand location fusion for vowel recognition in cued speech for french[END_REF][START_REF] Heracleous | Vowel and consonant automatic recognition in cued speech for french[END_REF], and using fusion [START_REF] Nefian | A coupled hmm for audio-visual speech recognition[END_REF][START_REF] Nakamura | Multi-modal temporal asynchronicity modeling by product hmms for robust[END_REF][START_REF] Adjoudani | On the integration of auditory and visual parameters in an hmm-based asr[END_REF] to integrate the lip shape and handshape components. In the case of a normal-hearing cuer, a vowel accuracy of 87.6% and a consonant accuracy of 79.6% were achieved. In the current study, a deaf cuer was also employed, and both cuer-dependent and multi-cuer isolated word recognition experiments in Cued Speech were conducted.

Methodology

Cued Speech Materials

In the data recording, a deaf and a normal-hearing female cuers were employed. The normal-hearing cuer was certified in transliteration speech into Cued Speech in the French language. She regularly cues in schools. The deaf speaker -also speech-impaired-uses Cued Speech to communicate with her family's members.

A camera with a zoom facility used to shoot the hand and face was connected to a betacam recorder. The speakers' lips were painted blue, and color marks were placed on the speakers' fingers. These constraints were applied in recordings in order to control the data and facilitate the extraction of accurate features. Fig. 2 shows the photos of the two cuers, and also the landmarks used in feature extraction. The data were derived from a video recording of the cuers pronouncing and coding in Cued Speech a set of 50 French isolated words, each one repeated 29 times. In the case of the deaf cuer, the vocabulary was extended up to 100 words by additionally recording another 50 words. Each new word was repeated 15 times.

An automatic image processing method was applied to the video frames in the lip region to extract their inner-and outer contours and to derive the corresponding characteristic parameters: lip width (A), lip aperture (B), and lip area (S) (i.e., six parameters in all). The automatic process resulted in a set of temporally coherent signals: the 2D hand information, the lip width (A), the lip aperture (B), and the lip area (S) values for both inner-and outer contours. In addition, two supplementary parameters relative to the lip morphology were extracted: the pinching of the upper lip (Bsup) and lower (Binf) lip. As a result, a set of eight parameters in all was extracted for modeling lip shapes. For hand position modeling, the xy coordinates of the two landmarks placed on the hand were used (i.e., 4 parameters). For handshape modeling the xy coordinates of the landmarks placed on the fingers were used (i.e., 10 parameters). Fig. 3 shows the lip shape parameters used in the current study, and Fig. 4 shows the x-coordinates of the fingers during the /V IN GT HU IT / word production, as tracked by the automatic image processing system.

Lip shape and handshape modeling

In the experiments, context-independent whole-word models were used. A 6-state, left-to-right with no skip HMM topology was used. Each state was modeled with 4 Gaussian distributions. In addition to the basic lip and hand parameters, the first (∆) and second derivatives (∆∆) were also used. For training and test 750 and 700 words were used, respectively.

In automatic speech recognition, a diagonal covariance matrix is often used because of the assumption that the parameters are uncorrelated. In lipreading, however parameters show a strong correlation. In this study, Principal Component Analysis (PCA) was applied to decorrelate the lip shape parameters, and then a diagonal covariance matrix was used. All 24 PCA lip shape components were used for HMM training. For training and recognition the HTK3.1 toolkit was used.

Concatenative feature fusion

The feature concatenation uses the concatenation of the synchronous lip shape and hand position features as the joint bimodal feature vector

O LH t = [O (L) T t , O (H) T t ] T ∈ R D (1)
where

O LH t
is the joint lip-hand feature vector, O (L) t the lip shape feature vector, O (H) t the hand feature vector, and D the dimensionality of the joint feature vector. In these experiments, the dimension of the lip shape stream was 24 (8 static parameters, 8 ∆, and 8 ∆∆ parameters). The dimension of the handshape stream was 30, and the D dimension was, therefore, 54.

Multistream HMM decision fusion

Decision fusion captures the reliability of each stream, by combining the likelihoods of single-modality HMM classifiers. Such an approach has been used in multi-band audio only ASR [START_REF] Bourlard | A new asr approach based on independent processing and recombination of partial frequency bands[END_REF] and in audio-visual speech recognition [START_REF] Potamianos | Recent advances in the automatic recognition of audiovisual speech[END_REF]. The emission likelihood of multistream HMM is the product of emission likelihoods of single-modality components weighted appropriately by stream weights. Given the O joint observation vector, i.e., lip shape and hand position component, the emission probability of multistream HMM is given by bj

(Ot) = S Y s=1 [ Ms X m=1 cjsmN (Ost; µjsm, Σjsm)] λs (2)
where N (O; µ, Σ) is the value in O of a multivariate Gaussian with mean µ and covariance matrix Σ, and S the number of streams. For each stream s, Ms Gaussians in a mixture are used, with each weighted with cjsm. The contribution of each stream is weighted by λs. In this study, we assume that the stream weights do not depend on state j and time t. However, two constraints were applied. Namely,

0 ≤ λ h , λ l ≤ 1 and λ h + λ l = 1 (3)
where λ h is the hand position stream weight, and λ l is the lip shape stream weight. The HMMs were trained using maximum likelihood estimation based on the Expectation-Maximization (EM) algorithm. However, the weights cannot be obtained by maximum likelihood estimation. In these experiments, the weights were adjusted to 0.4 and 0.6 values, respectively. The selected weights were obtained experimentally by maximizing the accuracy on a held-out data.

Experiments

In this section, cuer-dependent and multi-cuer isolated word recognition experiments in both normal-hearing and deaf subjects are presented.

Automatic recognition in the normal-hearing subject

Table 1 shows the results obtained in the case of the normalhearing cuer. The recognition rate when using lip parame- ters was 69.8%, and 90.7% when using handshape parameters. When lip shape and handshape components where integrated, a 94.8% recognition rate was obtained. The recognition rate was raised to 95.2% when multistream HMM decision fusion was applied. The obtained results are promising, and show the effectiveness of integrating handshape and lip shape in order to realize automatic recognition of Cued Speech.

Automatic recognition in the deaf subject

Table 2 shows the results obtained in the case of the deaf cuer. In this experiment, the same vocabulary as in the normal- hearing cuer's case was used. The recognition rate was 76.1% when using lip shape parameters, and 76.3% when handshape parameters were used. When lip shape and handshape components were fused, the recognition rate was 89% in the case of using feature fusion, and 92% in the case of multistream HMM decision fusion, respectively. The results show a variability between the two cuers. Concerning the differences in handshape recognition, a possible reason might be the fact that the normalhearing subject is a professional teacher of Cued Speech, and therefore cues more accurately. However, the recognition rates when fusion was applied are still closely comparable. Table 3 shows the results obtained, when a 100-word vocabulary was used. By increasing the vocabulary, the recognition rates remained almost unchanged. In fact, a modest increase in the recognition rate can be observed. It should be noted, however, that in this pilot study the vocabularies used are still small. It may be happened, that in the case of larger vocabularies significant differences will be obtained. 77.0 92.9

Multi-cuer automatic recognition

To investigate whether it is possible to train cuer-independent HMMs using a large number of subjects, an experiment based on concatenative feature fusion was conducted using data from both the normal-hearing and the deaf subjects. The vocabulary was 100 words, and for training and test 1900 and 1750 words were used, respectively. Table 4 shows the achieved results. When using multi-cuer HMMs, the recognition rate for the deaf cuer was 88.4%, slightly lower as compared to the 89.9% recognition rate when deaf HMMs were used. In the case of the normal-hearing cuer, the recognition rates remained equal. The results obtained indicate that there should not be particular difficulties in creating cuer-independent HMMs using a larger number of subjects.

Conclusions

In this study, isolated word recognition experiments in Cued Speech for French in both normal-hearing and deaf subjects were presented. Using feature fusion and multistream HMM decision fusion, lip shape and handshape components were integrated into a combined component, and automatic recognition experiments were conducted. In the case of a normal-hearing cuer, a 95.2% recognition rate, and in the case of a deaf cuer, a recognition rate of 92.0% was obtained. In addition to cuerdependent experiments, a multi-cuer experiment was also conducted showing a 91.1% recognition rate. The results obtained are promising, and show the effectiveness of the proposed methods in the automatic recognition of Cued Speech for French using visual information alone. Currently, data recording from several cuers are in progress in order to realize continuous Cued Speech recognition in the framework of the TELMA project.

Figure 2 :

 2 Figure 2: (a) Photo of the normal-hearing cuer; (b) photo of the deaf cuer

Figure 3 :

 3 Figure 3: Parameters used for lips shape modeling.

Figure 4 :

 4 Figure 4: Finger x-coordinates during the /V IN GT HU IT / word production by the deaf cuer.

Table 1 :

 1 Recognition rates for a 50-word vocabulary in the case of the normal-hearing cuer.

	Fusion		Component
		Lips Hand Lips + Hand
	Feature	69.8	90.7	94.8
	Multistream 69.8	90.7	95.2

Table 2 :

 2 Recognition rates for a 50-word vocabulary in the case of the deaf cuer.

	Fusion		Component
		Lips Hand Lips + Hand
	Feature	76.1	76.3	89.0
	Multistream 76.1	76.3	92.0

Table 3 :

 3 Recognition rates for a 100-word vocabulary in the case of the deaf cuer.

	Fusion		Component
		Lips Hand Lips + Hand
	Feature	68.4	77.0	89.9
	Multistream 68.4

Table 4 :

 4 Recognition rates for a multi-speaker experiment using a 100-word vocabulary.

	Test set		HMMs
		Deaf Normal Deaf + Normal
	Deaf	89.9	-	88.6
	Normal	-	94.8	94.8
	Deaf + Normal	-	-	91.1

Acknowledgments

The authors would like to thank the volunteer cuers Myriam Diboui, and Clémentine Huriez for their time spending on Cued Speech data recording, and also for accepting the recording constraints. Also the authors would like to thank Christophe Savariaux and Coriandre Vilain for their help in the Cued Speech material recording. This work is supported by the TELMA project (ANR, 2005 edition).


Pierre Cardaliaguet
email: pierre.cardaliaguet@univ-brest.fr

Catherine Rainer
email: catherine.rainer]@univ-brest.fr

Hölder regularity for viscosity solutions of fully nonlinear, local or nonlocal, Hamilton-Jacobi equations with super-quadratic growth in the gradient *

Keywords: Integro-Differential Hamilton-Jacobi equations, viscosity solutions, Hölder continuity, degenerate parabolic equations, reverse Hölder inequalities. MSC Subject classifications: 49L25, 35K55, 93E20, 26D15

Viscosity solutions of fully nonlinear, local or non local, Hamilton-Jacobi equations with a super-quadratic growth in the gradient variable are proved to be Hölder continuous, with a modulus depending only on the growth of the Hamiltonian. The proof involves some representation formula for nonlocal Hamilton-Jacobi equations in terms of controlled jump processes and a weak reverse inequality.

Introduction

In a previous paper [START_REF] Cannarsa | Hölder estimates in space-time for viscosity solutions of Hamilton-Jacobi equations[END_REF], the first author investigated the regularity of solutions to the Hamilton-Jacobi equation u t (x, t) -Tr a(x, t)D 2 u(x, t) + H(x, t, Du(x, t)) = 0 in R N × (0, T )

under a super-quadratic growth condition on the Hamiltonian H with respect to the gradient variable:

1 δ |z| q -δ ≤ H(x, t, z) ≤ δ|z| q + δ ∀(x, t, z) ∈ R N × (0, T ) × R N ,
for some δ ≥ 1, q ≥ 2. Under this assumption, it is proved in [START_REF] Cannarsa | Hölder estimates in space-time for viscosity solutions of Hamilton-Jacobi equations[END_REF] that any continuous, bounded solution u of ( 1) is Hölder continuous on R N × [τ, T ] (for any τ ∈ (0, T )), with 1 Hölder exponent and constant only depending on N, p, δ, q, τ and u ∞ . In particular, such a modulus of continuity is independent of the regularity of a and of H with respect to the variables (x, t). The result is somewhat surprizing since no uniform ellipticity on a is required.

The aim of this paper is to extend this regularity result to solutions of fully nonlinear, local or nonlocal, Hamilton-Jacobi equations which have a super-quadratic growth with respect to the gradient variable. Beside its own interest, such a uniform estimate is important in homogenization theory where, for instance, it is used to prove the existence of correctors.

Let us consider a fully nonlinear, nonlocal Hamilton-Jacobi equation of the form

u t + F (x, t, Du, [u]) = 0 in R N × (0, T ) . (2)
In the above equation we assume that the mapping F : R N × (0, T ) × R N × C 2 b (R N ) → R is nonincreasing with respect to the nonlocal variable, i.e., [φ ≤ ψ and φ(x) = ψ(x)] ⇒ F (x, t, ξ, [φ]) ≥ F (x, t, ξ, [ψ]) for any function φ, ψ ∈ C 2 b (R N ). Let us recall (see [START_REF] Alvarez | Viscosity solutions of nonlinear integro-differential equations[END_REF][START_REF] Arisawa | A remark on the definitions of viscosity solutions for the integrodifferential equations with Lvy operators[END_REF][START_REF] Barles | Backward stochastic differential equations and integral-partial differential equations[END_REF][START_REF] Barles | Second-order elliptic integro-differential equations: viscosity solutions' theory revisited[END_REF][START_REF] Pham | Optimal stopping of controlled jump diffusion processes: A viscosity solution approach[END_REF][START_REF] Sayah | Equations d'Hamilton-Jacobi du premier ordre avec termes intégrodifférentiels. I. Unicité des solutions de viscosité[END_REF][START_REF] Sayah | Equations d'Hamilton-Jacobi du premier ordre avec termes intégrodifférentiels. II. Existence de solutions de viscosité[END_REF][START_REF] Soner | Optimal control of jump-Markov processes and viscosity solutions. Stochastic differential systems, stochastic control theory and applications[END_REF] for instance) that a subsolution (resp. a supersolution) of equation ( 2) is a continuous map u : R N × [0, T ] → R such that, for any continuous, bounded test function φ : R N × (0, T ) → R, which has continuous second order derivatives and such that u -φ has a global maximum (resp. global minimum) at some point (x, t), one has φ t (x, t) + F (x, t, Dφ(x, t), [φ(•, t)]) ≤ 0 (resp. ≥ 0) .

Our main assumption on F is the following structure condition, which roughly says that F is super-quadratic with respect to the gradient variable:

-δM + [φ](x) + 1 δ |ξ| q -δ ≤ F (x, t, ξ, [φ]) ≤ -δM -[φ](x) + δ|ξ| q + δ (3)
for any (x, t, ξ, φ) ∈ R N × (0, T ) × R N × C 2 b (R N ), for some constants q > 2 and δ ≥ 1, where M -and M + are defined by and where B is the unit ball of R N . Let us note that, under the above assumption, a solution of ( 2) is a supersolution of

u t -δM -[u(•, t)](x) + δ|Du| q + δ = 0 in R N × (0, T ) (4)
and a subsolution of

u t -δM + [u(•, t](x) + 1 δ |Du| q -δ = 0 in R N × (0, T ) . (5)
We denote by p the conjugate exponent of q. We are interested in solutions which are bounded by some constant M:

|u(x, t)| ≤ M ∀(x, t) ∈ R N × [0, T ] . (6)
In what follows, a (universal) constant is a positive number depending on the given data q, δ, N and M only. Universal constants will be typically labeled with C, but also with different letters (e.g., θ, A, . . . ). Dependence on extra quantities will be accounted for by using parentheses (e.g., C(r) denotes a constant depending also on r). The constant C appearing in the proofs may change from line to line.

Theorem 1.1 Let u ∈ C(R N ×[0, T ]
) be a viscosity supersolution of (4) and a subsolution of [START_REF] Barles | Second-order elliptic integro-differential equations: viscosity solutions' theory revisited[END_REF],

such that |u| ≤ M in R N × [0, T ].
For any τ ∈ (0, T ), there are constants θ = θ(δ, M, N, q) > p and C(τ ) = (τ, δ, M, N, q) > 0 such that

|u(x 1 , t 1 ) -u(x 2 , t 2 )| ≤ C(τ ) |x 1 -x 2 | (θ-p)/(θ-1) + |t 1 -t 2 | (θ-p)/θ (7)
for any

(x 1 , t 1 ), (x 2 , t 2 ) ∈ R N × [τ, T ].
The main point of the above result is that (7) holds true uniformly with respect to F , as long as conditions [START_REF] Cannarsa | Hölder estimates in space-time for viscosity solutions of Hamilton-Jacobi equations[END_REF] and the bound |u| ≤ M are satisfied. In particular, θ and C(τ ) are independent of the continuity modulus of F . In contrast to [START_REF] Caffarelli | Regularity theory for fully nonlinear integro-differential equations[END_REF][START_REF] Silvestre | Hölder estimates for solutions of integro-differential equations like the fractional Laplace[END_REF], F can also be degenerate parabolic.

Let us note that the above result also applies to the solutions of the fully nonlinear, local equation

u t + F (x, t, Du, D 2 u) = 0 in R N × (0, T ) (8)
provided F : R N × (0, T ) × R N × S N → R is nonincreasing with respect to the matrix variable and satisfies the following structure condition:

-δΛ(X) + 1 δ |ξ| q -δ ≤ F (x, t, ξ, X) ≤ -δλ(X) + δ|ξ| q + δ (9)
for any (x, t, ξ, X) ∈ R N × (0, T ) × R N × S N , for some constants q > 2 and δ ≥ 1, where

Λ(X) = max |z|≤1 Xz, z and λ(X) = min |z|≤1 Xz, z ∀X ∈ S N ,
(S N being the set of N × N symmetric matrices). Indeed, since

M -[φ](x) ≤ λ(D 2 φ(x)) and M + [φ](x) ≥ Λ(D 2 φ(x)) ,
any solution of ( 8) is a supersolution of (4) and a subsolution of [START_REF] Barles | Second-order elliptic integro-differential equations: viscosity solutions' theory revisited[END_REF]. Note that F is neither required to be concave nor convex with respect to the matrix variable.

Here are some examples of nonlinear, nonlocal Hamilton-Jacobi equations satisfying the structure condition (3) (see [START_REF] Barles | Backward stochastic differential equations and integral-partial differential equations[END_REF] or [START_REF] Barles | Second-order elliptic integro-differential equations: viscosity solutions' theory revisited[END_REF] for instance): let us assume that

F (x, t, ξ, [φ]) = I[φ](x, t) + H(x, t, ξ)
where H is a first order term with superquadratic growth:

1 δ |ξ| q -δ ≤ H(x, t, ξ) ≤ δ|ξ| q + δ
and where the nonlocal term I can be of the form

I[φ](x, t) = inf α∈A sup β∈B R N φ(x + j α,β (x, t, e)) -φ(x) -Dφ(x), j α,β (x, t, e) dν(e)
where A, B are some sets, j α,β : R N × (0,

T ) × R N → R N is such that |j α,β (x, t, e)| ≤ C(|e| ∧ 1) ∀(x, t, e, α, β) ∈ R N × (0, T ) × R N × A × B
and the measure ν satisfies

R N |e| 2 ∧ 1 dν(e) ≤ C , (10)
or of the form

I[φ](x, t) = inf α∈A sup β∈B R N φ(x + j α,β (x, t, e)) -φ(x) -Dφ(x), j α,β (x, t, e) 1 B (e) dν(e)
where j α,β : R N × (0, T ) × R N → R N is now of linear growth

|j α,β (x, t, e)| ≤ C|e| ∀(x, t, e, α, β) ∈ R N × (0, T ) × R N × A × B
and the measure ν again satisfies the integrability condition [START_REF] Crandall | User's guide to viscosity solutions of second order Partial Differential Equations[END_REF]. In this later case, the part B φ(x + j α,β (x, t, e)) -φ(x) -Dφ(x), j α,β (x, t, e) 1 B (e) dν(e) can be estimated from above and below by M + [φ] and M -[φ], while the part

R N \B φ(x + j α,β (x, t, e)) -φ(x) dν(e)
is can be bounded-in equation (2)-by CM, where M = u ∞ and C is universal.

Some comments on the proof of Theorm 1.1 are now in order. As in [START_REF] Cannarsa | Hölder estimates in space-time for viscosity solutions of Hamilton-Jacobi equations[END_REF], the main ingredients are representation formulae for simplified Hamilton-Jacobi equations, existence of "nearly optimal trajectories", use of Brownian bridges and, finally, application of a reverse Hölder inequality. However, since we work with fully nonlinear, nonlocal equations, each step is technically more involved: the representation formulae (see Proposition 2.1 or the proof of Proposition 3.1) are inspired by a work on controlled structure equations by the second author [START_REF] Buckdahn | Stochastic control problems for systems driven by normal martingales[END_REF]. They involve controlled jump processes in a particular form. The estimates of the subsolutions in Proposition 3.1, which, in [START_REF] Cannarsa | Hölder estimates in space-time for viscosity solutions of Hamilton-Jacobi equations[END_REF], are obtained by controls issued from Brownian bridge techniques, have to be built here in a much more subtle way: indeed the Hamiltonian of equation ( 4) being non convex, the naturally associated control problem should actually be a differential game, which has never been investigated in this framework. We overcome this difficulty by building explicit feedbacks. Finally, the construction of optimal trajectories in Lemma 2.4, requires careful estimates because we are dealing with jump processes.

Notations : Throughout the paper, B denotes the closed unit ball of R N , B(B\{0}) the set of Borel measurable subsets of B\{0}, C(R N ×[0, T ]) the set of continuous functions on R N ×[0, T ] and C 2 b (R N ) the set of bounded continuous functions on R N with continuous second order derivatives.

Analysis of supersolutions

Let u be as in Theorem 1.1. Throughout the proof of Theorem 1.1 it will be more convenient to work with u(x, T -t) instead of u(x, t). We note that u(x, T -t) is a supersolution of

-v t -δM -[v(•, t)](x) + δ|Dv| q + δ = 0 in R N × (0, T ) (11)
and a subsolution of

-v t -δM + [v(•, t)](x) + 1 δ |Dv| q -δ = 0 in R N × (0, T ) . ( 12
)
To simplify the notation we will write u(x, t) instead of u(x, T -t).

In this part we are concerned with some monotonity property along particular trajectories of supersolution of equation [START_REF] Lio | Uniqueness results for convex Hamilton-Jacobi equations under p > 1 growth conditions on data[END_REF]. For this we have to give a representation formula for solutions of this equation in terms of controlled jump processes.

Let (Ω, F , P) be a complete probability space on which is defined a N-dimensional Poisson random measure µ. We assume that the Levy measure of µ, denoted by ν, is supported in B, has no atom and satisfies the conditions

We denote by μ(de, dt) = µ(de, dt) -ν(de)dt the compensated Poisson measure. For all t ∈ [0, T ], (F t,s , s ∈ [t, T ]) will denote the filtration generated by µ on the interval [t, T ], i.e. F t,s = σ{µ([t, r] × A), t ≤ r ≤ s, A ∈ B(R N )} completed by all null sets of P .

Let A(t) be the set of (F t,s )-adapted controls (a

s ) = (λ s , b s ) : [t, T ] → (0, 1] × B\{0} and let L p ad ([t, T ]) be the set of (F t,s )-adapted controls (ζ s ) : [t, T ] → R N such that E T t |ζ s | p ds < +∞.
To all control (a s ) = (λ s , b s ) ∈ A(t) we associate a (F t,s )-martingale M a in the following way: First we set

ρ s = inf r > 0 ; ν(B\B(0, r)) ≤ δ 1 |b s | 2 , s ∈ [t, T ].
By the assumptions (13) on the measure ν, the process (ρ s ) is well defined. It takes its values in [0, 1] and is adapted to the filtration (F t,s ). Then we introduce A s = A s (a) = B\B(0, ρ s ). For all s ∈ [0, T ], the set A s belongs to B(B\{0}) ⊗ F t,s and, since ν has no atoms, it satisfies

ν(A s ) = δ |b s | 2 for almost all s ∈ [t , T ], P-a.s. (14)
We finally denote by M a the controlled martingale

M a s = s t B λ r b r 1 Ar (e)μ(de, dr) . (15)
Let us precise Itô's formula satisfied by M a : for any smooth function

φ ∈ C 2 b , E[φ(M a s )] = φ(0) + E s t B (φ(M a r -+ λ r b r 1 Ar (e)) -φ(M a r -) -Dφ(M a r -), λ r b r 1 Ar (e) ) dν(e)dr = φ(0) + δE s t (φ(M a r -+ λ r b r ) -φ(M a r -) -Dφ(M a r -), λ r b r ) dr |b r | 2
Now we consider the controlled system

dY s = ζ s ds + dM a s , s ∈ [t, T ], Y t = x , (16)
where ζ ∈ L p ad ([t, T ]) and a ∈ A(t). The system ( 16) is related to equation ( 11) by the following proposition: Proposition 2.1 Let v : R N × [0, T ] be a continuous viscosity solution to [START_REF] Lio | Uniqueness results for convex Hamilton-Jacobi equations under p > 1 growth conditions on data[END_REF]. Then

v(x, t) = inf (ζ,a)∈L p ad ([t,T ])×A(t) E v(Y x,t,ζ,a T , T ) + C + T t |ζ s | p ds -δ(T -t) where (Y x,t,ζ,a s
) is the solution to ( 16) and C + > 0 is the universal constant given by

C + = δ -p/q pq p/q . ( 17
)
Proof : The proof relies on two arguments: first the map

w(x, t) = inf (ζ,a)∈L p ad ([t,T ])×A(t) E v(Y x,t,ζ,a T , T ) + C + T t |ζ s | p ds -δ(T -t)
is a viscosity solution of [START_REF] Lio | Uniqueness results for convex Hamilton-Jacobi equations under p > 1 growth conditions on data[END_REF]. This result has been proved-in a slightly different frameworkin [START_REF] Buckdahn | Stochastic control problems for systems driven by normal martingales[END_REF], and we omit the proof, which is very close to that of [START_REF] Buckdahn | Stochastic control problems for systems driven by normal martingales[END_REF]. Second, in order to conclude that w = v, we need a uniqueness argument for the solution of [START_REF] Lio | Uniqueness results for convex Hamilton-Jacobi equations under p > 1 growth conditions on data[END_REF] with terminal condition v(•, T ). This is a direct consequence of the following comparison principle.

Lemma 2.2 (Comparison)

Let u be a continuous, bounded subsolution of ( 11) on R N × [0, T ] and v be a continuous, bounded supersolution of [START_REF] Lio | Uniqueness results for convex Hamilton-Jacobi equations under p > 1 growth conditions on data[END_REF].

If u(•, T ) ≤ v(•, T ), then u ≤ v on R N × [0, T ].
Proof : We use ideas of [START_REF] Barles | Second-order elliptic integro-differential equations: viscosity solutions' theory revisited[END_REF] for the treatment of the nonlocal term and of [START_REF] Lio | Uniqueness results for convex Hamilton-Jacobi equations under p > 1 growth conditions on data[END_REF] for the treatment of the super-linear growth with respect to the gradient variable. Let M = max{ u ∞ , v ∞ }. Our aim is to show that, for any µ ∈ (0, 1),

ũ(x, t) := µu(x, t) -(1 -µ)M -(1 -µ)δ(T -t) ≤ v(x, t) in R N × [0, T ] . (18)
Note that this inequality holds at t = T . One also easily checks that ũ is a subsolution of equation

-w t -δM -[w(•, t)](x) + δµ 1-q |Dw| q + δ = 0 in R N × (0, T ) . ( 19
)
For any ǫ > 0, let ũǫ be the space-time sup-convolution of ũ and v ǫ be the space-time inf-convolution of v (see [START_REF] Crandall | User's guide to viscosity solutions of second order Partial Differential Equations[END_REF]):

ũǫ (x, t) = sup (y,s)∈R N ×[0,T ] ũ(y, s) - 1 ǫ |(x, t) -(y, s)| 2 and v ǫ (x, t) = inf (y,s)∈R N ×[0,T ] v(y, s) + 1 ǫ |(x, t) -(y, s)| 2 .
It is known that ũǫ is still a subsolution of [START_REF] Silvestre | Hölder estimates for solutions of integro-differential equations like the fractional Laplace[END_REF] in R N × ((2Mǫ)

1 2 , T -(2Mǫ) 1
2 ) while v ǫ a supersolution of [START_REF] Lio | Uniqueness results for convex Hamilton-Jacobi equations under p > 1 growth conditions on data[END_REF] and that u ǫ is semiconvex while v ǫ is semiconcave R N ×((2Mǫ)

1 2 , T - (2Mǫ) 1
2 ). In particular, u ǫ and v ǫ have almost everywhere a second order expansion and at such a point (x, t) ∈ R N × ((2Mǫ)

1 2 , T -(2Mǫ) 1 2 ) one has -ũ ǫ t (x, t) -δM -[ũ ǫ (•, t)](x) + δµ 1-q |Dũ ǫ (x, t)| q + δ ≤ 0 and -v ǫ t (x, t) -δM -[v ǫ (•, t)](x) + δ|Dv ǫ (x, t)| q + δ ≥ 0 .
As usual we prove (18) by contradiction and assume that sup x,t ũ(x, t)-v(x, t) > 0. Then, for any α > 0, σ > 0 sufficiently small, one can choose ǫ > 0 sufficiently small such that the map (x, t)

→ ũǫ (x, t) -v ǫ (x, t) -α|x| 2 + σt reaches its maximum on R N × [0, T ] at some point (x, t) ∈ R N × ((2Mǫ) 1 2 , T -(2Mǫ) 1 2 ). For any η > 0, the point (x, t) is a strict maximum of the map (x, t) → ũǫ (x, t) -v ǫ (x, t) -α|x| 2 + σt -η(|x -x| 2 + (t -t) 2 ).
Jensen's Lemma (see [START_REF] Crandall | User's guide to viscosity solutions of second order Partial Differential Equations[END_REF]) then states that one can find p n = (p n x , p n t ) ∈ R N +1 such that p n → 0 and the map

(x, t) → ũǫ (x, t) -v ǫ (x, t) -α|x| 2 + σt -η(|x -x| 2 + (t -t) 2 ) -p n x , x -p n t t
has a maximum at some point (x n , t n ) where ũǫ and v ǫ have a second order expansion. Note that (x n , t n ) → (x, t). At the point (x n , t n ) we have

ũǫ t (x n , t n ) = v ǫ t (x n , t n ) -σ + 2η(t n -t) + p n t , Dũ ǫ (x n , t n ) = Dv ǫ (x n , t n ) + 2αx n + 2η(x n -x) + p n x , -ũǫ t (x n , t n ) -δM -[ũ ǫ (•, t n )](x n ) + δµ 1-q |Dũ ǫ (x n , t n )| q + δ ≤ 0 , (20)
and

-v ǫ t (x n , t n ) -δM -[v ǫ (•, t n )](x n ) + δ|Dv ǫ (x n , t n )| q + δ ≥ 0 . ( 21
)
For the optimality conditions, we have, for any

x ∈ R N , ũǫ (x, t n ) ≤ v ǫ (x, t n )+ũ ǫ (x n , t n )-v ǫ (x n , t n )+α(|x| 2 -|x n | 2 )+η(|x-x| 2 -|x n -x| 2 )-p n x , x-x n so that M -[ũ ǫ (•, t n )](x n ) ≤ M -[v ǫ (•, t n )](x n ) + 2α + 2η .
Let us set ξ n = Dv ǫ (x n , t n ) and estimate the difference between ( 20) and ( 21): we get

σ -2η(t n -t) -p n t -2α -2η + δ µ 1-q |ξ n + 2αx n + 2η(x n -x) + p n x | q -|ξ n | q ≤ 0 .
When n → +∞, ξ n remains bounded since v ǫ is semi-concave. So we can assume that

ξ n → ξ α,ǫ with σ -2α -2η + δ µ 1-q |ξ α,ǫ + 2αx| q -|ξ α,ǫ | q ≤ 0 .
Since, ũ and v are bounded, so is α|x| 2 . So αx is bounded (in fact αx → 0 as α → 0). Then the above inequality implies that ξ α,ǫ is bounded, because since µ < 1 and q > 1. So, letting η, ǫ → 0 and then α → 0, we get that σ ≤ 0, which contradicts our assumption on σ.

Lemma 2.3 Let u ∈ C(R N × [0, T ]
) be a supersolution of [START_REF] Lio | Uniqueness results for convex Hamilton-Jacobi equations under p > 1 growth conditions on data[END_REF] satisfying |u| ≤ M in R N × (0, T ). Then, for all (x, t) ∈ R N × [0, T ], n ∈ N, R > 0 large and σ > 0 small, there exist a (Ft ,s )-adapted càdlàg process Y n and a control

ζ n ∈ L p ad ([ t, T ]) such that u(x, t) ≥ E u(Y n t , t) + C + t t |ζ n s | p ds -(δ + τ )(t -t) -c n (σ, R) ∀t ∈ [ t, T ] , ( 22
)
where

c n (σ, R) = C R -p + τ p-1 σ p + ω(σ) + (δ + τ )τ ,
with ω the modulus of continuity of u in B R (x) × [0, T ], and C an universal constant.

Proof: For any (y,

t) ∈ R N × [0, T ), ζ ∈ L p ad ([t, T ]), a ∈ A(t), let us denote by Y x,t,ζ,a the solution to dY s = ζ s ds + dM a s , t ≤ s ≤ T, Y t = x,
where the martingale M a is defined by [START_REF] Meyer | Tightness criteria for laws of semimartingales[END_REF]. Let us now fix an initial condition (x, t) ∈ R N × [0, T ]. For a large n ∈ N, we set τ = (T -t)/n and t k = t + kτ for k ∈ {0, . . . , n} .

We will build some controls ζ n ∈ L p ad ([ t, T ]) and a n ∈ A( t) such that the process Y n = Y x, t,ζ n ,a n satisfies the relation

u(x, t) ≥ E u(Y n t k , t k ) + C + t k t |ζ n s | p ds -(δ + τ )(t k -t) ∀k ∈ {1, . . . , n} , (23)
and then deduce from ( 23) that (Y n , ζ n ) also satisfy (22). We follow closely the construction in [START_REF] Cannarsa | Hölder estimates in space-time for viscosity solutions of Hamilton-Jacobi equations[END_REF]. For any k ∈ {1, . . . , n}, let v k be the solution of [START_REF] Lio | Uniqueness results for convex Hamilton-Jacobi equations under p > 1 growth conditions on data[END_REF], defined on the time interval [0, t k ], with terminal condition u(•, t k ). From the representation formula given in Proposition 2.1 we have, for all x ∈ R N ,

v k (x, t k-1 ) = inf (ζ,a)∈L p ad ([t k-1 ,t k ])×A(t k-1 ) E u(Y x,t k-1 ,ζ,a t k , t k ) + C + t k t k-1 |ζ s | p ds -δτ .
Since the filtration (F t k-1 ,s ) is generated by a random Poisson measure, the set

L p ad ([t k-1 , t k ])× A(t k-1
) is a complete separable space. Moreover u(•, t k ) is continuous. Therefore, thanks to the measurable selection theorem (see [START_REF] Aubin | Set-Valued Analysis[END_REF]), one can build Borel measurable maps

x → Z x,k and x → A x,k from R N to L p ad ([t k-1 , t k ]) and A(t k-1
) respectively, such that

v k (x, t k-1 ) ≥ E u(Y x,t k-1 ,Z x,k ,A x,k t k , t k ) + C + t k t k-1 |Z x,k s | p ds -(δ + τ )τ ∀x ∈ R N .
(24) We now construct ζ n , a n and Y n by induction on the time intervals [t k-1 , t k ): On [ t, t 1 ) we set ζ n t = Z x,1 t , a n = A t,1 and Y n = Y x, t,ζ n ,a n . Assume that ζ n , Y n and a n have been built on [ t, t k-1 ). Then we set

ζ n = Z Y n t k-1 ,k , a n = A Y n t k-1 ,k ,
and

Y n = Y x, t,ζ n ,a n on [t k-1 , t k ) .
(The process Y x, t,ζ n ,a n is P -a.s. continuous on each fixed t, so we have

Y n t k-1 -= Y n t k-1
P -a.s., which means that ζ n and A n are defined P -as surely. )

We remark that, on [t k-1 , t k ), we have

Y n = Y Y n t k-1 ,t k-1 ,ζ n ,a n
. Let us fix now some k ∈ {1, . . . , n}. Since the processes A x,k and Z x,k are (F t k-1 ,s )adapted and therefore independent of Ft ,t k-1 , the same holds also for

M A x,k t k and finally for Y x,t k-1 ,Z x,k ,A x,k t k , while Y n t k-1 is Ft ,t k-1 -measurable. It follows that E u(Y x,t k-1 ,Z x,k ,A x,k t k , t k ) + C + t k t k-1 |Z x,k s | p ds x=Y n t k-1 = E u(Y n t k , t k ) + C + t k t k-1 |ζ n s | p ds Ft ,t k-1 .
Using (24), the fact that u is a supersolution of (11) and the comparison Lemma 2.2, this leads to the relation

u(Y n t k-1 , t k-1 ) ≥ E u(Y n t k , t k ) + C + t k t k-1 |ζ n s | p ds -(δ + τ )τ Ft ,t k-1 P -a.s. .
Taking the expectation on both sides of the above inequality and summing up gives (23).

We now extend this inequality to the full interval [ t, T ] and prove (22). Let t ∈ [ t, T ] and k be such that t ∈ [t k-1 , t k ). From (23), we have

u(x, t) ≥ E u(Y n t , t) + C + t t |ζ n s | p ds -(δ + τ )(t -t) +E u(Y n t k-1 ) -u(Y n t , t) -(δ + τ )τ (25)
Let us fix R > 0 and σ > 0. Since u is bounded by M and from the definition of the modulus ω, we have

E u(Y n t k-1 ) -u(Y n t , t) ≤ ω(σ)P |Y n t k -x| ≤ R, |Y n t -x| ≤ R, |Y n t k -Y n t | ≤ σ +2M(P |Y n t k -x| > R + P [|Y n t -x| > R] + P |Y n t k -Y n t | > σ ) (26)
To estimate the right hand side term of (26), we first note that, for any 0 < s < t, it holds that

E [|Y n t -Y n s | p ] ≤ 2 p-1 E t s ζ n τ dτ p + E [|M an t -M an s | p ]
But, thanks to (23) again, we have

E T t |ζ n τ | p dτ ≤ 2M + (δ + τ )T ≤ C (27)
so that, by Hölder's inequality,

E t s ζ n τ dτ p ≤ C(t -s) p-1 .
Also by Hölder we have

E [|M an t -M an s | p ] ≤ E |M an t -M an s | 2 p/2
where, by Itô,

E |M an t -M an s | 2 = E t s B λ 2 s |b s | 2 1 As (e)dν(e)ds = δE t s λ 2 s ds ≤ δ(t -s) (28)
To summarize

E [|Y n t -Y n s | p ] ≤ C((t -s) p-1 + (t -s) p/2 ) ≤ C(t -s) p-1
since p < 2. Therefore we get

P[|Y n t k -x| > R] + P[|Y n t -x| > R] + P[|Y n t -Y n t k | > σ] ≤ C(R -p + |t -t k | p-1 σ p )
which, coming back to (25) and ( 26), proves claim (22).

Lemma 2.4 Let u ∈ C(R N × [0, T ]) be a supersolution of [START_REF] Lio | Uniqueness results for convex Hamilton-Jacobi equations under p > 1 growth conditions on data[END_REF] satisfying |u| ≤ M in R N × (0, T ). Then, for any (x, t) ∈ R N × (0, T ) there is a stochastic basis ( Ω, F, P), a filtration ( Ft ) t≥ t, a càdlàg process ( Ȳt ) adapted to ( Ft ) t≥ t and a process ζ

∈ L p ad ([ t, T ]) such that u(x, t) ≥ E u( Ȳt , t) + C + t t | ζs | p ds -δ(t -t) ∀t ∈ ( t, T ) , ( 29
)
where C + > 0 is the universal constant given by [START_REF] Sayah | Equations d'Hamilton-Jacobi du premier ordre avec termes intégrodifférentiels. I. Unicité des solutions de viscosité[END_REF], and

E Ȳt -x - t t ζs ds r ≤ δ r 2 |t -t| r/2 ∀t ∈ [ t, T ) ( 30
)
for any r ∈ (0, 2].

Proof: This Lemma will follow from Lemma 2.3 by passing to the limit as n → +∞ in (22). For this we set

Λ n t = t t ζ n s ds ∀t ∈ [ t, T ] .
From ( 27), the sequence of probability measures (P

Λ n ) on C([ t, T ], R N ) is tight. Let D( t)
be the set of càdlàg functions from [ t, T ] to R N , endowed with the Meyer-Zheng topology (see [START_REF] Meyer | Tightness criteria for laws of semimartingales[END_REF]). Since E[|M an T |] is uniformly bounded (thanks to (28)), Theorem 4 of [START_REF] Meyer | Tightness criteria for laws of semimartingales[END_REF] states that the sequence of martingale measures (P M an ) is tight on D( t). Then, from Prohorov's Theorem (Theorem 4.7 of [START_REF] Karatzas | Brownian motion and stochastic calculus[END_REF]), we can find a subsequence of (Y n , Λ n ), again labeled (Y n , Λ n ), and a measure m on C([ t, T ], R N ) × D( t) such that (P (Y n , Λ n ) ) weakly converges to m. Skorokhod's embedding Theorem (Theorem 2.4 of [START_REF] Prato | Stochastic equations in infinite dimensions[END_REF]) implies that we can find random variables ( Ȳ n , Λn ) and ( Ȳ , Λ) defined on a new probability space ( Ω, Ā, P), such that ( Ȳ n , Λn ) has the same law as (Y n , Λ n ) for any n, the law of ( Ȳ , Λ) is m and, P-almost surely, the sequence ( Λn ) converges to ( Λ) in C([ t, T ], R N ) while, for any t belonging to some set I ⊂ [ t, T ] of full measure in [ t, T ], the sequence ( Ȳ n t ) converges to Ȳt (Theorem 5 of [START_REF] Meyer | Tightness criteria for laws of semimartingales[END_REF]).

Since t → Λ n t is absolutely continuous P-a.s. and since Λn has the same law as Λ n , t → Λn t is absolutely continuous P-a.s.. Let us set ζn s = d ds Λn s . Then, by ( 27), E[ T t ζn s p ds] ≤ C for all n ≥ 0. Therefore, up to a subsequence again labeled in the same way, ( ζn ) converges weakly in L p ([ t, T ]) to some limit, ζ, which, P-a.s., satisfies Λt = t t ζs ds for all t ∈ [ t, T ]. Note that Mn t = Ȳ n tx -Λn t has the same law as M an t , so that by Hölder and (28), Ē Mn

t r ≤ δ r 2 (t -t) r/2
for all r ∈ (0, 2] and for all t ∈ [ t, T ]. Passing to the limit in the above inequality gives

Ē Ȳt -x -Λt r ≤ δ r 2 (t -t) r/2 ∀t ∈ I, ∀r ∈ (0, 2] .
We get the above inequality for all t ∈ [ t, T ) thanks to the càdàg property of the trajectories of Y . Recalling (22), a classical lower semicontinuity argument yields

u(x, t) ≥ Ē u( Ȳt , t) + C + t t ζs p ds -δ(t -t) ∀t ∈ I ,
and we conclude the proof by using again the càdàg property of the trajectories of Y .

Analysis of subsolutions

In this section we investigate properties of subsolution of equation ( 12).

Proposition 3.1 For any fixed (x, t) ∈ R N × (0, T ], there is a continuous supersolution w of [START_REF] Prato | Stochastic equations in infinite dimensions[END_REF] in

R N × [0, t) such that 1 C (t -s) 1-p |x -y| p -C(t -s) 1-p/2 ≤ w(y, s) ≤ C(t -s) 1-p |x -y| p + C(t -s) 1-p/2 (31)
for any (y, s) ∈ R N × [0, t) and for some universal constant C.

Proof : It relies on control interpretation of equation ( 12) as well as the construction of some Brownian bridges (see [START_REF] Fleming | Stochastic variational formula for fundamental solutions of parabolic PDE[END_REF]). Let us assume, without loss of generality, that x = 0. Having fixed α ∈ (1 -1/p, 1/2), (y, s) ∈ R N ×[0, t) and a ∈ A(s), let Y y,s,a τ be the solution to

dY τ = -α Yτ t-τ dτ + dM a τ Y s = y
Then one easily checks that

Y τ = (t -s) -α (t -τ ) α y + (t -τ ) α τ s (t -σ) -α dM a σ . (32)
Let us set We claim that there is a universal constant C > 0 such that

1 C (t -s) 1-p |x -y| p -C(t -s) 1-p/2 ≤ J(y, s, a) ≤ C(t -s) 1-p |x -y| p + C(t -s) 1-p/2 . (33) Indeed J(y, s, a) = E t s |Z y,s,a τ | p dτ ≤ 2 p-1 α p (t -s) -αp |y| p t s (t -τ ) p(α-1) dτ +2 p-1 α p t s (t -τ ) p(α-1) E τ s (t -σ) -α dM a σ p dτ
where, by Hölder and Itô,

E τ s (t -σ) -α dM a σ p ≤ C(t -s) p 2 (1-2α) So J(y, s, a) ≤ C(t -s) 1-p |y| p + C(t -s) 1-p/2 .
In the same way, = δ p/q pq p/q . Let us first show that w is continuous on R N × [0, t). The map y → J(y, s, a) being convex (since the map y → Y y,s,a τ is affine and p > 1) and locally uniformly bounded (thanks to (33)), it is has a modulus of continuity which is locally uniform with respect to s and a. The map s → J(y, s, a) being locally Hölder continuous on [0, t), locally uniformly with respect to y and a, this implies that the map (y, s) → J(y, s, a) has a modulus of continuity which is uniform with respect to a. Therefore w is continuous on

J(y, s, a) = E t s |Z y,s,a τ | p dτ ≥ 2 1-p α p (t -s) -αp |y| p t s (t -τ ) p(α-1) dτ -α p t s (t -τ ) p(α-1) E τ s (t -σ) -α dM a σ p dτ ≥ (1/C)(t -s) 1-p |y| p -C(t -s) 1-p/2 , Whence (33)
R N × [0, t).
Using the fact that w is continuous and arguments similar to the ones in [START_REF] Buckdahn | Stochastic control problems for systems driven by normal martingales[END_REF] one can prove that w satisfies the Hamilton-Jacobi equation w(y + λb, s) -w(y, s) -Dw(y, s), λb

|b| 2 = -M + [w(•, s)](y) while --α y t -s , Dw + C --α y t -s p ≥ 1 δ |Dw| q ,
w is a supersolution of [START_REF] Prato | Stochastic equations in infinite dimensions[END_REF]. We finally note that w satisfies (31) because the inequalities (33) are uniform with respect to a.

Lemma 3.2 Let u ∈ C(R N × [0, T ]
) be a subsolution of [START_REF] Prato | Stochastic equations in infinite dimensions[END_REF] satisfying |u| ≤ M. Then, for all (x, t) ∈ R N × (0, T ) and all (y, s

) ∈ R N × [0, t), u(y, s) ≤ u(x, t) + C |y -x| p (t -s) 1-p + (t -s) 1-p/2 (34)
for some universal constant C > 0.

Remark 3.3 In particular, if u = u(x) is a subsolution of the stationary equation

-δM + [u(•, t](x) + 1 δ |Du| q -δ = 0 in R N ,
then inequality (34) implies that, for any x, y ∈ R N and any τ > 0,

u(x) ≤ u(y) + C |y -x| p τ 1-p + τ 1-p/2 ,
for some universal constant C. Thus, choosing τ = |x-y| 2 yields u(x) ≤ u(y)+C |y-x| 2-p , that is, u is Hölder continuous. This extends to nonlocal equations one of the results of [START_REF] Docetta | Hölder estimates for degenerate elliptic equations with coercive Hamiltonians[END_REF].

Proof: According to Proposition 3.1 there is a supersolution w of ( 12) which satisfies

1 C (t -s) 1-p |x -y| p -C(t -s) 1-p/2 ≤ w(y, s) ≤ C(t -s) 1-p |x -y| p + C(t -s) 1-p/2
for any (y, s) ∈ R N × [0, t) and for some universal constant C. Since u is continuous and bounded and u(y, t) ≤ lim s→t w(y, s) + u(x, t) for any y ∈ R N , we get u ≤ w + u(x, t) on R N × [0, t) by comparison (Lemma 2.2). Whence the result.

Lemma 3.4 Let u ∈ C(R N × [0, T ]) be a subsolution of (12) satisfying |u| ≤ M. Fix (x, t) ∈ R N × (0, T ), ζ ∈ L p ad ([ t, T ]
) and let (X t , ζ t ) be stochastic processes satisfying (30). Then, for any x ∈ R N and t ∈ ( t, T ),

u(x, t) -E[u(X t , t)] ≤ C (t -t) 1-p E t t |ζ s |ds p + |x -x| p + (t -t) 1-p/2 (35)
for some constant C > 0.

Proof: Fix t ∈ ( t, T ) and apply Lemma 3.2 to (x, t) and (X t (ω), t). Then, for almost all ω ∈ Ω,

u(x, t) ≤ u(X t (ω), t) + C |X t (ω) -x| p (t -t) 1-p + (t -t) 1-p/2 . Hence, u(x, t) ≤ E [u(X t , t)] + C (E [|X t -x| p ] + |x -x| p )(t -t) 1-p + (t -t) 1-p/2 .
Since, on account of (30),

E [|X t -x| p ] ≤ C E t t |ζ s |ds p + (t -t) p 2
, the conclusion follows.

In order to proceed, we need to recall the following weak reverse Hölder inequality: Thanks to this inequality, we can estimate the L p norm of the process ζ appearing in Lemma 2.4.

Lemma 3.6 Let u ∈ C(R N × [0, T ]
) be a subsolution of [START_REF] Prato | Stochastic equations in infinite dimensions[END_REF] such that |u| ≤ M and let τ ∈ (0, T ). Then there is a universal constant θ ∈ (p, 2) and a constant C(τ, δ) > 0 such that, for every (x, t) ∈ R N × (0, T -τ ), and every stochastic processes (X t , ζ t ) satisfying ( 29) and (30), we have for any t ∈ ( t, T ). Since 1 > 1 -p/2 > (θ -p)/θ (recall that θ < 2), u(x, t) ≤ u(x, t) + C(τ )(t -t) (θ-p)/θ + C|x -x| p (t -t) 1-p .

E t t |ζ s |ds p ≤ C(τ )(t -t) p-p θ ∀t ∈ ( t, T ) . Proof: First, observe that, by Lemma 3.4 applied to x = x, u(x, t) ≤ E[u(X t , t)] + C (t -t)
Then, for |x -x| sufficiently small, choose t = t + |x -x| θ/(θ-1) to obtain u(x, t) ≤ u(x, t) + C(τ )|x -x| (θ-p)/(θ-1) . .

To derive the reverse inequality, one just needs to apply Lemma 3.2 with y = x = x to get u(x, t) ≤ u(x, t) + C(t -t) 1-p/2 .

This leads to the desired result since 1 -p/2 > (θ -p)/θ.

M

  -[φ](x) = inf λ∈(0,1], b∈B\{0} φ(x + λb) -φ(x) -Dφ(x), λb |b| 2 and M + [φ](x) = sup λ∈(0,1], b∈B\{0} φ(x + λb) -φ(x) -Dφ(x), λb |b| 2

B |e| 2

 2 dν(e) < +∞ and ν(B) = +∞ .

  . Next we introduce the value function w of the optimal control problem w(y, s) = C sup a∈A(s) J(y, s, a) -δ(t -s) , where C -.

2 -

 2 + λb, s) -w(y, s) -Dw(y, s), λb |b|

Lemma 3 . 5 (

 35 [START_REF] Cannarsa | Hölder estimates in space-time for viscosity solutions of Hamilton-Jacobi equations[END_REF]) Let (Ω, A, P) be a probability space. Let p ∈ (1, 2) and assume that the function ξ ∈ L p ((a, b); R N ) satisfies the inequality for some positive constants A and B. Then there are constants θ ∈ (p, 2) and C > 0, depending only on p and A, such that E

  t) 1-p/2for all t ∈ [ t, T ). Moreover, in view of (29),E [u(X t , t)] ≤ u(x, t) -C + E t t |ζ s | p ds + δ(t -t) ∀t ∈ [ t, T ) . Hence, taking into account that t -t ≤ C(t -t) 1-p/2 , E t t |ζ s | p ds ≤ C (t -t) 1-p E C(t -t) 1-p/2 ∀t ∈ [ t, T ) .Then, owing to Lemma 3.5, there are universal constants θ ∈ (p, 2) and C > 0 such that Since u is bounded by M, assumption (29) implies that ζ p ≤ C. So, we finally get E Let u : R N × [0, T ] → R be a continuous supersolution of (11), a subsolution of (12) and such that |u| ≤ M. Space regularity: Fix (x, t) ∈ R N × (0, T -τ ) and let x ∈ R N . By Lemma 2.4 there is a control ζ ∈ L p ad ([ t, T ]) and an adapted process X such that (29) and (30) hold. So, u(x, t) ≥ E [u(X t , t)] -δ(t -t) ∀t ∈ [ t, T ) .

							(37)
	Also, Lemma 3.6 ensures that
		E	t t |ζ s |ds	p	≤ C(τ )(t -t) p-p θ	∀t ∈ ( t, T )	(38)
							t
							t |ζ s |ds
	E	t t |ζ s |ds	p	≤ C ζ p p + 1	(t -t) p-p θ θ (T -t) 1-p	∀t ∈ ( t, T ) .

1-p E t t |ζ s |ds p + (t p + t t |ζ s |ds p

≤ C(τ )(t -t) p-p θ ∀t ∈ ( t, T ) , because t ≤ T -τ .

4 Proof of Theorem 1.1 for some universal constant θ ∈ (p, 2) and some constant C(τ ) > 0. Furthermore, applying Lemma 3.4, for any t ∈ ( t, T ) we have

u(x, t) -E[u(X t , t)] ≤ C (t -t) 1-p E t t |ζ s |ds p + |x -x| p (t -t) 1-p + (t -t) 1-p/2 .

Plugging (37) and (38) into the above inequality leads to

u(x, t) ≤ u(x, t) + δ(t -t) + C(τ )(t -t) (θ-p)/θ + C|x -x| p (t -t) 1-p + C(t -t) 1-p/2

  Time regularity : Let now t ∈ (0, T -τ ). Then, in light of (37),u(x, t) ≥ E [u(X t , t)] -δ(t -t).Now, applying the space regularity result we have just proved, we obtainE [u(X t , t)] ≥ u(x, t) -C(τ )E |X t -x| Notice that (θ -p)/(2(θ -1)) > (θ -p)/θ since θ < 2. So, u(x, t) ≥ u(x, t) -C(τ )(t -t)

									θ-p
									θ
									θ-p θ-1	.
	Moreover, since (θ -p)/(θ -1) < 1, by (30) we get
		E |X t -x|	θ-p θ-1	≤ CE	t t |ζ s |ds	θ-p θ-1	+ C(t -t)	θ-p 2(θ-1) .
	Also, by Hölder's inequality and (38),
	E	t t |ζ s |ds	θ-p θ-1	≤ C E	t t |ζ s |ds	p	θ-p p(θ-1)	θ-p θ	.

≤ C(τ )(t -t)

This work was partially supported by the French ANR (Agence Nationale de la Recherche) through MICA project (ANR-06-06-BLAN-0082) and KAMFAIBLE project (ANR-07-BLAN-0361).